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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K/k\) be a field extension and let \(G\) be a finite group of exponent \(e\). A theorem of Brauer states that in the ``non-modular'' setting (where characteristic of \(k\) does not divide \(|G|\)), any representation \(\rho : G\to\mathrm{GL}_n(K)\) can be defined over \(k\) provided it contains a primitive \(e\)-th root of unity. If \(k\) does not contain such a root, the classically defined Schur index of \(\rho\) (when the representation is absolutely irreducible) measures how far away \(\rho\) is from being defined over \(k\).
In this paper, the authors study the notion of essential dimension of a representation \(\mathrm{ed}(\rho)\) which provides another numerical invariant to address the same question. \(\mathrm{ed}(\rho)\) is defined to be the smallest transcendence degree of an intermediate field \(k \subset K_0 \subset K\) such that \(\rho\) is defined over \(K_0\). A naive upper bound for \(\mathrm{ed}(\rho)\) is \(rn^2\) where \(G\) is generated by \(r\) elements. The authors present much improved upper bounds (for instance \(n^2/4\), \(|G|/4\),...etc) in the case when \(k\) has characteristic 0.
They do this by relating \(\mathrm{ed}(\chi)\) (the maximal value of essential dimensions of representations with a given character \(\chi\)) with the ``canonical dimension'' of a product of Weil transfers of generalized Severi Brauer varieties.
To further develop the analogy between the Schur index and the essential dimension of a representation, the authors prove a variant of another of Brauer's theorem. Namely, they show that for any integer \(l\geq 1\), there exist a number field \(K/\mathbb{Q}\) and a linear representation \(\rho : G\to \mathrm{GL}_{2l}(K)\) with essential dimension \(l\) over \(\mathbb{Q}\).
Computations of canonical dimensions of a large class of Weil transfers of generalized Severi Brauer varieties extending previous results of Karpenko are made. These lead to a nice formula of the essential \(p\) dimension of a character in terms of its absolutely irreducible components. A variant of a theorem of Schilling is also proved.
The authors finally investigate the essential dimensions of modular representations (when \(\mathrm{char}(k)\) divides \(|G|\)) and prove that it can be arbitrarily large. In the appendix, a constructive version of this result is given. Schur index; essential dimension; canonical dimension; representations of finite groups; Severi-Brauer varieties; central simple algebras Algebraic cycles, Brauer groups (algebraic aspects), Group rings of finite groups and their modules (group-theoretic aspects) A numerical invariant for linear representations of finite groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The problem of the construction of antisymmetric paramodular forms of canonical weight 3 has been open since 1996. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to \((1,t)\)-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight 3 as automorphic Borcherds products whose first Fourier-Jacobi coefficient is a theta block. Siegel modular forms; automorphic Borcherds products; theta functions and Jacobi forms; moduli space of abelian and Kummer surfaces; affine Lie algebras and hyperbolic Lie algebras Theta series; Weil representation; theta correspondences, Fourier coefficients of automorphic forms, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Jacobi forms, Other groups and their modular and automorphic forms (several variables), Theta functions and abelian varieties Antisymmetric paramodular forms of weight 3 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(M\) be a reductive (algebraic) monoid with zero and with unit group \(G\) defined over an algebraically closed field \(K\). A natural \(G\times G\) action on \(M\) is \(G\times M\times G\to M\) defined by \((x,a,y)\mapsto xay^{-1}\). There are only finitely many orbits, i.e., \(\mathcal J\)-classes, which form a lattice. The general problem of finding the lattice remains open.
The authors study a class of reductive monoids as the multilined closure of a representation of a (non-abelian) simple algebraic group, in particular the \((\mathcal J,\sigma)\)-irreducible monoids of Suzuki or Ree type [cf. \textit{Z. Li, L. E. Renner}, J. Algebra 190, No. 1, 172-194 (1997; Zbl 0879.20035)], and obtain a general theorem to determine the lattices for these monoids. cross sections; Dynkin diagrams; idempotents; irreducible representations; \(\mathcal J\)-classes; \(({\mathcal J},\sigma)\)-irreducible monoids; lattices; multilined closures; one parameter closures; orbits; reductive groups; reductive monoids of Ree type; simple algebraic groups Li, Zhuo; Putcha, M., Types of reductive monoids, J. algebra, 221, 102-116, (1999) Semigroups of transformations, relations, partitions, etc., Linear algebraic groups over arbitrary fields, Representation theory for linear algebraic groups, Other algebraic groups (geometric aspects) Types of reductive monoids | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper generalizes Noether's theorem: If \(A\) is affine then \(A^G\) is affine also, where \(A\) is a commutative algebra and \(G\) is a finite group of automorphisms acting on \(A\). This theorem has been generalized to actions of any finite dimensional cocommutative Hopf algebra \(H\) on a commutative algebra \(A\), and the third author has shown that cocommutativity of \(H\) can be replaced by semisimplicity of \(H\). In this paper the authors follow their philosophy that states that often properties that hold for a cocommutative Hopf algebra \(H\) and a commutative \(H\)-module \(A\) should hold for appropriate generalizations: a triangular Hopf algebra \((H,R)\) and a quantum-commutative \(H\)-module \(A\) (i.e. \(A\) is commutative in the category of \(H\)-modules). Indeed, they generalize the theorem to this set-up, and also to its ``dual'': \((H,\langle \mid \rangle)\) a cotriangular Hopf algebra and \(A\) a quantum-commutative \(H\)-comodule algebra.
In order to prove the theorem the authors construct a new, non-commutative, determinant function for each of the cases mentioned above. This construction involves the action of the symmetric group that is defined by the symmetric braiding of the twist map in the category of \(H\)-modules; this gives rise to a kind of Grassmann algebra. The determinant is also computed explicitly for some examples of group gradings. affine algebras; actions of finite dimensional cocommutative Hopf algebras; Noether's theorem; finite groups of automorphisms; triangular Hopf algebras; quantum-commutative modules; non-commutative determinant functions; symmetric braidings; twist maps; categories of modules; Grassmann algebras; group gradings Cohen, M.; Westreich, S.; Zhu, S., Determinants, integrality and Noether's theorem for quantum commutative algebras, Israel J. math., 96, 185-222, (1996) Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Automorphisms and endomorphisms, Geometric invariant theory, Determinants, permanents, traces, other special matrix functions Determinants, integrality and Noether's theorem for quantum commutative algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be an algebraic group of type \(D_4\) over a field \(F\) of characteristic not equal to \(2\). The paper under review describes the \(F\)-points of twisted flag varieties associated with \(G\). The author's starting point is the fact that these varieties depend only on the \(F\)-isogeny class of \(G\), whence one may consider only the special case in which \(G\) is simply connected. It is known that when this occurs one can associate with \(G\) a triality \(T\) [see \textit{M.-A. Knus, A. Merkurjev, M. Rost} and \textit{J.-P. Tignol}, The book of involutions. Colloq. Publ. 44, AMS, Providence, RI (1998; Zbl 0955.16001)]. The notion of a triality is too technical to be presented here (the corresponding definition can be found in the book referred to). We only note that a triality \(T\) is a \(4\)-tuple \((E,L,\sigma,\alpha)\) satisfying certain conditions, where \(L\) is a cubic étale \(F\)-algebra, \(E\) is a central simple \(L\)-algebra, \(\sigma\) is an orthogonal involution on \(E\), and \(\alpha\) is an isomorphism of the even Clifford algebra \((C_0(E,\sigma),\overline\sigma)\) on the algebra \(^\rho((E,\sigma)\otimes_F\Delta(L))\), \(\overline\sigma\) being the involution of \(C_0(E,\sigma)\) canonically induced by \(\sigma\), \(\Delta(L)\) the discriminant \(F\)-algebra of \(L\), and \(\rho\) a generator of the Galois group of \(L\otimes_F\Delta(L)\) over \(\Delta(L)\). This means that \(L\) is presentable as a direct sum \(\bigoplus_{i=1}^n L_i\) of separable field extensions of \(F\) of degree \(3\), \(E\) is isomorphic to a direct sum \(\bigoplus_{i=1}^n E_i\) of central simple algebras \(E_i\) of degree \(8\) over the field \(L_i\), \(\sigma\) is an involution of \(E\) inducing on \(E_i\) an orthogonal involution \(\sigma_i\), for each index \(i\), and \((C_0(E,\sigma),\overline\sigma):=\bigoplus_{i=1}^n(C_0(E_i,\sigma_i),\overline\sigma_i)\), where \(\overline\sigma_i\) is the involution of the Clifford algebra \((C_0(E_i,\sigma_i),\overline\sigma_i)\) canonically induced by \(\sigma_i\), and \(\overline\sigma\) is a prolongation of each \(\overline\sigma_i\).
The author gives a method of constructing an isotropic right ideal of \((C_0(E,\sigma),\overline\sigma)\) from an arbitrary isotropic right ideal of \((E,\sigma)\) and describes the \(F\)-points of \(G\) in terms involving this correspondence and the triality \(T\). flag varieties; algebraic groups of type \(D_4\); triality; central simple algebras; orthogonal involutions; Clifford algebras Garibaldi, R. S.: Twisted flag varieties of trialitarian groups. Commun algebra 27, No. 2, 841-856 (1999) Finite-dimensional division rings, Linear algebraic groups over arbitrary fields, Galois cohomology of linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Rings with involution; Lie, Jordan and other nonassociative structures, Clifford algebras, spinors Twisted flag varieties of trialitarian groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities See the preview in Zbl 0579.14015. Hilbert's fourteenth problem; algebraic group; finite; generation of algebra of invariant functions Grosshans, F.D.: Hilbert's fourteenth problem for non-reductive groups. Math. Z.193, 95--103 (1986) Geometric invariant theory, Group actions on varieties or schemes (quotients), Linear algebraic groups and related topics Hilbert's fourteenth problem for non-reductive groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper establishes new connections between the representation theory of finite groups and sandpile dynamics. Two classes of avalanche-finite matrices and their critical groups (integer cokernels) are studied from the viewpoint of chip-firing/sandpile dynamics, namely, the Cartan matrices of finite root systems and the McKay-Cartan matrices for finite subgroups \(G\) of general linear groups. In the root system case, the recurrent and superstable configurations are identified explicitly and are related to minuscule dominant weights. In the McKay-Cartan case for finite subgroups of the special linear group, the cokernel is related to the abelianization of the subgroup \(G\). In the special case of the classical McKay correspondence, the critical group and the abelianization are shown to be isomorphic. chip firing; toppling; sandpile; avalanche-finite matrix; Z-matrix; M-matrix; McKay correspondence; McKay quiver; root system; Dynkin diagram; minuscule weight; highest root; numbers game; abelianization Root systems, Combinatorial aspects of representation theory, McKay correspondence Chip firing on Dynkin diagrams and McKay quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(p\). A \(k\)-group \(\mathcal G\) is uniserial if \(\mathcal G\) has a unique composition series. Uniserial groups play an important role in determining if an infinitesimal group is representation-finite, that is it admits only finitely many isomorphism classes of finite-dimensional indecomposable modules.
This paper gives a complete classification of isomorphism classes of non-trivial infinitesimal unipotent commutative uniserial \(k\)-groups. It turns out that there are six different types of isomorphism classes, all of which can be described as kernels of Witt vectors (or their duals).
The classification is facilitated by the use of (classical) Dieudonné modules. A Dieudonné module \(M\) corresponds to a uniserial group if and only if either \(M/FM\) or \(M/VM\) is a simple module (over the Dieudonné ring \(\mathbb{D}\)). In this case, \(M\) is also called uniserial. The authors construct a list of Dieudonné modules with \(M/VM\) simple, and then pass to Cartier duality to get the others.
The results of this classification enable an examination of representation-finite infinitesimal groups, and the article concludes with such a study, as well as with a discussion of the classification problem for unipotent groups of complexity 1. uniserial groups; infinitesimal groups; finite representation type; Witt vectors; Dieudonné modules; simple modules; group schemes Rolf Farnsteiner, Gerhard Röhrle, and Detlef Voigt, Infinitesimal unipotent group schemes of complexity 1, Colloq. Math. 89 (2001), no. 2, 179 -- 192. Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Modular Lie (super)algebras, Group schemes Infinitesimal unipotent group schemes of complexity 1 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of algebraic monoids was founded in the 1980s and has been well developed mainly by Putcha and the author. It culminates in a natural blend of algebraic groups, torus embeddings and semigroups. In studying algebraic monoids, Putcha mainly takes the algebraic point of view, while the author mainly takes the geometric point of view, and they meet at algebraic combinatorics. The monograph by \textit{M. S. Putcha} [Linear algebraic monoids, Cambridge Univ. Press, Cambridge (1988; Zbl 0647.20066)] is an algebraic version of the theory; this one by the author is essentially a geometric version of algebraic monoids. As far how Putcha's algebraic and the author's geometric approaches meet in algebraic combinatorics, one is advised to read \textit{L. Solomon} [``An introduction to reductive monoids'', in NATO ASI Ser., Ser. C, Math. Phys. Sci. 466, 295-352 (1995; Zbl 0870.20047)], which is equipped with many illustrative examples.
The book under review is a survey based monograph (or monograph style survey) of algebraic monoids, with focus on the geometry, classifications and representations of normal reductive algebraic monoids, plus a survey of their extensions to monoids of Lie type and spherical embeddings.
The book consists of 15 chapters. Chapter 1 is an introduction to the history of algebraic monoids; Chapter 2 is for introducing tools of algebraic geometry, linear algebraic groups and semigroups; Chapter 3 is for introducing relative algebraic monoids. Chapters 4-14 form the main body, surveying the geometry of algebraic groups or algebraic monoids, classifications and representations of normal reductive algebraic monoids, the monoids of Lie type. Chapter 15 is a brief survey of closely related developments in other branches. algebraic transformation groups; classification; convex geometry; geometry of algebraic groups; invariant theory; linear algebraic semigroups; linear algebraic monoids; linear semigroups; monoids of Lie type; normal algebraic monoids; Putcha lattices of cross-sections; reductive monoids; regular semigroups; Renner monoids; representation theory; spherical embeddings; strongly \(\pi\)-regular semigroups; Tits systems; torus embeddings Renner, L. E.: Linear algebraic monoids, Encyclopædia math. Sci. 134 (2005) Semigroups of transformations, relations, partitions, etc., Linear algebraic groups over arbitrary fields, Research exposition (monographs, survey articles) pertaining to group theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Group actions on varieties or schemes (quotients), Classical groups (algebro-geometric aspects) Linear algebraic monoids. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We fix an error on a \(3\)-cocycle in the original version of our paper [ibid. 8, Paper No. e12, 93 p. (2020; Zbl 1484.20084)]. We give the corrected statements of the main results. Hecke category; finite groups of Lie type; character sheaves; endoscopic group Linear algebraic groups over finite fields, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Hecke algebras and their representations, Representations of finite groups of Lie type Corrigendum to: ``Endoscopy for Hecke categories, character sheaves and representations'' | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper considers the geometry of \(E_8\) from a Clifford point of view in three complementary ways. Firstly, in earlier work, I had shown how to construct the four-dimensional exceptional root systems from the 3D root systems using Clifford techniques, by constructing them in the 4D even subalgebra of the 3D Clifford algebra; for instance the icosahedral root system \(H_3\) gives rise to the largest (and therefore exceptional) non-crystallographic root system \(H_4\). Arnold's trinities and the McKay correspondence then hint that there might be an indirect connection between the icosahedron and \(E_8\). Secondly, in a related construction, I have now made this connection explicit for the first time: in the 8D Clifford algebra of 3D space the 120 elements of the icosahedral group \(H_3\) are doubly covered by 240 8-component objects, which endowed with a `reduced inner product' are exactly the \(E_8\) root system. It was previously known that \(E_8\) splits into \(H_4\)-invariant subspaces, and we discuss the folding construction relating the two pictures. This folding is a partial version of the one used for the construction of the Coxeter plane, so thirdly we discuss the geometry of the Coxeter plane in a Clifford algebra framework. We advocate the complete factorisation of the Coxeter versor in the Clifford algebra into exponentials of bivectors describing rotations in orthogonal planes with the rotation angle giving the correct exponents, which gives much more geometric insight than the usual approach of complexification and search for complex eigenvalues. In particular, we explicitly find these factorisations for the 2D, 3D and 4D root systems, \(D_6\) as well as \(E_8\), whose Coxeter versor factorises as \(W=\exp(\frac{\pi}{30}B_C)\exp(\frac{11\pi}{30}B_2)\exp(\frac{7\pi}{30}B_3)\exp(\frac{13\pi}{30}B_4)\). This explicitly describes 30-fold rotations in 4 orthogonal planes with the correct exponents \({\{1, 7, 11, 13, 17, 19, 23, 29\}}\) arising completely algebraically from the factorisation. \(E_8\); exceptional phenomena; Clifford algebras; icosahedral symmetry; Coxeter groups; root systems; spinors; Coxeter plane; Lie algebras; Lie groups; representation theory; quantum algebras; trinities; McKay correspondence Three-dimensional polytopes, Special polytopes (linear programming, centrally symmetric, etc.), Symmetry properties of polytopes, Clifford algebras, spinors, Reflection and Coxeter groups (group-theoretic aspects), Root systems, McKay correspondence The \(E_8\) geometry from a Clifford perspective | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is a survey on the representations of algebraic monoids over an algebraically closed field, mainly normal reductive algebraic monoids and a class of solvable algebraic monoids, and finite monoids of Lie type. The concerns of representations of algebraic monoids are the following three: representations of an algebraic monoid \(M\) in its own interest, the extensions of its unit group \(G(M)\) to \(M\), and the restriction to \(G(M)\) of a representation of \(M\).
There are many similarities with representations of corresponding algebraic groups, since the unit groups exert a lot of influence on the outcome. Other suggestions come from the representation theories of associative algebras and Lie algebras. Proofs of some fundamental results are presented or outlined in the paper. The whole survey is presented in a geometric flavor. The interested reader is also advised to read \textit{M. S. Putcha}'s complementary paper in these Proceedings [ibid. 113-124 (2004; Zbl 1065.20076)]. blocks; conjugacy classes; extension principle; finite monoids of Lie type; irreducible representations; linear algebraic monoids; modular representations; Morita equivalences; normal reductive monoids; Putcha lattices of cross-sections; representation theory; semisimple elements; solvable algebraic monoids Renner, L. E.: Representations and blocks of algebraic monoids, Fields inst. Commun. 40 (2004) Representation of semigroups; actions of semigroups on sets, Representation theory for linear algebraic groups, Homogeneous spaces and generalizations, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Representations of finite groups of Lie type, Semigroups of transformations, relations, partitions, etc. Representations and blocks of algebraic monoids | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For hypersurfaces in characteristic 0, the rings of finite CM type (i.e. having only finitely many indecomposable Cohen-Macaulay modules) are those corresponding to the simple hypersurface singularities, as was shown by \textit{H. Knörrer} [Invent. Math. 88, 153-164 (1987; Zbl 0617.14033)] and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and \textit{F.-O. Schreyer} [ibid. 165-182 (1987; Zbl 0617.14034)]. This paper is in two parts about (I) scrolls and (II) fixed rings.
A scroll of type \((m_ 1,\ldots,m_ r)\) is a polynomial ring modulo the ideal generated by all \(2\times 2\) determinants from a certain \(2\times(m_ 1+\cdots+m_ r)\) matrix whose entries are indeterminates. --The fixed rings are of form \(S^ G\) where, usually, \(S\) is taken to be a complete local CM ring with algebraically closed residue field and \(G\) is a finite group with faithful linear action on \(S\) such that \(| G|\) is invertible in \(S\).
The main aim is to show that, among these rings, the only ones of dimension \(\geq 3\) which have finite CM type are the scroll of (2,1) type \(k[[X_ 0,X_ 1,X_ 2,Y_ 0,Y_ 1]]/(X_ 0X_ 2-X^ 2_ 1\), \(X_ 0Y_ 1-X_ 1Y_ 0,\) \(X_ 1Y_ 1-X_ 2Y_ 0)\) and the fixed ring \(S^ G\) where \(\hbox{char}(k)\neq 2\), \(S=k[[X,Y,Z]]\), \(G=\mathbb{Z}_ 2\) and the generator of \(G\) acts by sending each indeterminate to its negative. Use is made of almost split sequences and, in part(II), of reflexive modules over \(S^ G\) and over the skew group ring \(SG\). rings of finite CM type; finitely many indecomposable Cohen-Macaulay modules; simple hypersurface singularities; scrolls; fixed rings; almost split sequences Auslander, M., Reiten, I.: The Cohen--Macaulay type of Cohen--Macaulay rings. Adv. Math. 73(1), 1--23 (1989) Cohen-Macaulay modules, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Complete rings, completion, Polynomial rings and ideals; rings of integer-valued polynomials, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) The Cohen-Macaulay type of Cohen-Macaulay rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a reductive algebraic group \(G\) and a \(G\)-representation \(V\) we can construct the algebraic quotient \(V//G\), which is the affine variety corresponding to the ring of invariant polynomial functions \(\mathbb C[V]^G\). The embedding \(\mathbb C[V]^G\subset\mathbb C[V]\) gives rise to a quotient map \(V\to V//G\).
The main problem in invariant theory is to describe the geometry of such a quotient. There are several questions that one can try to answer: What is the dimension of \(V//G\)? How do the fibers of \(V//G\) look like? Is \(V//G\) a smooth variety?
In complete generality a solution for these problems is unattainable but given restrictions on the groups or the representations one can expect some interesting partial results. In many cases one can find a certain class of couples \((V,G)\) which share the same geometrical properties for their quotients. More precisely one can try to find classes that are closed under local behavior. By this we mean that if we have a couple \((V,G)\) and a point \(p\in V//G\) we can find another couple \((V_p,G_p)\) of the same class such that there is an étale neighborhood of \(p\) that is locally isomorphic to an étale neighborhood of the zero point in \(V_p//G_p\). Such a result simplifies the questions a lot because we can use this local result to reduce the questions about complicated representations to more simple representations.
For \((G,V)\) a representation space of a quiver this was done by \textit{C. Procesi} and \textit{L. Le Bruyn} [in Trans. Am. Math. Soc. 317, No. 2, 585-598 (1990; Zbl 0693.16018)]. Similar results have been obtained for representation spaces of preprojective algebras by \textit{W. Crawley-Boevey} [Math. Ann. 325, No. 1, 55-79 (2003; Zbl 1063.16016)].
In this paper we will study the case of supermixed quivers. These were introduced and studied by \textit{A. N. Zubkov} and \textit{A. A. Lopatin} [in Transform. Groups 12, No. 2, 341-369 (2007; Zbl 1159.16012); J. Algebra Appl. 4, No. 3, 245-285 (2005; Zbl 1082.16022); ibid. 4, No. 3, 287-312 (2005; Zbl 1082.16023)] and are closely related to generalized quivers which were studied by \textit{H. Derksen} and \textit{J. Weyman} [in Colloq. Math. 94, No. 2, 151-173 (2002; Zbl 1025.16010)]. First we will give a coordinate free description of a representation space of a supermixed quiver by means of involutions on semisimple algebras. Then we will extend the results by Derksen and Weyman to obtain a representation theoretic interpretation of the points in the representation spaces and in the quotient. This will enable us to formulate an extension of the result on local quivers by Procesi and Le Bruyn to supermixed quivers. To make full use of this result we will also determine which supermixed settings have simple representations. representations of quivers; invariant theory of classical groups; Luna slice theorem; representation spaces; supermixed quivers; dimension vectors; simple supermixed representations Bocklandt, R., A slice theorem for quivers with an involution, J. algebra appl., 9, 3, 339-363, (2010) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Geometric invariant theory A slice theorem for quivers with an involution. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0601.00002.]
This paper is an exposition for the work of G. Lusztig concerning the classification of irreducible characters of finite groups of Lie type such as \(GL_ n({\mathbb{F}}_ q)\), where \({\mathbb{F}}_ q\) is a finite field of q-elements. Let G be a connected reductive group defined over \({\mathbb{F}}_ q\), and \(F: G\to G\) be the Frobenius map. Fix an F-stable maximal torus T in an F-stable Borel subgroup of G, and let \(W=N_ G(T)/T\). After some preliminary description in the case of \(GL_ n({\mathbb{F}}_ q)\), the author explains the construction of the Deligne- Lusztig virtual characters \(R_ w^{\theta}\) (parametrized by \(w\in W\) and \(\theta \in Hom(T^{wF},{\bar {\mathbb{Q}}}^*_{\ell}))\), which is defined in terms of \(\ell\)-adic cohomology on a certain variety on which \(G^ F\times T^{wF}\) acts.
Let \(\hat G{}^ F\) be the set of irreducible characters of \(G^ F\). Then any \(\phi\in \hat G^ F\) appears in some \(R_ w^{\theta}\). Using the notion of the dual group \(G^*\) of G, \(\hat G{}^ F\) is partitioned into subsets \({\mathcal E}(G^ F,(s^*))\), where \((s^*)\) is a semisimple class in \(G^*\) related to \((T^{wF},\theta)\) and runs over all the F- stable semisimple classes in \(G^*\). Then he explains the ''Jordan decomposition'' of \(G^ F\), i.e., \({\mathcal E}(G^ F,(s^*))\) is in bijective correspondence with \({\mathcal E}(Z_{G^*}(s^*)^{*^ F},(1))\) in a natural way, subject to the condition that G has connected center.
Finally the author explains Lusztig's main result, i.e., the decomposition of \(R_ w^{\theta}\) into irreducible constituents. For this, the notion of ''families'' in \(\hat G{}^ F\) is introduced, which is a refinement of \({\mathcal E}(G^ F,(s^*))\) and is described by a Weyl group of \(Z_{G^*}(s*)\) and F-action on it. Then the multiplicity of \(\phi\in \hat G^ F\) in \(R_ w^{\theta}\) is described in a uniform way in terms of ''Fourier transforms'' with respect to a certain finite group \(\Gamma_{{\mathcal J}}\) associated to each family \({\mathcal J}\). irreducible characters; finite groups of Lie type; connected reductive group; Frobenius map; maximal torus; Borel subgroup; Deligne-Lusztig virtual characters; \(\ell \)-adic cohomology; semisimple classes; Jordan decomposition; irreducible constituents Linear algebraic groups over finite fields, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Simple groups: alternating groups and groups of Lie type, Ordinary representations and characters, Group actions on varieties or schemes (quotients) Détermination des caractères des groupes finis simples: Travaux de Lusztig. (Determination of characters of finite simple groups: Work of Lusztig) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Each infinitesimally faithful representation of a reductive complex connected algebraic group \(G\) induces a dominant morphism \(\Phi\) from the group to its Lie algebra \(\mathfrak g\) by orthogonal projection in the endomorphism ring of the representation space. The map \(\Phi\) identifies the field \(Q(G)\) of rational functions on \(G\) with an algebraic extension of the field \(Q(\mathfrak g)\) of rational functions on \(\mathfrak g\). For the spin representation of \(\text{Spin}(V)\) the map \(\Phi\) essentially coincides with the classical Cayley transform. In general, properties of \(\Phi\) are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find \(\text{Harm}(G)\) so that for the coordinate ring \(A(G)\) of \(G\) we have \(A(G)=A(G)^G\otimes\text{Harm}(G)\). As a consequence of a partial solution to this problem and a complete solution for \(\text{SL}(n)\) one has in general the equality \([Q(G):Q({\mathfrak g})]=[Q(G)^G:Q({\mathfrak g})^G]\) of the degrees of extension fields. Among other results, \(\Phi\) yields (for the complex case) a generalization, involving generic regular orbits, of the result of Richardson showing that the Cayley map, when \(G\) is semisimple, defines an isomorphism from the variety of unipotent elements in \(G\) to the variety of nilpotent elements in \(\mathfrak g\). In addition if \(G\) is semisimple the Cayley map establishes a diffeomorphism between the real submanifold of hyperbolic elements in \(G\) and the space of infinitesimal hyperbolic elements in \(\mathfrak g\). Some examples are computed in detail. infinitesimally faithful representations; reductive complex connected algebraic groups; Lie algebras; representation spaces; fields of rational functions; Cayley transforms; coordinate rings; regular orbits; varieties of unipotent elements Kostant, B.; Michor, P.; Christian, Duval, The generalized Cayley map from an algebraic group to its Lie algebra, \textit{Prog. Math.}, 213, 259-296, (2003), Birkhäuser, Boston, MA Representation theory for linear algebraic groups, Simple, semisimple, reductive (super)algebras, Lie algebras of linear algebraic groups, Classical groups (algebro-geometric aspects), Linear algebraic groups over the reals, the complexes, the quaternions, Representations of Lie and linear algebraic groups over local fields The generalized Cayley map from an algebraic group to its Lie algebra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A_6\) be the normal subgroup of the symmetric group consisting of all even permutations on 6 letters. The aim of this note is to study the relationship of \(A_6\) in a finite geometry based on the Leech lattice and in the complex algebraic geometry of \(K3\) surfaces. The authors' main results are stated as theorem 2.3 characterizing \(A_6\) as the pointwise stabilizer subgroup of some uniquely determined pentagon in the Leech lattice which is an analogue of results by \textit{R. T. Curtis} [J. Algebra 27, 549--573 (1973; Zbl 0297.20028)] and \textit{L. Finkelstein} [J. Algebra, 25, 58--89 (1973; Zbl 0263.20010)] and theorems 3.1 and 5.1 which show the existence and uniqueness of the triplet \((F, G, {\rho}_F)\) of a \(K3\) surface \(F\) its finite group action \({\rho}_F\) of \(G\) on \(F\) up to isomorphism, where \(G\) is an extension of \(A_6 \) by \({\mu}_4\), the multiplicative group of four roots of unity. This last result motivated for example by \textit{S. Kondo} [Duke Math. J. 92, 593--598 (1998; Zbl 0958.14025)]. The paper's main objectives are outlined as follows.
In section 2 theorem 2.3 the alternating group \(A_6 \) is characterized as the subgroup of the group of affine isometries of the Leech lattice fixing the six vertices of a Coxeter-Dynkin diagram of type \({A_2}^{\oplus 2} \oplus {A_1}^{\oplus 2} \). In section 3 theorem 3.1 a triplet \(( F, \tilde{A_6}, {\rho}_F)\) consisting of a \(K3\) surface \(F\) and a faithful \(A_6\) -action: \({\rho}_F: {\tilde A_6} \times F \rightarrow F \) is constructed, where \({\tilde A}_6 \) is the group defined in definition 2.7. This section ends with proposition 3.5 giving an explicite equation of a canonical model of \(F\) in \(P_1 \times P_2 \). In section 4, proposition 4.1 it is shown the maximality of the extension of \(A_6\) by \( {\mu}_4 \) and in proposition 4.5 it is proved that the \(K3\) surface admitting an \(A_6 \cdot {\mu}_4\)-action is unique. The paper concludes with section 5, theorem 5.1 proving the uniqueness of the \( A_6 \cdot {\mu_4} \)-action. K3 surfaces and Enriques surfaces; Lattices and Convex bodies; Simple groups and groups of Lie type; simple groups of Lie type; simple groups: sporadic groups Keum J.H., Oguiso K., Zhang D.-Q.: The alternating group of degree 6 in the geometry of the Leech lattice and K3 surfaces. Proc. Lond. Math. Soc. (3) 90(2), 371--394 (2005) Lattices and convex bodies (number-theoretic aspects), Simple groups: alternating groups and groups of Lie type, Simple groups: sporadic groups, \(K3\) surfaces and Enriques surfaces The alternating group of degree 6 in the geometry of the Leech lattice and \(K3\) surfaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For \(G\subset \text{SL}(2,{\mathbb C})\) a finite group, the quotient variety \(X={\mathbb C}^2/G\) is called a Klein quotient singularity. The resolution of singularities \(Y\rightarrow X\) has exceptional locus consisting of \(-2\)-curves \(E_i\) (i.e. isomorphic to \({\mathbb P}_{{\mathbb C}}^1\), with self-intersection \(E_i^2=-2\)), and whose intersections \(E_iE_j\) are given by one of the Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\) or \(E_8\). The classical McKay correspondence begins in the late 1970s with the observation that the same graph arises in connection with the representation theory of \(G\), i.e. there is a one-to-one correspondence between the components of the exceptional locus of \(Y\rightarrow X\) and the nontrivial irreducible representations of \(G\subset \text{SL}(2,{\mathbb C})\). The paper explains this coincidence in several ways, and discusses higher dimensional generalizations. group action; \(K\)-theory; derived category; quotient variety; resolution of singularity; motivic integration; McKay correspondence; Hilbert schemes of \(G\)-orbits; crepant resolution; discrepancy divisor; Klein quotient singularity Reid, Miles, La correspondance de McKay, Astérisque, 276, 53-72, (2002) Global theory and resolution of singularities (algebro-geometric aspects), Linear algebraic groups over arbitrary fields, Homogeneous spaces and generalizations McKay's correspondence | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We study a triangulated category of graded matrix factorizations for a polynomial of type \(ADE\). We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading.
For part I see the third author [``Matrix factorizations and representations of quivers. I'', preprint, \\url{arxiv:math/0506347}]. matrix factorizations; triangulated categories; representations of Dynkin quivers; \(ADE\) singularities; Landau-Ginzburg orbifolds; mirror symmetries; path algebras; quiver representations; Auslander-Reiten quivers Kajiura, H.; Saito, K.; Takahashi, A., Matrix factorization and representations of quivers. II. type \textit{ADE} case, Adv. Math., 211, 1, 327-362, (2007) Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Matrix factorizations and representations of quivers. II: Type \(ADE\) case. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a classical simple algebraic group (type \(A\), \(B\), \(C\), \(D\)) over a field of characteristic different from two. The author gives a description of generators for the subring of invariants in the ring of polynomial functions on a direct sum of \(r\) copies of the adjoint representation. The generators involve taking traces, an unspecified module endomorphism of a specified \(G\)-module, and generic matrices. For type \(A\) the result is covered by work of \textit{S. Donkin} [Invent. Math. 110, No. 2, 389-401 (1992; Zbl 0826.20036)], who generalized results obtained in characteristic zero by \textit{C. Procesi} [Adv. Math. 19, 306-381 (1976; Zbl 0331.15021)]. Just like Donkin the author deals with the difficulties of characteristic \(p\) by showing that the modules that are needed have good filtration. For types \(B\), \(C\), \(D\) he views the adjoint representation as a direct summand of \(\text{gl}(V)\), where \(V\) is the defining representation, and then studies the invariant polynomials on \(\text{gl}(V)^{\oplus r}\). The analysis also shows that the Poincaré series of the ring of invariants is the same as in characteristic zero. simple algebraic groups; generators; subring of invariants; ring of polynomial functions; adjoint representation; traces; generic matrices; good filtrations; direct summands; invariant polynomials; Poincaré series; ring of invariants Zubkov, AN, \textit{on the procedure of calculation of the invariants of an adjoint action of classical groups}, Comm. Algebra, 22, 4457-4474, (1994) Representation theory for linear algebraic groups, Vector and tensor algebra, theory of invariants, Geometric invariant theory On the procedure of calculation of the invariants of an adjoint action of classical groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let G be a semisimple, simply connected algebraic group over an algebraically closed field K of characteristic \(p>0\). For \(n>0\), let \(G_ n\) be the schemetheoretic kernel of the n-th power of the Frobenius morphism and let \(G_ nT\) be the related group scheme for a fixed maximal torus T of G. For each \(\lambda\) in the character group X(T), L(n,\(\lambda)\) (resp. L(\(\lambda)\)) denotes the simple \(G_ nT\)-module (resp. G-module) of highest weight \(\lambda\) and Q(n,\(\lambda)\) the injective hull \((=projective\) cover) of L(n,\(\lambda)\). Denote by \(X_ n(T)\) the set of dominant weights with coordinates between 0 and \(p^ n- 1\). Each \(\lambda\in X(T)\) can be written uniquely as \(\lambda^ 0+p^ n\lambda^ 1\) with \(\lambda^ 0\in X_ n(T)\). When \(p\geq 2h-2\) (h being the Coxeter number of G), it is known that there is a G-module, for each \(\lambda \in X_ n(T)\), restricting to the \(G_ nT\)-module Q(n,\(\lambda)\), this G-module is also called Q(n,\(\lambda)\). The group of rational points of G over a field of \(p^ n\) elements is denoted by G(n). The simple KG(n)-modules are the restrictions of the L(\(\lambda)\) for \(\lambda \in X_ n(T)\). When \(p\geq 2h-2\), the G-module Q(n,\(\lambda)\) is injective and projective for KG(n) with the injective hull \((=\) projective cover) U(n,\(\lambda)\) of L(\(\lambda)\) occurring once as a summand. Given \(\lambda \in X_ n(T)\), the Cartan invariant \(\hat c_ n(\lambda,\mu)\) is the multiplicity of L(n,\(\mu)\) as a factor of the \(G_ nT\)-module Q(n,\(\lambda)\). Define \(c_ n(\lambda,\mu)\) to be the multiplicity of the G(n)-module L(\(\mu)\) as a factor of U(n,\(\lambda)\). Let \(W_ p\) be the affine Weyl group associated with the Weyl group of G and the corresponding alcoves in X(T)\(\otimes {\mathbb{R}}\). \(W_ p\) acts on weights via the dot action: \(w\cdot \lambda =w(\lambda +\rho)-\rho\), where \(\rho\) is the sum of fundamental dominant weights.
In the paper under review, the author regards (A)-(D) below as the conditions on Cartan invariants for the KG(n)-modules U(n,\(\lambda)\) to be generic.
(A) The weight \(\lambda\) lies in the interior of some alcove for \(W_ p.\)
(B) As a G(n)-module, Q(n,\(\lambda)\) remains indecomposable and is therefore isomorphic to U(n,\(\lambda)\).
(C) Let \(L(\mu^ 0)\otimes L(\mu^ 1)^{[n]}\) be any G composition factor of Q(n,\(\lambda)\), where [n] denotes twisting by the n-th power of the Frobenius map. If A is the alcove containing \(\mu^ 0\), then \(\mu^ 0\) lies at a distance of at least \(r(\mu^ 1)\) from all walls of A, where \(r(\mu^ 1)\) denotes half the diameter of the weight diagram of \(L(\mu^ 1).\)
(D) Let both L(\(\mu)\) and L(\(\nu)\) be G composition factors of Q(n,\(\lambda)\). If \(\mu^ 0\) and \(\nu^ 0\) lie in the same alcove but are distinct, then \(\mu^ 0+\pi \neq \nu^ 0+\sigma\) for all weights \(\pi\), \(\sigma\) of \(L(\mu^ 1)\), \(L(\nu^ 1)\), respectively.
Then the main result of this paper is stated as follows. Suppose \(\lambda \in X_ n(T)\) satisfies conditions (A)-(D). Then there is a natural 1-1 correspondence between the distinct \(G_ nT\) composition factors of Q(n,\(\lambda)\) and the distinct G(n) composition factors of \(U(n,\lambda)=Q(n,\lambda)\). The correspondence sends \(L(n,\mu^ 0+p^ n\pi)\) to \(L(\mu^ 0+\pi)\). Moreover, the composition factor multiplicities agree, i.e. the respective Cartan invariants are equal: \(\hat c_ n(\lambda,\mu^ 0+p^ n\pi)=c_ n(\lambda,\mu^ 0+\pi).\)
Note that L. Chastkofsky and Ye Jia-Chen have described the generic Cartan invariants for finite groups of Lie type in two different ways. The present paper offers a more conceptual treatment on these Cartan invariants which avoids the messy explicit calculations. semisimple, simply connected algebraic group; group scheme; maximal torus; character group; dominant weights; Coxeter number; G-module; group of rational points; injective hull; projective cover; affine Weyl group; fundamental dominant weights; Cartan invariants; composition factors; finite groups of Lie type Humphreys, J. E.: Generic Cartan invariants for Frobenius kernels and Chevalley groups. J. algebra 122, 345-352 (1989) Representation theory for linear algebraic groups, Group schemes, Linear algebraic groups over finite fields, Linear algebraic groups over arbitrary fields Generic Cartan invariants for Frobenius kernels and Chevalley groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Conformal field theories are among the recent leading candidates for a general theory encompassing all fundamental physical forces. The various approaches to construct, to investigate and to classify appropriate conformal field theories have revealed, over the past 15 years, deep connections between the underlying physical requirements and the mathematical framework needed for their modelling. One of the most recent and fascinating examples for this close interrelation occurs in two-dimensional conformal quantum field theory, where physical intuition and some current, highly advanced topics of pure mathematics intertwine in a mutually challenging and inspiring manner. Alas, scientific communication between physicists and (pure) mathematicians is traditionally rather complicated, due to the different understanding of many mathematical concepts and to the discrepancy between physical intuition and indispensible mathematical rigor. The present book aims at contributing towards a better understanding of the mathematical background and framework of two-dimensional conformal field theory.
More precisely, the main goal of the text is to describe and to explain, in a mathematically rigorous and comprehensive way, the fundamental role of the Virasoro algebra in the quantization process of conformal symmetries in dimension two. In this vein, the book under review should be seen as a ``mathematical'' preparation for a profound study of conformal field theory, esspecially designed for mathematicians, as well as a particular mathematical reference book adapted to needs of physicists engaged in conformal field theory.
According to these objectives, the text consists of two major parts of approximately equal length. Part I is entitled ``Mathematical Preliminaries'' and provides a rather elementary, detailed and largely self-contained account on classical conformal symmetry in arbitrary dimensions and its quantization in two dimensions.
Section 1 discusses the conformal transformations on semi-Riemannian manifolds, together with their classification theory, while Section 2 is devoted to the structure of the corresponding conformal transformation groups. Central extensions of groups and the quantization of symmetries are the topics in Section 3, which are then extended to Lie groups in the following Section 4, including a complete proof of V. Bargmann's fundamental theorem (1954) on liftings of projective representations to unitary representations of certain Lie groups. Section 5 describes how the Witt algebra and the Virasoro algebra, the latter one being a central extension of the former one, do appear in the study of conformal symmetries. This concludes Part I of the text, which is throughout comparatively elementary, detailed and enhanced by various concrete examples.
In opposition, Part II, entitled ``First Steps Towards Conformal Field Theory'', is more like a survey on some topics in conformal field theory and the mathematical methods and results related to them. Section 6 sketches the representation theory of the Virasoro algebra via Verma modules and the Kac determinant, Section 7 touches upon the projective representations of the diffeomorphism group of the circle, and Section 8 briefly discusses the conformal symmetry in bosonic string theory with an application to the representation theory of the Virasoro algebra.
The physical aspects are emphasized in Section 9, where the author explains a two-dimensional conformally invariant quantum field theory along the axiomatic approach by K. Osterwalder and R. Schrader (1973-1975). The concluding Section 10 takes up another spectacular aspect of the link between conformal field theory and mathematics, namely the geometry encoded in the celebrated Verlinde formula. The author surveys the character of the Verlinde formula as both a fusion rule for conformal fields and a dimension formula for spaces of generalized theta functions on moduli spaces of vector bundles on compact Riemann surfaces, with a strong emphasis on the mathematical significance.
Part II is by far less elementary than the introductory first part of the text, and only few proofs are given here. However, the systematic overview of the recently obtained insights into the striking interplay between quantum field theory and complex geometry is extremely enlightening, and very useful, appetizing and guiding for both mathematicians and physicists interested in the subject. The rich bibliography provides valuable hints to the current research literature for further studies.
Altogether, the author's contribution to the improvement of communication between physicists and mathematicians, embodied by this booklet, is highly rewarding and welcome. central extensions of Lie algebras; conformal groups; Witt algebra; conformal field theories; central extensions of groups; two-dimensional conformal field theory; Virasoro algebra; conformal symmetries in dimension two; representation; Verma modules; Kac determinant; diffeomorphism group of the circle; bosonic string theory; Verlinde formula; fusion rule; dimension formula; spaces of generalized theta functions; moduli spaces of vector bundles; compact Riemann surfaces; bibliography Schottenloher, M.: A mathematical introduction to conformal field theory. (1997) Virasoro and related algebras, Vector bundles on curves and their moduli, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to quantum theory, Axiomatic quantum field theory; operator algebras, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations A mathematical introduction to conformal field theory. Based on a series of lectures given at the Mathematisches Institut der Universität Hamburg | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0575.00010.]
The series of six lectures given by the author at the meeting dealt largely with the author's paper with the above title which appeared in J. Algebra 89, 437-499 (1984; Zbl 0552.12004).
The remainder of this contribution deals with related topics, and includes remarks designed to facilitate the reading of the J. Algebra paper. inverse problem of Galois theory; Fischer-Griess monster as Galois group over \({\mathbb{Q}}\); finite simple groups; fundamental group; rigid simple groups; cyclotomic field; discrete subgroups of \(PSL_ 2({\mathbb{R}})\); congruence subgroup; modular curve; Puiseux-series; group of covering transformations; compact Riemann surface; algebraic function field; ramification points; cusps; lectures Galois theory, Simple groups: sporadic groups, Representations of groups as automorphism groups of algebraic systems, Arithmetic theory of algebraic function fields, Simple groups: alternating groups and groups of Lie type, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Compact Riemann surfaces and uniformization, Algebraic functions and function fields in algebraic geometry, Separable extensions, Galois theory Some finite groups which appear as Gal L/K, where \(K\subseteq {\mathbb{Q}}(\mu _ n)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers representations of group schemes in characteristic \(0\) and \(p\), and gives applications to representations of groups and Lie algebras. It is the first part of his study on the existence of Auslander-Reiten sequences of group representations. representations of group schemes; representations of groups; Lie algebras; Auslander-Reiten sequences DOI: 10.1023/A:1009968402535 Representation theory for linear algebraic groups, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Group schemes, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) On the existence of Auslander-Reiten sequences of group representations. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Brauer-Severi varieties were introduced in 1944 by F. Châtelet to obtain a deeper understanding of the relation between a central simple algebra and the splitting fields of the algebra. Brauer-Severi schemes were introduced by M. Van den Bergh to study a wider class of algebras. In particular, he defined the Brauer-Severi schemes for the free algebra with \(m\) generators over a commutative ring \(R\) which the authors of the paper under review call generic Brauer-Severi schemes and show that they can be realized as moduli spaces of representations of a specific quiver.
For an order in a central simple algebra, the Brauer-Severi scheme is a projective fiber bundle over the variety of the centre. As the generic Brauer-Severi scheme is in a sense the most general one, it is important to study the geometry of its fibers and to compute their dimensions in order to determine the flat locus. In the paper under review the authors continue the study of the local structure of generic Brauer-Severi schemes started in a recent work by one of the authors. The main result is the following. For the Brauer-Severi scheme \(\eta\colon BS_{m,n}\to V_{m,n}\) of the trace ring of \(m\) generic \(n\times n\) matrices over the variety of matrix invariants, for a point \(\xi\in V_{m,n}\) of representation type \((m_1,n_1;\ldots;m_r,n_r)\), the reduced variety of \(\eta^{-1}(\xi)\) has \((\sum m_i)!/\prod m_i!\) irreducible components, each of the same explicitly given dimension. Brauer-Severi schemes; matrix invariants; moduli spaces of representations of quivers; fibers; orders in central simple algebras; projective fiber bundles; trace rings of generic matrices L. Le Bruyn and G. Seelinger, Fibers of generic Brauer--Severi schemes, J. Algebra, 214 (1999), 222--234.Zbl 0932.16025 MR 1684876 and Applied Mathematics, 290, Chapman and Hall, 2008.Zbl 1131.14006 MR 2356702 Trace rings and invariant theory (associative rings and algebras), Representations of quivers and partially ordered sets, Brauer groups (algebraic aspects), Rings arising from noncommutative algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Fibers of generic Brauer-Severi schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The finite-dimensional complex matrix Lie supergroups are realized as affine group superschemes, i.e., as algebraic supergroups. There are given explicit formulas for the structure of Hopf algebras of polynomial functions on these supergroups. Certain real forms are also considered.
The formulas for the Hopf algebra structure follow easily from those for the general linear supergroup (first published by \textit{D. Leites} and \textit{B. Zupnik} [Symmetries between bosons and fermions and superfields (Fan, Tashkent, 1976)], see also \textit{F. Berezin} [Introduction to superanalysis (Kluwer, 1987; Zbl 0659.58001)]).
The transition to any of the real forms (all of which are described by \textit{V. Serganova} [Funkts. Anal. Prilozh. 17, No.3, 46-54 (1983; Zbl 0545.17001)]) is straightforward with the help of Serganova's tables; the case of Poincaré and Heisenberg supergroups, also considered in the paper easily follows from their matrix realizations (for the Heisenberg supergroup this can be found in [\textit{D. Leites}, Usp. Mat Nauk 29, No.3(177), 209-210 (1974; Zbl 0326.14001)]). finite-dimensional complex matrix Lie supergroups; affine group superschemes; Hopf algebras of polynomial functions; real forms; Heisenberg supergroups; matrix realizations H. Boseck, Classical Lie supergroups, Math. Nachr. 148 (1990), 81--115. Infinite-dimensional Lie groups and their Lie algebras: general properties, Superalgebras, Supermanifolds and graded manifolds, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group schemes, Graded Lie (super)algebras, Loop groups and related constructions, group-theoretic treatment Classical Lie supergroups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The pioneering work of \textit{G. Shimura} [Ann. Math., II. Ser. 91, 144-222 (1970; Zbl 0237.14009), ibid. 92, 528-549 (1970; Zbl 0237.14010)] and Taniyama brought forth a far-reaching generalization of the classical theory of complex multiplication (and reciprocity laws for special values of the elliptic modular invariant), based on the theory of abelian varieties and their fields of moduli; further development resulted from Shimura's general theory of canonical models and a quite general reciprocity law for special values of modular functions of several variables on classical domains (of symplectic type).
The object of this important paper - the first in a projected series of three papers - is to pave the way for a ''relatively elementary treatment of the reciprocity laws for special values of Hilbert modular functions'', by an investigation of the relevant arithmetic groups and Eisenstein series along with number-theoretic ramifications, with a view to obtain new results for these modular functions; these results are then applied to the ideas in Hecke's dissertation (and Habilitationsschrift), following also some ideas of Eichler and Hasse, in order eventually to ''recover'', in an ''elementary'' way (i.e. independent of abelian varieties etc.), some results pertaining to ''reciprocity laws for special values of arithmetic modular functions of several variables with respect to arithmetic groups acting on tube domains'' and certainly, in the first instance, for groups commensurable with the Hilbert modular group \(\Gamma\).
Section 1 contains a list of necessary facts on central simple algebras over number fields k and Section 2, devoted to a description of maximal arithmetic subgroups of \(PGL_ 2(k)\), provides Helling's results (with alternative proofs for some) on the classification of conjugacy classes of maximal arithmetic groups commensurable with \(\Gamma\). ''Saturated'' arithmetic groups (with principal congruence subgroups constituting an important class of such groups) and maximality of discrete groups of holomorphic automorphisms of the Cartesian product of upper half planes are studied in the next sections.
The last section contains a discussion of Eisenstein series for the corresponding adèle group, which induce, on each component, holomorphic modular forms generalizing the familiar Eisenstein series due to Kloosterman. Arithmetic groups for which the Eisenstein series above generate the field of automorphic functions are realised more or less explicitly, viz. as group extensions of the usual arithmetic subgroups of \(PGL^+_ 2(k)\) by a (number-theoretically defined) subgroup of the group of automorphisms of k. complex multiplication; reciprocity laws for special values of Hilbert modular functions; arithmetic groups; Eisenstein series; maximal arithmetic groups; maximality of discrete groups of holomorphic automorphisms; adèle group; holomorphic modular forms Baily W L Jr, On the theory of Hilbert modular functions I, Arithmetic groups and Eisenstein series,J. Algebra 90 (1984) 567--605 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Complex multiplication and moduli of abelian varieties, Global ground fields in algebraic geometry On the theory of Hilbert modular functions. I: Arithmetic groups and Eisenstein series | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a finite dimensional associative \(K\)-algebra with unity over an algebraically closed field \(K\). Then the category \(\text{mod}_A\) of all finite-dimensional left \(A\)-modules is equivalent to the category of all finite-dimensional \(K\)-linear representations \(V=(V_i,\varphi_\alpha)\) of the Gabriel quiver \(Q\) of \(A\), where \(V_i\), \(i\in Q_0\), are finite dimensional vector spaces and \(\varphi_\alpha\colon\varphi_{s(\alpha)}\to\varphi_{t(\alpha)}\), \(\alpha\in Q_1\), are linear maps satisfying certain relations [see \textit{P.Gabriel}, Lect. Notes Math. 831, 1-71 (1980; Zbl 0445.16023)]. Denote by \(\mathbf{dim}V=(\dim_KV_i)_{i\in Q_0}\) the dimension-vector in the Grothendieck group \(K_0(A)\) of \(\text{mod}_A\). Let \(\text{mod}_A(\mathbf d)\) be the affine variety of \(\mathbf d\)-dimensional \(A\)-modules. It is proved that the condition \(\text{Ext}^1_A(X,X)=0\) for all indecomposable \(A\)-modules of dimension-vector \(\leq{\mathbf d}\) implies that the degeneration partial order \(\leq_{\text{deg}}\) in \(\text{mod}_A\) coincides with the extension partial order \(\leq_{\text{ext}}\) [cf. \textit{A. Skowroński, G. Zwara}, Can. J. Math. 48, No. 5, 1091-1120 (1996; Zbl 0867.16006)]. In the case when the latter condition is fulfilled the structure of minimal degenerations is described completely. finite dimensional modules; finitely generated algebras; Gabriel quivers; indecomposable Auslander-Reiten quivers; degenerations of modules; quiver representations; Grothendieck groups; affine varieties A. Skowronski and G. Zwara, ?Degenerations in module varieties with finitely many orbits,? Contemporary Mathematics 229 (1998), 343-356. Representations of associative Artinian rings, Group actions on varieties or schemes (quotients), Finite rings and finite-dimensional associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Module categories in associative algebras, Formal methods and deformations in algebraic geometry Degenerations in module varieties with finitely many orbits | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Seien \(X=\{x_ 1,...,x_ n\}\), \(Y=\{y_ 1,...,y_ n\}\) zwei freie Erzeugendensysteme der freien Gruppe \(F_ n\) vom Rang n. Ferner sei \(x_ i\) in \(F_ n\) konjugiert zu \(y_ i\) für \(i=1,...,n\). Es ist wohlbekannt, daß sich dann X aus Y herleiten läßt durch eine Folge von elementaren Transformationen des Typs \(\{u_ 1,...,u_ n\}\mapsto \{u'_ 1,...,u'_ n\}\) mit \(u'_ i=u_ j^{\epsilon}u_ iu_ j^{-\epsilon}\), \(\epsilon =\pm 1\), für ein Paar (i,j) und \(u'_ k=u_ k\) für \(k\neq i\). Der Autor gibt hierfür einen Beweis und beschreibt dann einige Anwendungen. monodromy groups of isolated singularities of hypersurfaces; free generators; free group; transformations; Dynkin diagram S. P. Humphries, ''On weakly distinguished bases and free generating sets of free group,''Quart. J. Math.,2, No. 36, 215--219 (1985). Free nonabelian groups, Generators, relations, and presentations of groups, Local complex singularities, Singularities of surfaces or higher-dimensional varieties On weakly distinguished bases and free generating sets of free groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The topic of this paper is the famous 'inverse problem' of Galois theory: Which groups occur as Galois groups over a fixed ground field k? \(k={\mathbb{Q}}\) or k a small extension of \({\mathbb{Q}}\) are of special interest. The most striking consequence of this work is the realization of the Fischer-Griess monster as a Galois group over \({\mathbb{Q}}.\)
The author deals more general with non-abelian finite simple groups F that are 'rigid'. This means: there are conjugacy classes \(C_ 1,...,C_ k\) in F, such that
i) F is generated by \(x_ 1,...,x_ k\) for every \((x_ 1,...,x_ k)\in C_ 1\times...\times C_ k\) with \(x_ 1\cdot...\cdot x_ k=1,\)
and ii) F operates (by conjugation) transitively on these generators \((x_ 1,...,x_ k).\)
This is the fundamental group theoretical concept. Examples of rigid simple groups given by the author are the Fischer-Griess monster and the groups \(PSL(2,2^ n)\).
The author then constructs Galois extensions \(L| K\) of some cyclotomic field K with a given rigid group F as Galois group, and more general: Galois extensions \(N| {\mathbb{Q}}\) with a Galois group built out of F in an explicit form.
The construction of these field extensions is based on the theory of Riemann surfaces and of discrete subgroups of \(PSL_ 2({\mathbb{R}})\). The proof is rather long and involved, containing a study of the congruence subgroup \(\Gamma_ 0(12)\) with its corresponding modular curve \(X_ 0(12)\), and explicit calculations in terms of Puiseux-series.
To fix the ideas behind all this one should add: A group G generated by \(x_ 1,...,x_ k\) \(with\quad x_ 1\cdot...\cdot x_ k=1\) can be realized as the group of covering transformations of some compact Riemann surface \(X\to {\mathbb{P}}^ 1({\mathbb{C}})\), unramified outside k points in \({\mathbb{P}}^ 1({\mathbb{C}})\). Hence G is the Galois group of an algebraic function field L over the field of rational functions \({\mathbb{C}}(T)\). Here \({\mathbb{C}}\) may be replaced by \({\bar {\mathbb{Q}}}\), the field of algebraic numbers. The main problem then is, to reduce \({\bar {\mathbb{Q}}}\) further to some small subfield K.
In the authors exposition \(X_ 0(12)\) plays the role of \({\mathbb{P}}^ 1({\mathbb{C}})\), and the ramification points are the six cusps of \(X_ 0(12)\). To reduce the field K of definition for \(L| {\mathbb{C}}(T)\) he uses explicit calculations in terms of Puiseux-series.
One should add that in general - by the work of B. H. Matzat - one can get information about K in purely group theoretical terms of F, and at this point the rigidity of F comes in. inverse problem of Galois theory; Fischer-Griess monster as Galois group over \(\mathbb{Q}\); finite simple groups; fundamental group; rigid simple groups; cyclotomic field; discrete subgroups of \(PSL_2(\mathbb{R})\); congruence subgroup; modular curve; Puiseux series; group of covering transformations; compact Riemann surface; algebraic function field; ramification points; cusps J. Thompson , Some finite groups which appear as Gal (L/K) where K \subset Q(\mu n) , J. Alg. 89 (1984) 437-499. Galois theory, Simple groups: sporadic groups, Representations of groups as automorphism groups of algebraic systems, Arithmetic theory of algebraic function fields, Simple groups: alternating groups and groups of Lie type, Unimodular groups, congruence subgroups (group-theoretic aspects), Finite automorphism groups of algebraic, geometric, or combinatorial structures, Compact Riemann surfaces and uniformization, Algebraic functions and function fields in algebraic geometry, Separable extensions, Galois theory Some finite groups which appear as \(\mathrm{Gal } L/K\), where \(K\subseteq \mathbb{Q}(\mu_n)\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities During the last few decades, invariant theory of finite groups became an extensive research subject. There are several excellent references available, e.g. [\textit{D. J. Benson}, Polynomial invariants of finite groups. London Mathematical Society Lecture Note Series. 190. Cambridge: Cambridge University Press (1193; Zbl 0864.13001)] and [\textit{L. Smith}, Polynomial invariants of finite groups. Research Notes in Mathematics (Boston, Mass.). 6. Wellesley, MA: A. K. Peters (1995; Zbl 0864.13002)].
In the non-modular case, that is, when the characteristic of the ground field does not divide the order of the group, there is -- for instance -- the Hochster-Eagon Theorem saying that the ring of invariants is always a Cohen-Macaulay ring. This is not true in the modular case by the famous example of \textit{M.-J. Bertin} [C. R. Acad. Sci., Paris, Sér. A 264, 653--656 (1967; Zbl 0147.29503)], which provides a presentation of \(\mathbb Z_4,\) the cyclic group of order 4, acting on a polynomial ring of four variables over a the field \(\mathbb F_2\), such that the ring of invariants is a unique factorization domain that is not a Cohen-Macaulay ring. Another example in the non-modular situation is the characterization of Sheppard and Todd which asserts that the ring of invariants \({\mathbb K}[V]^G\) of the action of a finite group \(G\) is a polynomial ring if and only if the action of \(G\) on \(V\) is generated by (pseudo)-reflections. In the modular case there is no characterization (besides of special cases) of representations of finite groups with polynomial rings of invariants.
The main reason for the book under review is the authors' intention to present a broad picture of modular invariant theory of finite groups with particular focus on the big difference to the non-modular case. This is done by various techniques for determining the structure of generators for modular rings of invariants, avoiding too much overlap with existing references. In the reviewer's opinion, this is done in an excellent way.
The authors start in their first section ``First steps'' with basic material on actions of groups on vector spaces and their coordinate rings, followed by illustrative examples with focus on the modular case. Then there is a short review of background material from commutative algebra and algebraic geometry, with applications to invariant theory. Modular aspects of the Noether bound, Molien's Theorem, and others are discussed. A separate chapter is devoted to examples, containing, among others, examples related to Göbel's Theorem, the ring of invariants of the regular representation of the cyclic group of order 4, and to non-Cohen-Macaulay rings of invariants. Other chapters of the book are related to computational aspects (SAGBI bases), hypersurface rings of invariants, Cohen-Macaulay invariant rings for \(p\)-groups, invariant rings via localization, etc. One chapter is devoted to the study of representations of finite groups which have polynomial rings of invariants. Note that in characteristic zero, this happens only if the group is generated by (pseudo-)reflections. The book contains an overwhelming amount of examples, discussed focusing on various aspects in order to enlighten problems specific for modular invariant theory.
The authors quote 117 references, among them a lot of those covering up-to-date research on the subject. Altogether, the book is a good source for examples and inspirations in modular invariant theory. In the reviewer's opinion, it is well suited for researchers who aim to to get a feeling for recent problems in modular invariant theory and related problems. It can also be used as a companion book for a graduate course in invariant theory of finite groups with a view towards the differences to the modular case. modular invariant theory; polynomial invariants; representations of finite groups; group action Campbell, H.E.A., Wehlau, D.L.: Modular Invariant Theory. Encyclopaedia of Mathematical Sciences, vol. 139. Springer, New York (2011) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to commutative algebra, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Group actions on varieties or schemes (quotients) Modular invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this survey, noncommutative algebraic geometry is interpreted as the category of quasi-coherent sheaves on an affine noncommutative scheme, having in view the category \(\operatorname{Mod}_A\) of (right) \(A\)-modules. That is, one considers directly the categories of sheaves of modules as this is an effective way to study algebraic varieties in the ordinary commutative situation. The author emphasises that quasi-coherent sheaves are independent of the choice of the topology on schemes, they are sheaves in any natural topology, and so they are claimed to be the objects that best reflect the algebraic structures of schemes. The next task is to glue together the noncommutative affine schemes, and there are different possibilities. The author finds that the most fruitful possibility is connected with the category of sheaves together with natural generalizations dictated by homological algebra. This leads to the derived category of quasi-coherent sheaves and the category of perfect complexes on a noncommutative scheme.
Let \(X\) be a scheme over a field \(\Bbbk\) that is quasi-compact, quasi-separated, has a finite covering by affine schemes whose intersections are affine with the same properties. To \(X\) is associated the unbounded derived category of complexes of \(\mathcal O_X\)-modules with quasi-coherent cohomology \(\mathcal D_\text{Qcoh}.\) It is proved that this category has enough compact objects, and that the triangulated subcategory of compact objects coincides with the category \(\mathit{Perf}-X\) of perfect complexes: A complex is perfect if it is locally isomorphic to a bounded complex of locally free sheaves of finite type. This category can be generated by a single object, called a classical generator. Then the minimal full triangulated subcategory of \(\mathit{Perf}-X\) containing this object and closed under taking direct summands coincides with the whole category \(\mathit{Perf}-X\).
Given the existence of a classical generator \(E\in\mathit{Perf}-X,\) the author gives a new view on the derived category \(\mathcal D_\text{Qcoh}(X),\) as well as the triangulated category \(\mathit{Perf}-X\). With this, \(\mathcal D_\text{Qcoh}(X)\) is equivalent to the unbounded derived category of differential graded (DG) modules \(D(\mathcal R)\) over some differential graded algebra \(\mathcal R\), and the triangulated category of perfect complexes
\(\mathit{Perf}-X\) is equivalent to the category of perfect DG modules \(\mathit{Perf}-\mathcal R.\) The algebra \(\mathcal R\) is given as the DG algebra if endomorphisms \(\operatorname{End}(E)\) of the given generator as its lift to a differential graded category \(\mathscr{P}\mathit{erf}-X.\) This is in some sense the game-changer in this paper: The differential graded category \(\mathscr{P}\mathit{erf}-X\) is a natural enhancement of the category \(\mathit{Perf}-X,\) in particular \(\mathit{Perf}-X\) is equivalent to the homotopy category \(\mathcal H^0(\mathscr{P}\mathit{erf}-X).\) A differential graded (DG) category \(\mathcal A\) is a category whose morphisms have the structure of complexes of \(\Bbbk\)-vector spaces. Passing to their zero cohomology spaces, one obtain a \(\Bbbk\)-linear category \(\mathcal H^0(\mathcal A)\) with the same set of objects, called the homotopy category for the DG category \(\mathcal A.\) With an equivalence \(\epsilon:\mathcal H^0(\mathcal A)\overset\sim\rightarrow\mathcal T,\;(\mathcal A,\epsilon)\) is called a DG enhancement for the category \(\mathcal T.\)
The triangulated category \(\mathcal D_\text{Qcoh}(X)\) has several natural enhancements, all naturally quasi-equivalent to each other. A convenient model can be chosen from the xlass of quasi-equivalent DG categories. A DG enhancement of the category \(\mathcal D_\text{Qcoh}(X)\) induces a DG enhancement of the triangulated subcategory \(\mathit{Perf}-X\), and this is denoted \(\mathscr P\mathit{erf}-X.\)
It is proved that \(\mathscr P\mathit{erf}-X\) is quasi-equivalent to a category of the form \(\mathscr P\mathit{erf}-\mathcal R\) with \(\mathcal R\) the DG algebra of endomorphisms of some generator \(E\in\mathscr P\mathit{erf}-X.\) Note that when \(X\) is quasi-compact and quasi-separated, \(\mathcal R\) is cohomologically bounded; it has only a finite number of non-trivial cohomology spaces. This leads to the following definition, stated verbatim:
By a derived noncommutative scheme \(\mathscr X\) ia meant a \(\Bbbk\)-linear DG category of the form \(\mathit{Perf}-\mathcal R,\) where \(\mathcal R\) is a cohomologically bounded DG algebra over \(\Bbbk.\) \(D(\mathcal R)\) is called the derived category of quasi-coherent sheaves on this noncommutative scheme, and the triangulated category \(\mathcal P\mathit{erf}-\mathcal R\) is called the category of perfect complexes on \(\mathscr X.\) There is an equivalence of categories \(\mathcal H^0(\mathscr P\mathit{erf}-\mathcal R)\cong \mathit{Perf}-\mathcal R,\) and by considering the DG categories of the form \(\mathscr P\mathit{erf}-\mathcal R\) the author is able to glue together noncommutative schemes from affine pieces and to obtain noncommutative derived schemes.
The paper studies noncommutative derived schemes, and properties as smoothness, regularity, and properness are extended to such. Morphisms between NC schemes are defined as quasi-functors between DG categories, so there are more morphisms between schemes in the noncommutative sense, forming a DG category where the morphisms can be added and one can define maps between morphisms. Also, for noncommutative schemes the concepts of compactification, resolution of singularities and the Serre functor, are defined.
As a preparation, this article studies some properties of derived noncommutative schemes and compare them to the corresponding properties for commutative schemes. Much space is devoted to a glueing together of noncommutative schemes. This concept does not exist in the commutative world, and gives a lot of applications. The most important is a geometric realization of derived noncommutative schemes, arising naturally. First of all as a natural geometric realization of an abstract algebraic structure, and also because many noncommutative schemes are constructed from a usual geometry with a given geometric realization. These are sometimes linked to admissible subcategories \(\mathcal N\subset\mathscr P\mathit{erf}-\mathcal X\) of categories of perfect complexes on smooth projective schemes \(X.\) Then \(\mathcal N\) is a DG category \(\mathcal N\subset\mathscr P\mathit{erf}-\mathcal R\) and \(\mathcal N\) is smooth and proper. The embedding \(\mathcal N\subset\mathscr P\mathit{erf}-\mathcal X\) is a particular case of a geometric realization of smooth, proper, noncommutative schemes, and is called pure.
Given two DG categories \(\mathscr A,\mathscr B\) and a \(\mathscr B^\circ-\mathscr A\)-bimodule \(\mathsf T\) a DG category \(\mathscr A\vdash_{\mathsf T}\mathscr B\) is defined, and called the gluing via \(\mathsf T.\) This defines gluing of noncommutative schemes \(\mathscr X, \mathscr Y\) under conditions on \(\mathsf T\) and base extension conditions are considered.
In previous work [\textit{D. Orlov}, Adv. Math. 302, 59--105 (2016; Zbl 1368.14031)], the author proved that if \(\mathscr X, \mathscr Y\) comes from admissible subcategories in categories of perfect complexes on smooth projective schemes, then their gluing \(\mathscr X\vdash_{\mathsf T}\mathscr Y\) via a perfect bimodule \(\mathsf T\) can be realized in the same way, and the article gives conditions for when geometric realizations are stable under base change. Also, noncommutative schemes give more information than commutative schemes. Phenomena called phantoms and quasi-phantoms are discussed. They are smooth and proper noncommutative schemes \(\mathscr X\) for which the \(K\)-theory \(K_\ast(\mathscr X)\) is completely trivial. Krull-Schmidt partners are discussed; these are smooth and proper noncommutative schemes \(\mathscr X\) and \(\mathscr X'\) for which there exists a smooth proper noncommutative scheme \(\mathscr Y\) with the condition that some gluings \(\mathscr X\vdash_{\mathsf T}\mathscr Y\) and \(\mathscr X'\vdash_{\mathsf T}\mathscr Y\) are isomorphic. The paper gives a procedure for constructing smooth and proper noncommutative schemes that are Krull-Schmidt partners for the usual schemes and have the same additive invariants.
The final section of the article studies geometric realization of finite-dimensional algebras. An explicit construction is given, and the particular case of algebras that are endomorphism of a vector-bundle on a (projective) scheme gives really nice results, linking the definitions in this article to other versions (tilting theory) of noncommutative geometry.
This article gives an important survey of Orlov's noncommutative geometry, it gives a very thorough illustration of the ideas and the complete theory, and it includes new techniques and thereby new important results. differential graded categories; triangulated categories; derived noncommutative schemes; finite-dimensional algebras; geometric realizations; noncommutative algebraic geometry; quasi-coherent sheaves; homological algebra; perfect complexes; unbounded derived category; enough injectives; classical generator; homotopy category; enhanced category; noncommutative scheme; noncommutative derived scheme; compactification; resolution of singularities; Serre functor; geometric realization; pure geometric realization; phantoms; quasi-phantoms; Krull-Schmidt partners Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.), Noncommutative algebraic geometry, Differential graded algebras and applications (associative algebraic aspects), Derived categories, triangulated categories, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Chain complexes (category-theoretic aspects), dg categories Derived noncommutative schemes, geometric realizations, and finite dimensional algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This article gives a solution to a problem of [\textit{J. Kollár}, ``Semi log resolutions'', \url{arXiv:0812.3592}, problem 19] on resolution of singularities of pairs except for semi-snc singularities.
Let \(X\) be a reduced algebraic variety over a field of characteristic 0 and \(D\) be a \(\mathbb Q\)-Weil-divisor on \(X\). In the sence of Kollar, the pair \((X,D)\) is semi-snc (``semi-simple normal crossings'') at \(a\in X\), if \(X\) is simple normal crossings at \(a\) (i.e. locally a simple normal crossings hypersurface in a smooth variety) and \(D\) is induced by the restriction to \(X\) of a hypersurface that is simple normal crossings with respect to \(X\).
Main result of the article is the following theorem on partial resolution of singularities of \((X,D)\):
``Let \(U\subseteq X\) be the largest open subset such that \((U,D|_U)\) is semi-snc. Then there is a morphism \(f:\tilde X \to X\) given by a composite of blowings-up with smooth (admissible) centers, such that [the total transform] \((\tilde X,\tilde D)\) [of \((X,D)\)] is semi-snc and \(f\) is an isomorphism over \(U\).''
The theorem has two special cases previously known already: If \(X\) is smooth, the resolution obtained is the snc-strict log-resolution (i.e. the log resolution of singularities of \(D\) by a morphism that is an isomorphism over the snc locus). Another important case is \(D=0\). Both of these cases are ingredients of the proof of the main theorem, where the reduction to the case that \(X\) has only snc-singularities is used, followed by an induction on the number of components of \(X\).
Functoriality of the desingularization morphism holds for algebraic varieties in characteristic 0 with fixed ordering of the components of \(X\) and with respect to etale morphisms that preserve the number of irreducible components of \(X\) and \(D\) passing through every point. Functoriality for general etale morphisms does not hold.
The paper is essentially self-contained (apart from resolution of singularities which is used as a ``black box''). It is based on an idea from papers of the first named author and J. Milman, E. Lairez respectively [\textit{E. Bierstone} and \textit{P. D. Milman}, Adv. Math. 231, No. 5, 3022--3053 (2012; Zbl 1257.14002); \textit{E. Bierstone, P. Lairez} and \textit{P. D. Milman}, Adv. Math. 231, No. 5, 3003--3021 (2012; Zbl 1262.14003); \textit{E. Bierstone} and \textit{P. D. Milman}, Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)] on characterizing local normal forms of mild singularities.
It should be noted, that the mild singularities treated in the previous papers [\textit{E. Bierstone, S. Da Silva, P. M. Milman} and \textit{F. V. Pacheco}, ``Desingularization by blowing up avoiding, simple normal crossings'', \url{arXiv:1206.5316v1}; Zbl 1257.14002] are hypersurface singularities, contrary to the semi-snc case. Therefore, the desingularization invariant is more complicated here: It is the Hilbert-Samuel function and the invariant based on it which is used to characterize semi-snc singularities. resolution of singularities; simple normal crossings; semisimple normal crossings; desingularization invariant; Hilbert-Samuel function; singularities of pairs; log-resolution of singularities Bierstone, Edward; Vera Pacheco, Franklin, Resolution of singularities of pairs preserving semi-simple normal crossings, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matematicas. RACSAM, 107, 159-188, (2013) Global theory and resolution of singularities (algebro-geometric aspects), Modifications; resolution of singularities (complex-analytic aspects), Divisors, linear systems, invertible sheaves, Invariants of analytic local rings Resolution of singularities of pairs preserving semi-simple normal crossings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities algebraic geometry; zeta functions; L-functions; schemes of finite type; analytic continuation; rationality; Artin-Chebotarev density theorem . J-P. Serre, ''Zeta and \(L\) functions,'' in Arithmetical Algebraic Geometry, New York: Harper & Row, 1965, pp. 82-92. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Zeta and \(L\) functions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This book, which seems to me very readable, is meant to give its reader an introduction to the representation theory of reductive algebraic groups G over an algebraically closed field k. In the case \(char(k)=0\) the subject is well understood: each G-module is semisimple, the simple G-modules are classified by their highest weights, and one has a character formula for these simple modules - in fact, Weyl's formula holds. The situation in prime characteristic is much worse: except for the case of a torus, there are non-simple G-modules, except for low rank cases, it is not known a character formula for simple modules, and Weyl's formula will certainly not carry over (though, according to Chevalley, one property survives: the simple modules are still classified by their highest weights, and the possible highest weights are the ``dominant'' weights of the group of characters X(T) of a maximal torus T of G). Nowadays this subject is the field of activity of many specialists who made significant progress and developed a beautiful theory. The reader will find this theory in the book under review. Modern developments of the theory are based on many different techniques which have been introduced into the theory, especially during the last fifteen years. This is why the book is divided to two parts: Part I contains a general introduction to the representation theory of algebraic group schemes, whereas Part II then deals with the representations of reductive groups.
The contents of Part I is as follows. Chapters I1 and I2 contain an introduction to schemes and to affine group schemes and their representations. The ``functorial'' point of view for schemes is adopted. The reader is supposed to have a reasonably good knowledge of varieties and algebraic groups. In Chapter I3, induction functors are defined in the context of group schemes, their elementary properties are proved, and they are used in order to construct injective modules and injective resolutions. These in turn are applied in Chapter I4 to the construction of derived functors, especially to that of the Hochschild cohomology groups and of the derived functors of induction. The values of the derived functors of inductions are interpreted in Chapter I5 as cohomology groups of certain associated bundles on the quotient G/H. Before doing that, one has to understand the construction of the quotient G/H. The situation gets simpler if H is normal in G. This is discussed in Chapter I6. The algebra of distributions on a group scheme G (called also hyperalgebra of G) is described in Chapter I7. The representation theory of a finite group scheme G is discussed in Chapter I8. A special case of this subject, when G arises as Frobenius kernels of algebraic groups over an algebraically closed field k of characteristic \(p\neq 0\), is discussed in Chapter I9. In the final chapter of Part I (Chapter I10) the general properties of the reduction mod p procedure are proved.
The contents of Part II is as follows. The purpose of Chapter II1 is to introduce split reductive group schemes and their most important subgroups, to fix a lot of notations and to mention without proof the main properties of these objects. The algebra of distributions on such a group scheme is described, and the relationship between the representation theories of the group and its algebra of distributions is discussed. Further G is assumed to be reductive, \(T\subset B\) are a fixed maximal torus and Borel subgroup, the ordering of X(T) is chosen in such a way that the weights of T in Lie(B) are negative, L(\(\lambda)\) denotes the simple module with the highest weight \(\lambda\). The main content of Chapter II2 is the discussion of the G-module \(H^ 0(\lambda)\), \(\lambda\in X(T)\), induced by a one-dimensional representation of B defined by \(\lambda\) : it is nonzero iff \(\lambda\) is dominant and L(\(\lambda)\) is the unique simple submodule of \(H^ 0(\lambda)\). Chapter II3 contains a rather simple proof of Steinberg's tensor product theorem (i.e. the existence of a decomposition of L(\(\lambda)\) into a tensor product of the form \(L(\lambda_ 0)\otimes L(\lambda_ 1)^{(1)}\otimes...\otimes L(\lambda_ r)^{(r)}\) induced by a suitable p-adic expansion of \(\lambda)\), discovered by Cline, Parshall, and Scott. This is based on consideration of some irreducible representations of the Frobenius kernels. Chapter II4 contains the proof (due to Haboush and Andersen) of Kempf's vanishing theorem, which shows that one special case of the Borel-Bott-Weil theorem holds for any field k (in full generality this theorem does not carry over to prime characteristic): if \(\lambda\) is dominant, then \(H^ i(\lambda)=0\) for all \(i>0\) (here \(H^ i(\mu)\) denotes the i-th cohomology group of a linear bundle of G/B associated to \(\mu)\). This theorem is crucial for the representation theory. In Chapter II5, it is given Demazure's proof of the Borel-Bott-Weil theorem in the case \(char(k)=0:\) it describes explicitly all \(H^ i(\mu)\) with \(i\in {\mathbb{N}}\), \(\mu\in X(T)\), i.e. for each \(\mu\) there is at most one i with \(H^ i(\mu)\neq 0\), and this \(H^ i(\mu)\) can then be identified with a specific L(\(\lambda)\). Furthermore, it is proved (following Donkin) that Weyl's character formula yields the alternating sum (over i) of the characters of all \(H^ i(\lambda)\). Chapter II6 is devoted to the linkage principle: if L(\(\mu)\) and L(\(\lambda)\) are both composition factors of a given indecomposable G- module, then \(\mu \in W_ p\cdot \lambda\) (here \(W_ p\) is the group generated by the Weyl group W and all translations by \(p\alpha\) with \(\alpha\) a root; the dot is to indicate a shift in the operation by \(\rho\), the half sum of the positive roots (i.e., \(w.\lambda =w(\lambda +\rho)-\rho))\). The general proof appeared only in 1980 (Andersen). Chapter II7 is devoted to the translation principle, a special case of which was conjectured by Verma: if two dominant weights \(\lambda\), \(\mu\) belong to the same ``facet'' with respect to the affine reflection group \(W_ p\), then the multiplicity of any L(w.\(\lambda)\) with \(w\in W_ p\) as a composition factor of \(H^ 0(\lambda)\) should be equal to that of L(w.\(\mu)\) in \(H^ 0(\mu)\). This was proved by the author. Kempf's vanishing theorem implies that one can construct for any k the \(H^ 0(\lambda)\) with \(\lambda\) dominant by starting with the similar object over \({\mathbb{C}}\), taking a suitable lattice stable under a \({\mathbb{Z}}\)-form of G, and then tensoring with k. This approach allows the construction of certain filtrations of \(H^ 0(\lambda)\). One can compute the sum of the characters of the terms in the filtration for large p and use this information to get information about composition factors. All this is considered in Chapter II8. Chapter II9 contains information on representation theory of the group scheme \(G_ rT\) and the applications to some weak version of the Borel-Bott-Weil theorem and the construction of the composition factors of \(H^ 0(\lambda +p\nu)\) from those of \(H^ 0(\lambda)\) if \(\lambda\) and \(\lambda +p\nu\) for some \(\nu\in X(T)\) are the weights that are ``small'' with respect to \(p^ 2\) and are ``sufficiently dominant''. Chapter II10 is devoted to the Steinberg modules \(St_ r\), which are both simple and injective as \(G_ r\)- modules, and some applications. It is proved (following Humphreys) that G is geometrically reductive. One may wonder whether any injective \(G_ r\)-module can be extended to a G-module; for large p this was proved by Ballard, and this is the subject of Chapter II11. The Hochschild cohomology groups \(H^ n(G,M)\) are discussed in Chapter II12, where the theorem due to Friedlander and Parshall is proved: for large p the cohomology ring \(H^.(G_ 1,k)\) is isomorphic to the ring of regular functions on the nilpotent cone in Lie(G). Chapters II13 and II14 are devoted to Schubert schemes and line bundles on these schemes: one can find there a proof of Kempf's vanishing theorem for Schubert varieties, as well as a proof of the normality of such varieties and a character formula for the space of global sections. representation theory; reductive algebraic groups; simple G-modules; highest weights; character formula; Weyl's formula; affine group schemes; injective modules; injective resolutions; derived functors; Hochschild cohomology groups; hyperalgebra; split reductive group schemes; Steinberg's tensor product theorem; irreducible representations; Kempf's vanishing theorem; Borel-Bott-Weil theorem; characters; linkage principle; dominant weights; filtrations; Steinberg modules; cohomology ring; ring of regular functions; Schubert schemes; line bundles [6] Jantzen J.\ C., Representations of Algebraic Groups, Academic Press, Orlando, 1987 Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Research exposition (monographs, survey articles) pertaining to group theory, Group schemes, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Affine algebraic groups, hyperalgebra constructions, Linear algebraic groups over arbitrary fields Representations of algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0619.00007.]
Let \(A=\mathbb{C}[[ X_1,\ldots,X_ n]]\) and let \(G\) be a finite group acting linearly on \(A\) with ring of invariants \(R=A^ G\). It has been proved by Auslander and Reiten that \(R\) has a finitely generated Grothendieck group \(G(R)\) and that the reduced Grothendieck group \(\widetilde G(R)\) is finite, at least when \(G\) acts freely on the linear space \(L = \sum_{i=1} \mathbb{C} X_ i\) of 1-forms. In the present note, which is essentially completely self-contained, the authors show that \(\widetilde G(R)\) is finite if \(G\) is just assumed to be abelian. Moreover, the reduced Grothendieck group of all simple hypersurface singularities is explicitly calculated.
For the second part of the paper (the explicit calculation of \(\widetilde G(R)\)), the authors make use of the fact that for any two-dimensional normal local domain \((R,m,k)\) with \([k]=0\), \(\widetilde G(R) = Cl(R)\), the class group of \(R\). Indeed, this result implies that if \(G\) acts faithfully and linearly on \(A=\mathbb{C}[[ X,Y]]\), then \(\widetilde G(A^ G)=(G/H)^*\), where \(H\) is the subgroup of \(G\), which is generated by the pseudoreflections on \(G\). The explicit description of \(\widetilde G(R)\) referred to above, makes use of this as well as some results of Knörrer, which say that (i) if \(f\in \mathbb{C}[[ X_1,\ldots,X_ n]]\), then the stable \(AR\)-quivers of \(\mathbb{C}[[ X_1,\ldots,X_ n]]/(f)\) and \(\mathbb{C}[[ X_1,\ldots,X_ n,Y,Z]]/(f+Y^ 2+Z^ 2)\) coincide, and (ii) simple hypersurface singularities are of finite Cohen-Macaulay representation type. Indeed, in htis case the stable \(AR\)-quivers of \(M(R)\) determine \(G(R)\), so one only has to consider the Grothendieck groups of simple hypersurface singularities in dimension 1 and 2.
Note: As pointed out by the authors, Auslander and Reiten have proved independently that \(\widetilde G(R)\) is finite for any action of a finite, not necessarily abelian group. Moreover, the \(AR\)-quivers of simple hypersurface singularities have been calculated by Dieterich and Wiedemann in dimension~1 and by Auslander in dimension~2. ring of invariants; reduced Grothendieck group; simple hypersurface singularities; stable AR-quivers; exponential-type operators; uniform approximation Jürgen Herzog and Herbert Sanders, The Grothendieck group of invariant rings and of simple hypersurface singularities, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 134 -- 149. Singularities of surfaces or higher-dimensional varieties, Geometric invariant theory, Grothendieck groups (category-theoretic aspects), Grothendieck groups, \(K\)-theory and commutative rings, Representation theory of associative rings and algebras, Group actions on varieties or schemes (quotients) The Grothendieck group of invariant rings and of simple hypersurface singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper under review is about ``partial'' resolution of singularities of an algebraic variety (over a characteristic zero field).
More precisely, the authors say that a variety \(X\) has a \textit{stable simple normal crossing} (s-snc) at a point \(a\) if, locally near \(a\), there is a closed embedding of \(X\) into a smooth variety \(Z\) such if \(X^{(1)}, \ldots, X^{(m)}\) are the irreducible components of \(X\) that contain \(a\) and \(X_a = \bigcap _{i=1}^{m}X^{(i)}\), then \(X_a\) is smooth at \(a\) and \(\sum_{i=1} ^{r} {\mathrm{codim}} (X^{(i)},Z)={\mathrm{codim}} (X_a,Z)\). Then the embedding dimension of \(X\) at \(a\) equals the dimension of \(Z\). They give several equivalent versions of this definition.
They also work with pairs \((X,D)\), where \(X\) is as above and \(D\) a \({\mathbb Q}\)-Weil divisor on \(X\) and they similarly define the notion ``a pair \(((X,D)\) has s-snc at \(a \in X\)''. Given a pair \((X,D)\) and a smooth closed subvariety \(C\) of \(X\), there is a new pair \((X_1,D_1)\), the \textit{transform} of \((X,D)\) with center \(C\), where \(X_1\) is the the blowing-up of \(X\) with center \(C\) and \(D_1=D'+E'\), with \(D'\) the birational transform of \(D\) and \(E'\) the exceptional divisor. Their main result is:
{Theorem 1.} Given a pair \((X,D)\) as above, there is a sequence of blowings-up \(X=X_0 \leftarrow \cdots \leftarrow X_t\) with smooth centers \(C_i \subset X_i\), \(i=0, \ldots, t-1\), such that if \((X,D)=(X_0,D_0)\) and for all \(i \geq 1\) \((X_i,D_i)\) is the transform of \((X_{i-1},D_{i-1})\) with center \(C_{i-1}\), then \((X_t,D_t)\) has only s-snc singularities, and each blowing-up \(X_{i-1} \leftarrow X_i\) is an isomorphism over the set of s-snc points of \(X_{i-1}\). The assignment of the sequence above to a pair \((X,D)\) is functorial with respect to smooth morphisms \(X' \to X\) that preserve the number of irreducible components at each point.
In the special case where \(X\) is smooth, Theorem 1 gives a log resolution of the divisor \(D\).
They also give a definition of {simple normal crossings singularity} and present an example showing that the analog of Theorem 1 is not valid for snc singularities.
Theorem 1 generalizes the main theorem of \textit{E. Bierstone} and \textit{F. V. Pacheco} [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 107, No. 1, 159--188 (2013; Zbl 1285.14014)]. The notion of snc point given in the present paper is more general than that found in the article just mentioned article, or in [\textit{E. Bierstone} and \textit{P. D. Milman}, Adv. Math. 231, No. 5, 3022--3053 (2012; Zbl 1257.14002)], where a a snc point is necessarily a hypersurface singularity, which is not the case with the new definition.
Due to the structure of the proof, the authors find it more convenient to show a similar theorem but for triples \((X,D,E)\), where \((X,D)\) is a pair as above and \(E = a_1 H_1 + \cdots + a_r H_r\) is snc divisor on \(X\) (the \(H_i\) taken in a given order), which easily implies Theorem 1. The proof of this result involves, among other things, the use of the function inv, introduced by Bierstone and Milman in their mentioned article, the Hilbert-Samuel function and, several times, the technique of \textit{presentations}, i.e., the introduction of an auxiliary object involving a sheaf of ideals, whose cosupport is related to values of an invariant. Of course, these techniques must be adapted to the present situation, and a number of subtle points are addressed by the authors.
In spite of its technical nature, the paper is relatively self contained and well written. resolution of singularities; stable simple normal crosssings; desingularization invariant; Hilbert-Samuel function; presentations Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Divisors, linear systems, invertible sheaves, Invariants of analytic local rings, Modifications; resolution of singularities (complex-analytic aspects) Desingularization preserving stable simple normal crossings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Block and Wilson have shown that a nonabelian simple Lie algebra of restricted type over an algebraically closed field \(k\) of characteristic \(p\geq 7\) is of classical or of Cartan type. The Lie algebras of Cartan type consists of the four series \(W_n\), \(S_n\), \(H_n\) and \(K_n\). For example,
\[
W_n= \text{Der}_k (k[x_1,\dots, x_n]/ (x_1^p,\dots, x_n^p)).
\]
The author constructs nonreduced group schemes for all Lie algebras of Cartan type and studies their irreducible and injective representations. irreducible representations; group schemes; Lie algebras of Cartan type; injective representations DOI: 10.1080/00927879608825553 Lie algebras of linear algebraic groups, Group schemes, Modular Lie (super)algebras Representations of group schemes of Cartan type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{A. V. Geramita, T.Harima} and \textit{Y. S. Shin} introduced in the paper reviewed above [The curves seminar at Queen's, Vol. VII, Kingston 1998, Queen's Pap. Pure Appl. Math. 114, 67-96, Exposé II C (1998; Zbl 0943.13012)] the type vector of a finite set of points in \(\mathbb{P}^n\). Type vectors are in 1-1 correspondence with the possible Hilbert functions of such point sets. Thus they encode the same numerical informations, but in a different way, so that certain geometrical informations about the point set may be more accessible. The authors also introduced \(k\)-configurations of points in \(\mathbb{P}^n\) which are special sets of points lying on certain arrangements of linear subspaces such that they have a prescribed type vector. In the paper under review, the authors continue their investigations of those notions and obtain a number of useful and interesting results.
(1) The number of minimal generators of the homogeneous vanishing ideal \(I_{\mathbb{X}} \subset k[x_0,\dots, x_n]\) of a \(k\)-configuration \(\mathbb{X}\subset \mathbb{P}^n\) and the degrees of those generators do not depend on the particular choice of \(\mathbb{X}\), but only on the type vector of \(\mathbb{X}\). More generally, all graded Betti numbers in the resolution of \(k[x_0,\dots, x_n]/ I_{\mathbb{X}}\) as a \(k [x_0,\dots, x_n]\)-module depend only on the type vector of \(\mathbb{X}\).
(2) For each type vector \({\mathcal T}\), one can construct a \(k\)-configuration \(\mathbb{X}\) (called a standard configuration of type \({\mathcal T}\)) such that the vanishing ideal of \(\mathbb{X}\) in the affine space \(D_+(x_0)\subset \mathbb{P}^n\) is the lex-segment ideal corresponding to the Hilbert function \({\mathbf H}(\mathbb{X}, -)\).
(3) Consequently, using the results of \textit{A. M. Bigatti} [Commun. Algebra 21, No. 7, 2317-2334 (1993; Zbl 0817.13007)], \textit{H. A. Hulett} [ibid. 21, No. 7, 2335-2350 (1993; Zbl 0817.13006)] and \textit{K. Pardue} [Ill. J. Math. 40, No. 4, 564-585 (1996; Zbl 0903.13004)], the authors show that, over an infinite base field \(k\), the minimal resolution of the homogeneous coordinate ring \(k[x_0,\dots, x_n]/I_{\mathbb{X}}\) of a \(k\)-configuration \(\mathbb{X}\) has the maximal graded Betti numbers possible for its Hilbert function \({\mathbf H} (\mathbb{X},-)\). In particular, this describes those extremal resolutions in a new and more geometrical way.
(4) Next the authors embed a \(k\)-configuration \(\mathbb{X}\) in a set of points \(\mathbb{Z}\) which is a complete intersection defined by products of distinct linear forms (called a basic configuration \(\mathbb{Z}\subset \mathbb{P}^n\)), and they let \(\mathbb{Y}= \mathbb{Z}\setminus \mathbb{X}\) be the complementary configuration (which need not be a \(k\)-configuration). In this liaison situation, they show that if \(\alpha(\mathbb{X})= \alpha(\mathbb{Z})\), then the Hilbert function of \(I_{\mathbb{Y}}\), the number of its minimal generators and their degrees can be computed easily.
(5) One can use this liaison construction to get a Gorenstein Artinian \(k\)-algebra \(k [x_0,\dots, x_n]/ I_{\mathbb{X}}\) \(+I_{\mathbb{Y}}\). If \(\mathbb{X}\) is a standard configuration and \(2\sigma (\mathbb{X})\leq\sigma (\mathbb{Z})\), then the graded Betti numbers of that ring depend solely on the type vector of \(\mathbb{X}\) and on \(\sigma(\mathbb{Z})\). Using this method, one gets examples for all Hilbert functions which are possible for Gorenstein Artinian \(k\)-algebras of codimension three. These examples have the extremal graded Betti numbers allowed by a result of \textit{S. J. Diesel} [Pac. J. Math. 172, No. 2, 365-397 (1996; Zbl 0882.13021)].
(6) If one applies the above liaison construction, then the Hilbert function of the resulting Gorenstein Artinian ring \(k [x_0,\dots, x_n]/ I_{\mathbb{X}}+ I_{\mathbb{Y}}\) is always unimodal. Thus, by a result of \textit{D. Bernstein} and \textit{A. Iarrobino} [Commun. Algebra 20, No. 8, 2323-2336 (1992; Zbl 0761.13001)], one cannot get all possible Hilbert functions of Gorenstein Artinian algebras of codimension \(n\geq 4\) in this way.
(7) Finally, the authors order the multi-sets of graded Betti numbers of Gorenstein Artinian algebras with fixed Hilbert function partially in the natural way. They ask whether there exist extremal graded Betti numbers under this ordering and conjecture that (at least for unimodal Hilbert functions arising as above) their liaison construction yields Gorenstein Artinian algebras having those extremal graded Betti numbers. To support their conjecture (or at least what the reviewer conjectures to be their conjecture, since the formulation of conjecture 8.1 is incomplete), they prove it under the additional assumptions that \(\sigma(\mathbb{Z})- 2\sigma(\mathbb{X})\geq n\) and one considers only Gorenstein Artinian algebras having the weak Lefschetz property studied by \textit{J. Watanabe} [in: Commutative algebra and combinatorics, US.-Jap. joint Semin., Kyoto 1985, Adv. Stud. Pure Math. 11, 303-312 (1987; Zbl 0648.13010)].
The paper is written in a lively, informal, and very readable manner and suggests numerous questions and problems awaiting further research. type vector; finite set of points in \(\mathbb{P}^n\); Hilbert functions; \(k\)-configurations; arrangements; number of minimal generators; lex-segment ideal; maximal graded Betti numbers; liaison Geramita, A. V.; Harima, T.; Shin, Y. S.: Extremal point sets and Gorenstein ideals. Queen's papers in pure and appl. Math. 114, 99-140 (1998) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Configurations and arrangements of linear subspaces, Linkage, Linkage, complete intersections and determinantal ideals Extremal point sets and Gorenstein ideals | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Simple singularities (also called ADE because the different types \(A_ k\), \(D_ k\) and \(E_ 6\), \(E_ 7\), \(E_ 8\) in which they are classified) appear in a natural way in problems of classification of singularities from different points of view. In dimension 1, simple singularities have been characterized in terms of their resolution procedure by \textit{W. P. Barth}, \textit{C. A. M. Peters} and \textit{A. J. H. M. Van de Ven} [``Compact complex surfaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4 (1984; Zbl 0718.14023)] in characteristic 0 and in any characteristic by \textit{K. Kiyek} and \textit{G. Steinke} [Arch. Math. 45, 565-573 (1985; Zbl 0553.14012)] who obtained the normal forms of them.
In dimension 2 simple singularities are just the rational double points [see \textit{M. Artin}'s paper in Complex Anal. Algebr. Geom., Collect. Papers dedic. K. Kodaira, 11-22 (1977; Zbl 0358.14008)]. On the other hand in characteristic 0 and arbitrary dimension \textit{V. I. Arnol'd} showed [cf. Funct. Anal. 6(1972), 254-272 (1973); translation from Funkts. Anal. Prilozh. 6, No.4, 3-25 (1972; Zbl 0278.57011)] that they are exactly the hypersurface singularities of finite deformation type (i.e. singularities from which one can obtain, by deformation, only a finite number of nonequivalent singularities).
According to previous works of \textit{H. Knörrer} [Invent. Math. 88, 153- 164 (1987; Zbl 0617.14033)] and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and \textit{F.-O. Schreyer} [ibid. 165-182 (1987; Zbl 0617.14034)] it is known that the simple singularities are the hypersurface singularities with finite Cohen-Macaulay type (i.e. singularities for which there exists only a finite number of non isomorphic indecomposable maximal Cohen-Macaulay modules over their local ring). However, the relationship with the point of view of Arnol'd (using deformation theory) remains essentially unknown in arbitrary characteristic.
This point constitutes the main result in the paper, more precisely, if \(f\in k[[x_ 1,...,x_ n]]\), k being an algebraically closed field of arbitrary characteristic, the authors define f to be simple if and only if f is contact-equivalent with one of the normal forms given in dimension 1 by Kiyek and Steinke, in dimension 2 by Artin and for dimension greater than two by double suspension of curves or surfaces.
Then, the main theorem asserts that the following statements are equivalent: (a) f is simple; (b) f is of finite deformation type; and (c) f is of finite Cohen-Macaulay type. - In particular a complete list of the normal forms for simple singularities is given, and, although the calculations in order to obtain the normal form of a given hypersurface are not completely described, a useful list of the main subcases is included. Also the adjacencies between the different types of simple singularities are completely given except for the case of surface singularities in characteristic 2 in which only partial information appears [see also \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002)]. The characterizations of the singularities \(A_{\infty}, D_{\infty}\) are also obtained by the same methods. The differences to the well known case of characteristic 0 are adequately explained and some useful examples are given in order to make clear the exceptions appearing only in positive characteristic. ADE singularities; hypersurface singularities of finite deformation type; hypersurface singularities with finite Cohen-Macaulay type; normal forms for simple singularities Greuel, G.-M., Kröning, H.: Simple singularities in positive characteristic. Math. Z. 203(2), 339-354 (1990). Zbl 0715.14001 Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local deformation theory, Artin approximation, etc., Local ground fields in algebraic geometry, Complex surface and hypersurface singularities Simple singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper deals with a geometric study of finite dimensional modules over a finite dimensional associative algebra. The author proves a cancellation theorem for degenerations of modules and considers its applications to the theory of preprojective modules and modules over tame concealed algebras, the theory of matrix pencils and others. In fact, the main results of the article under review have been improved and generalized in two papers written later but published earlier [\textit{K. Bongartz}, Comment. Math. Helv. 69, No. 4, 575-611 (1994; Zbl 0832.16008) and Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 5, 647-668 (1995; Zbl 0844.16007)]. regular modules; stretched modules; tame concealed algebras; tame quivers; indecomposable quivers; simple quivers; path algebras; finite dimensional modules; finite dimensional associative algebras; cancellation theorems; degenerations of modules; preprojective modules; modules over tame concealed algebras; matrix pencils Bongartz, K., On degenerations and extensions of finite dimensional modules, \textit{Adv. Math.}, 121, 245-287, (1996) Representations of quivers and partially ordered sets, Finite rings and finite-dimensional associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Group actions on varieties or schemes (quotients), Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers On degenerations and extensions of finite dimensional modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The simple-elliptic singularities \(\tilde E_ 7\) and \(\tilde E_ 8\) are studied by means of the derivation algebra of the moduli algebra. These singularities occur in 1-parameter families, which constitute (\(\mu\),\(\tau)\)-constant deformations. The authors introduce the notion of a derivation liftable to a vector field on the parameter space. For bad parameter values the liftable subalgebra is a proper subalgebra. For all parameter values the nilradical is liftable. This is conjectured to be true for arbitrary weighted-homogeneous isolated hypersurface singularities.
Unlike the example of the \(\tilde E_ 6\) singularities, the isomorphism class of the nilradical is an analytic invariant of the singularity. This is shown by computing the cross-ratio of four invariant lines in a 2-dimensional subquotient of the nilradical. variation of complex structures; variation of Lie algebras; simple-elliptic singularities; 1-parameter families Seeley C, Yau Stephen S-T. Variation of complex structures and variation of Lie algebras. Invent Math, 1990, 99: 545--566 Deformations of singularities, Deformations of complex singularities; vanishing cycles, Algebraic moduli problems, moduli of vector bundles, Simple, semisimple, reductive (super)algebras Variation of complex structures and variation of Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of arrangements of hyperplanes allows to attach to every parabolic subgroup of a finite Coxeter group ``exponents of the Coxeter group''-like numbers. In the case of Weyl groups similar numbers arise from the theory of Springer's representations. \textit{G. I. Lehrer} and \textit{T. Shoji} [J. Aust. Math. Soc., Ser. A 49, 449-485 (1990; Zbl 0729.20017)] have shown that in characteristic zero these numbers coincide for classical Weyl groups. In the present paper, the complete determination of the multiplicity of the reflection representation in the Springer representations for classical groups with arbitrary characteristic is given. Both unipotent elements in the group and nilpotent elements in the Lie algebra are considered.
Let \(\mathcal G\) be the Lie algebra of a connected reductive algebraic group defined over an algebraically closed field \(k\). Let \(\mathcal B\) be the variety of all Borel subgroups of \(G\). For \(A \in {\mathcal G}\) let \({\mathcal B}_ A = \{B \in {\mathcal B} \mid A \in \text{Lie} (B)\}\). The Springer representation for \(A\) is a natural action of the Weyl group \(W\) of \(G\) on the cohomology \(H^* ({\mathcal B}_ A)\). The cohomology theory used is \(l\)-adic cohomology. Let \(\rho\) be the reflection representation of \(W\). In the present paper, the multiplicity of \(\rho\) in the cohomology groups \(H^ i ({\mathcal B}_ A)\) (\(A\) nilpotent, \(G\) a classical group) is determined.
Let \(\mathcal P\) be a conjugacy class of parabolic subgroups of \(G\), and for \(A \in {\mathcal G}\), let \({\mathcal P}_ A = \{P \in {\mathcal P} \mid A \in \text{Lie} (P)\}\). Also, let \(W({\mathcal P})\) be a subgroup of \(W\) which can be thought of as the Weyl group of \(P \in {\mathcal P}\). The following theorem due to Borho-MacPherson is essential; Theorem. The homomorphism \(\pi^*_ A: H^*({\mathcal P}_ A) \to H^*({\mathcal B}_ A)\) induces an isomorphism of \(W^ P\)-modules \(H^*({\mathcal P}_ A) \cong H^* ({\mathcal B}_ A)^{W(P)}\), where \(W^ P = N(P)/W(P)\), \(N(P) = {\mathcal N}_ W (W({\mathcal P}))\).
A proof of this theorem is outlined in the present paper. In the case of \(\text{GL}_ n\), \(\text{Sp}_{2n}\) and \(\text{SO}_{2n + 1}\), results of Lehrer and Shoji for characteristic zero are reproved and also proved for arbitrary characteristic. In the case of \(\text{SO}_{2n}\) and for arbitrary characteristic, the same results are proved. multiplicity of reflection representation; arrangements of hyperplanes; finite Coxeter groups; Weyl groups; Springer representations; classical groups; Lie algebras; connected reductive algebraic groups; Borel subgroups; \(l\)-adic cohomology; cohomology groups; parabolic subgroups N. Spaltenstein, On the reflection representation in Springer's theory, Comment. Math. Helv. 66 (1991), no. 4, 618--636. Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Cohomology theory for linear algebraic groups, Group actions on varieties or schemes (quotients) On the reflection representation in Springer's theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Lambda\) be a finite-dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of \(\Lambda\) with fixed dimension \(d\) and fixed squarefree top \(T\). Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations of \(\Lambda\). In the case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety \(\mathfrak{Grass}^T_d\) of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety \(\mathfrak{Grass}^T_d\) is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple \(T\)', the radical layering \((J^lM/J^{l+1}M)_{l\geq 0}\) is shown to be a classifying invariant for the modules with top \(T\). This relies on the following general fact obtained as a byproduct: proper degenerations of a local module \(M\) never have the same radical layering as \(M\). finite-dimensional algebras; moduli spaces; simple modules; Grassmannians; projective varieties; quivers; finite local representation type; degenerations B. Huisgen-Zimmermann, ''Classifying representations by way of Grassmannians,'' Trans. Am. Math. Soc., 359, 2687--2719 (2007). Representations of associative Artinian rings, Representations of quivers and partially ordered sets, Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Representation type (finite, tame, wild, etc.) of associative algebras Classifying representations by way of Grassmannians. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a finite dimensional module \(V=V_0\oplus V_1\) over a semisimple complex Lie group \(G\), where each \(V_i\) is a \(G\)-submodule, the supersymmetric algebra is the algebra \(P(V)=S(V_0)\times\bigwedge(V_1)\) and is a \(G\)-module in a natural way. The author classifies indecomposable pairs \((G,V_0\oplus V_1)\) such that the action of the saturation \(\widetilde G\) on \(P(V)\) is multiplicity-free. Here, given a pair \((G,V)\) of a module \(V\) over a semisimple complex Lie group \(G\) which decomposes in a sum \(V=U_1\oplus\cdots\oplus U_k\) of irreducible \(G\)-modules, the saturation is the pair \((\widetilde G,V)\) where \(\widetilde G=G\times(\mathbb C^*)^k\), the \(i\)-th copy of \(\mathbb C^*\) acting as scalars on \(U_i\) and trivially elsewhere. A pair \((G,V)\) is decomposable if \(G\) is an almost direct product \(G=G_1G_2\) and \((G,V)\) is equivalent to \((G_1,V_1)\oplus(G_2,V_2)\) and indecomposable otherwise.
The special cases where either \(V_1\) or \(V_0\) is trivial reduce to multiplicity-free or skew multiplicity-free modules, respectively, that were already classified by \textit{V. G. Kac} [J. Algebra 64, 190-213 (1980; Zbl 0431.17007)] and \textit{C. Benson} and \textit{G. Ratcliff} [J. Algebra 181, No. 1, 152-186 (1996; Zbl 0869.14021)] in the first and by \textit{R. Howe} [Isr. Math. Conf. Proc. 8, 1-182 (1995; Zbl 0844.20027)] and the author [Transform. Groups 17, No. 1, 233-257 (2012; Zbl 1257.20048)].
Reviewer's remark: In the paper, Definition 2.1 of geometric equivalence of representations as stated reduces to the ordinary notion of equivalence since the author insists that it be induced by an underlying isomorphism of the vector spaces in question. Hence, the claim that \((G,V)\) is always geometrically equivalent to \((G.V^*)\) is incorrect. This problem does not seem to affect the results, though. semisimple complex Lie groups; invariant theory; skew multiplicity-free actions; supersymmetric algebras; finite-dimensional representations; skew multiplicity-free representations; exterior algebras Representation theory for linear algebraic groups, Semisimple Lie groups and their representations, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Vector and tensor algebra, theory of invariants Multiplicity-free super vector spaces. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We survey old and new results on semi-invariant polynomial functions on affine varieties of quivers with respect to conjugate actions of products of general linear groups over an algebraically closed field and the geometry of the sets of common zeros of the nonconstant semi-invariants. semi-invariants of quivers; semi-invariant polynomial functions; affine varieties of quivers; complete intersections; tame algebras; Auslander-Reiten quivers Bobiński, G.; Riedtmann, Ch.; Skowroński, A.: Semi-invariants of quivers and their zero sets, EMS ser. Congr. rep., 49-99 (2008) Representations of quivers and partially ordered sets, Geometric invariant theory, Actions of groups on commutative rings; invariant theory, Complete intersections, Representation type (finite, tame, wild, etc.) of associative algebras Semi-invariants of quivers and their zero sets. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a central simple algebra of dimension \((2m)^ 2\) over a field \(F\) of characteristic different from 2 and let \(\sigma\) be an involution of orthogonal type on \(A\), i.e. an \(F\)-linear anti-automorphism of period 2 on \(A\) such that the space of \(\sigma\)-symmetric elements has dimension \(m(2m + 1)\). Let \(\delta \in F^ \times\) be the reduced norm of some skew-symmetric unit in \(A\). The elements \(g \in A\) such that \(\sigma(g)g \in F^ \times\) are called similitudes of \((A,\sigma)\). The reduced norm of such an elements is \((\sigma(g)g)^ m\) or \(-(\sigma(g)g)^ m\). The similitude \(g\) is called proper in the former case and improper in the latter. In the first part of the paper, the following result is proved: if \(g\) is a proper similitude of \((A,\sigma)\), then the quaternion algebra \((\delta, \sigma(g)g)_ F\) is split; if \(g\) is an improper similitude, then the quaternion algebra \((\delta, \sigma(g) g)_ F\) is Brauer-equivalent to \(A\). This result generalizes a classical theorem of Dieudonné, which asserts that the multipliers of similitudes of a quadratic space of even dimension are norms from the discriminant extension. For algebras of degree 4 or 6, the multipliers of proper similitudes are explicitly determined in terms of the associated Clifford algebra. The method relies on several structures associated to central simple algebras with involution, such as Clifford bimodules and Clifford groups. A different proof, based on Galois cohomology, has been found by \textit{E. Bayer-Fluckiger} [C. R. Acad. Sci., Paris, Sér. I 319, 1151- 1153 (1994)].
The second part of the paper deals with homogeneous varieties, i.e. varieties of parabolic subgroups of semisimple linear algebraic groups. The main result is an explicit description of the Brauer group of a homogeneous variety and of the Brauer group kernel of the scalar extension map from the base field to the function field of a homogeneous variety. central simple algebras; involution of orthogonal type; \(\sigma\)- symmetric elements; reduced norm; similitudes; quaternion algebras; discriminant extensions; Clifford algebras; algebras with involution; Clifford bimodules; Clifford groups; homogeneous varieties; semisimple linear algebraic groups; Brauer groups Merkurjev, A.; Tignol, J., The multipliers of similitudes and the Brauer group of homogeneous varieties, Journal für die Reine und Angewandte Mathematik, 461, 13-47, (1995) Finite-dimensional division rings, Rings with involution; Lie, Jordan and other nonassociative structures, Brauer groups of schemes, Clifford algebras, spinors, Linear algebraic groups over arbitrary fields, Group actions on varieties or schemes (quotients) The multipliers of similitudes and the Brauer group of homogeneous varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G_ i\), \(i=1,2\) be complex analytic groups and let \(R(G_ i)\) denote their complex Hopf algebras of representative functions (an analytic function on \(G\) is representative if its \(G\) translates span a finite dimensional complex vector space). Suppose that \(R(G_ 1)\) and \(R(G_ 2)\) are isomorphic as complex Hopf algebras. The author's main theorem establishes that then there must be a complex algebraic group \(F\) and a torus \(Y\) in the radical of \(F\) which contains no non-trivial central element of \(F\) and analytic embeddings \(\rho_ i: G_ i\to F\) with Zariski dense images such that \(F\) is the semi-direct product \(F=Y\cdot \rho_ i(G_ i)\) and such that if \(M\supseteq Y\) is any maximal reductive subgroup then \(M\cap\rho_ 1(G_ 1)=M\cap \rho_ 2(G_ 2)\). Conversely, if such an \(F\) exists then the Hopf algebras \(R(G_ 1)\) and \(R(G_ 2)\) are isomorphic.
A triple \((F,Y,\rho_ i)\) as above is called a normal reduced split hull for \(F_ i\). Split hulls \((F,Y,\sigma)\) and \((H,Z,\tau)\) for the analytic group \(G\) are said to be equivalent if there is an algebraic group isomorphism \(\theta: F\to H\) such that \(\theta(Y)=Z\) and \(\tau\theta=\sigma\). Recall that a nucleus of \(G\) is a simple connected solvable normal subgroup with linearly reductive quotient, and that the intersection of the kernels of all the semi-simple representations of \(G\) is the representation radical of \(G\). The author shows that if the representation radical of \(G\) is a nucleus then \(G\) is the unique reduced split hull of \(G\). If not, then there are equivalent normal reduced split hulls unless the connected component of the center of any maximal reductive subgroup of \(G\) is also central in \(G\) itself. complex analytic groups; complex Hopf algebras of representative functions; complex algebraic group; analytic embeddings; semi-direct product; maximal reductive subgroup; normal reduced split hull; semi- simple representations General properties and structure of complex Lie groups, Affine algebraic groups, hyperalgebra constructions, Complex Lie groups, group actions on complex spaces, Other representations of locally compact groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act Complex analytic groups and Hopf algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A ring \(R\) is said to be an MM (short for ``Morita matrix'') ring if every ring which is Morita-equivalent to \(R\) is isomorphic to a full matrix ring over \(R\). Examples include rings for which all finitely-generated projective modules are free, but there are also many more interesting examples. In general an MM ring need not be indecomposable, but an MM ring which satisfies a polynomial identity (in particular, a commutative MM ring) is necessarily indecompossable. The paper is not restricted to commutative rings (for instance the final section is concerned with the class of von Neumann regular algebras known as ultramatricial algebras), but many of the results are for commutative rings and so for the remainder of this review we shall consider only the commutative case. A ring \(R\) is MM if and only if so also is the ring of formal power series over \(R\), but the MM property does not in general pass from \(R\) to the polynomial ring \(R[X]\) or to the partial quotient rings of \(R\). There is a lengthy discussion of the MM property for pullbacks, and in particular this gives an important method for constructing examples and counter-examples. If \(R\) is an MM ring then necessarily \(R\) is indecomposable and the Picard group \(\text{Pic}(R)\) is divisible, but these two conditions together are not sufficient for \(R\) to be MM. On the other hand, if \(R\) is Noetherian of Krull dimension at most one, then \(R\) is an MM ring if and only if \(R\) is indecomposable and \(\text{Pic}(R)\) is divisible. Thus a Dedekind domain is an MM ring if and only if its ideal class group is divisible. Morita matrix rings; commutative MM rings; full matrix rings; finitely-generated projective modules; polynomial identities; von Neumann regular algebras; rings of formal power series; partial quotient rings; Picard groups; ideal class groups P. Merisi and P. Vámos, On rings whose Morita class is represented by matrix rings , J. Pure Appl. Alg. 126 (1998), 297-315. Module categories in associative algebras, Endomorphism rings; matrix rings, Semiprime p.i. rings, rings embeddable in matrices over commutative rings, Dedekind, Prüfer, Krull and Mori rings and their generalizations, Picard groups On rings whose Morita class is represented by matrix rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper continues the author's series of surveys [Itogi Nauki Tekh., Ser. Probl. Geom. 11, 203-240 (1980; Zbl 0473.51013); ibid. 16, 195-229 (1984; Zbl 0592.51008); ibid. 21, 155-208 (1989; Zbl 0731.20033)]. In Sec. 1 (Secs. 1.1-1.9), we describe, mainly, investigations in which finite groups of symmetry that belong to a real space \(E^m\) and their invariant algebras are studied; papers are selected (Secs. 1.10-1.12) in which some related problems are considered (the structure of invariant algebras of finite linear groups, the action of reductive groups on manifolds, ect.). Section 2 is devoted to new results in the geometric theory of invariants of infinite groups that are generated by oblique reflections in the space \(E^m\). finite groups; invariant algebras; finite linear groups; actions; reductive groups; geometric theory of invariants; oblique reflections V. F. Ignatenko, ''Invariants of finite and infinite groups generated by reflections,''J. Math. Sci.,76, No. 3, 334--361 (1996). Other geometric groups, including crystallographic groups, Reflection groups, reflection geometries, Reflection and Coxeter groups (group-theoretic aspects), Classical groups (algebro-geometric aspects) Invariants of finite and infinite groups generated by reflections | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities For a natural number \(N\), denote by \(\mu_N\) the group of complex \(N\)-th roots of unity and consider the variety
\[
X_N:= \mathbb{P}_\mathbb{C}^1\setminus\bigl\{\{0,\infty\} \cup\mu_N\bigr\}=\mathbb{C}^* \setminus\mu_N.
\]
In the study of higher cyclotomy, understood as the motivic theory of multiple polylogarithms at roots of unity, it has turned out that there should be some fundamental link between the structure of the motivic fundamental group of the variety \(X_N\), on the one hand, and the geometry of certain modular varieties for the congruence subgroups \(\Gamma_1(m:N)\) of \(\text{GL}_m(\mathbb{Z})\), \(m\geq 1\), on the other. In the case of \(N=1\), this mysterious link had been sketchily foretold by A. Grothendieck in the early 1980s, and was then completely established by the work of P. Deligne (l987), Y.Ihara (1986--1999), V. G. Drinfeld (1991), and others in the sequel.
The paper under review is devoted to an extensive study of this link for \(\dot N>1\) from a particular viewpoint. Namely, according to P. Deligne's motivic philosophy, the motivic fundamental group of \(X_N\) can be interpreted as a Lie algebra object in the category of mixed motives, appearing there as a pro-nilpotent completion of the topological fundamental group of \(X_N\) equipped with various additional structures of analytic, geometric, and arithmetic nature. Moreover, the \(\ell\)-adic realization of the motivic fundamental group is obtained from the action of the absolute Galois group \(\text{Gal}(\overline \mathbb{Q}/\mathbb{Q})\) on the pro-\(\ell\) completion \(\pi_1^{(\ell)}(X_N)\) of \(\pi_1(X)\), and that is why a detailed study of this Galois action constitutes one main part of the present paper.
The author's approach is based on a fine analysis of certain Lie algebras constructed from the \(\ell\)-adic pro-Lie algebra \(\mathbb{L}_N^{(\ell)}\) corresponding via Deligne-Maltsev theory to the group \(\pi_1^{(\ell)} (X_N)\). The central object in the author's building up is a Lie algebra bigraded by negative integers \(-w\) and \(-m\), called the weight and the depth indices, which is called the ``level \(N\) Galois Lie algebra associated to \(\pi_1^{(\ell)}(X_N)\)''. In this context, the author's main goal is prove the following general statement:
Let \(V\) be the standard \(m\)-dimensional representation of the linear group \(\text{GL}_m\). Then the depth \(m\) and weight \(w\) part of the standard cochain complex of the level \(N\) Galois Lie algebra associated to \(\pi_1^{(\ell)}(X_N)\) is explicitely related to the geometry of the local system with fibre \(S^{w-m}V_m\) over the level \(N\) modular variety for the congruence subgroup \(\Gamma_1(m;N)\) of \(\text{GL}_m(\mathbb{Z})\). In particular, the depth \(m\) quotient of the level \(N\) Galois Lie algebra can be completely described by the geometry of that modular variety.
In the present paper, this is made explicit (and then-proved) in the special cases \(m=2\) and \(m=3\). The case \(m=1\) can be deduced from the motivic theory of classical polylogarithms due to P. Deligne and A. Beilinson, whereas there are indications that the next complicated case \((m=4)\) could also be tackled by the author's methods developed in the present paper. As for the novel and subtle techniques established here, the author builds upon his previous related results obtained in a series of foregoing papers the author [Math. Res. Lett. 4, No. 5, 617--636 (1997; Zbl 0916.11034); ibid. 5, No. 4, 497--516 (1998; Zbl 0961.11040)], this time emphasizing the Galois-theoretic part of his global program. Finally, the (partial) results of the present paper lead the author to a number of conjectures, in this present paper lead the author to a number of conjectures, in this context, which are explained and illustrated in detail. fundamental groups; modular varieties; Galois groups; motives; Lie algebras; complexes; polylogarithms; cohomology of arithmetic groups Goncharov, AB, The dihedral Lie algebras and Galois symmetries of \( \uppi_1^{(1)}\left( {{{\text{P}}^1}-\left( {\left\{ {0,\infty } \right\}\cup {\mu_{\text{N}}}} \right)} \right) \), Duke Math. J., 110, 397, (2001) Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Polylogarithms and relations with \(K\)-theory, Coverings of curves, fundamental group The dihedral Lie algebras and Galois symmetries of \(\pi_1^{(l)}(\mathbb P^1-(\{0,\infty\}\cup\mu-\{N\}))\). | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors define `generalized quiver representations' as quadruples \((G,R,V,Rv)\), where \(G\) is a reductive algebraic group, \(R\) is the centralizer of an Abelian reductive subgroup of \(G\) (hence \(R\) itself is reductive), \(V\) is a finite dimensional representation of \(R\) whose irreducible summands all appear in the Lie algebra of \(G\) viewed as an \(R\)-module via the adjoint action, and \(Rv\) is the orbit of \(v\in V\). When \(G\) is the general linear group, one obtains the classical notion of quiver representations. It is shown in the paper that when \(G\) is the orthogonal (respectively, symplectic) group, then the generalized quiver representations can be interpreted as the orthogonal (respectively, symplectic) representations of quivers with involution.
Extending results of \textit{P. Gabriel} [Manuscr. Math. 6, 71-103 (1972; Zbl 0232.08001)], \textit{L. A. Nazarova} [Izv. Akad. Nauk SSSR, Ser. Mat. 37, 752-791 (1973; Zbl 0298.15012)], \textit{P. Donovan} and \textit{M. R. Freislich} [The representation theory of finite graphs and associated algebras, Carleton Mathematical Lecture Notes No. 5. Ottawa, Ont., Canada: Carleton University. 83 p. (1973; Zbl 0304.08006)] on ordinary quiver representations, the finite or tame type quivers with involution are classified, and their indecomposable orthogonal (symplectic) representations are described. Finally, the authors discuss the relation of their results to related prior work of A. V. Rojter, V. V. Sergejchuk, and S. A. Kruglyak. generalized quiver representations; classical groups; quivers with involution; Dynkin graphs; tame representation type; orthogonal groups; symplectic groups; reductive algebraic groups; general linear groups; finite representation type Derksen, H., Weyman, J.: Generalized quivers associated to reductive groups. Colloq. Math. 94(2), 151--173 (2002) Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Classical groups (algebro-geometric aspects), Representation theory for linear algebraic groups Generalized quivers associated to reductive groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a status report on a century-old discovery that is still now making interesting progress. A retrospect is given at first on the emergence and history of the so-called Capelli identity. The text traces back to George Boole's work on the relative invariance of the discriminant of a general \(m\)-ary form of degree \(n\) about 150 years ago, and Arthur Cayley's systematic investigation on invariants a few years later. The theoretical background of the ``symbolic method'' stimulated Cayley to introduce the omega process -- a fundamental differential operator. But it was until D. Hilbert, who, giving a solution to the finiteness problem of the generating system of invariants, raised the curtain for modern abstract mathematics, and the pre-Hilbert era of invariant theory ended.
Meanwhile, another set of precision instruments, the Capelli identities, was brought into the invariant theory which can be understood as the equality to connect the two important processes: a) the omega process of Cayley and b) the polarization process of Aronhold.
Since these two processes transform an invariant into another, the Capelli identity was helpful in various transformations or reductions of invariants. While the main concern was the general linear group \(\text{GL}_n\) in the pre-Hilbert era, the same transformations are still effective for the invariant theory of other classical groups. It was H. Weyl who proved the first fundamental theorem (description of the generating system) and the second fundamental theorem (description of the relations) of vector invariants for the classical groups explicitly, and thus revived invariant theory and built the foundation of representation theory. But the treatment of the Capelli identity in his famous book ``The classical groups'' was, however, rather formal, and its meaning was not clear enough. It was Roger Howe, who recognized the deep relationship between invariant theory and representation theory and proposed it as the theory of (reductive) dual pairs. From this representation-theoretic point of view, a new light was also shed on the Capelli identities. Thus it took a century after Capelli for us to get ready for what could be clarified under its developments.
Next, the author discusses the Capelli identities from several different points of view, and looks at the related problems.
The classical Capelli identities are explicitly stated as an equality of differential operators. Let \(t_{ij}\) be the coordinates on the space of \(n\times n\) matrices, and \(\partial_{ij}=\partial/\partial t_{ij}\) the corresponding partial differential operators. Cayley's omega process is \(\Omega=\text{det}(\partial_{ij})\) of order \(n\) with constant coefficients. Under the identification of an \(n\times n\) matrix as a set of \(n\) column- (or row-) vectors, the polarization operators (or Aronhold process) are those which interchange its columns (or rows):
\[
E_{ij}=\sum^n_{i=1}t_{li} {\partial\over\partial t_{lj}},\quad E^0_{ij}=\sum^n_{l=1}t_{jl} {\partial\over\partial t_{il}}.
\]
The Capelli identity is given in the following formula:
\[
\text{det}(E_{ij}+(n-i)\delta_{ij})=\text{det}(t_{ij})\text{det}(\partial_{ij})
\]
or the equality obtained by putting \(E^0_{ij}\)'s in this equality instead of \(E_{ij}\)'s. A famous application to a concrete calculation is the so-called Cayley formula:
\[
\Omega(\text{det}(t_{ij})^s)=s(s+1)\cdots(s+n+1)\text{det}(t_{ij})^{s-1}.
\]
On the significance of the Capelli identity, the author points out that the Capelli identity is the way to describe an invariant differential operator by a central element in the enveloping algebra. A closer look at this situation enables the author to predict that any invariant operator is always obtained as a homomorphic image of a central element in the enveloping algebra of Lie algebra. As one gets the Capelli identity from the invariant differential operator \(\text{det}(t_{ij})\text{det}(\partial_{ij})\) from other differential operators one should expect to get other identities that describe other invariant differential operators by elements in the center. Identities of this kind were indeed obtained by Alfredo Capelli himself in 1890. These lower order Capelli identities are indispensable for the clear meanings of the Capelli identity from the representation-theoretic point of view, so that it is remarkable that Capelli himself already reached that high point.
With the background of the Capelli identities clarified, the author gives an outline of the proof from the representation-theoretic point of view.
The fact that the center of the universal enveloping algebra is mapped onto the algebra of invariant differential operators is deeply related to the validity of the Capelli identities. The author brings this to a more general context as follows: Considering a (complex) algebraic group \(G\) and its (rotational) representation \((\rho,V)\), one is naturally led to the following Capelli problems: (1) When is \(\rho\) surjective? (2) Describe \(\rho\) concretely, when it is surjective. The case considered in the subsequent text is that \(G\) is connected and reductive. And if the surjectivity holds in the problem (1) above, then the multiplicity of each irreducible component of the \(G\)-module is at most one. Such a representation \((\rho,V)\) of \(G\) is called, according to V. Kac, a ``multplicity-free action''. A classification of such actions is given by him and the classification list falls into 13 series, for each of them, the above Capelli problems are treated by R. Howe and T. Umeda almost a century after Capelli published his paper.
The main result for the Capelli problems are given then in this report, followed by a brief sketch of the general theory of the multiplicity-free actions from the view point of invariant theory and summarized in a theorem.
By the general theory stated above, the description of the decomposition of the algebra \(P(V)\) of polynomials and the structure of the algebra \(PD(V)\) of invariant differential operators are all reduced to the determination of the fundamental highest weight vectors. This principle is applied to concrete examples in several ways. It is hard to say that a uniform way to understand the essence of the Capelli identities has been obtained, because they are established through case-by-case analysis. Anyhow, the understanding of the classical Capelli identities is almost satisfactory.
After reviewing and extending the Capelli identities from the representation-theoretic point of view, the author takes up the quantum version of the Capelli identities, thus still another new aspect is found finally. Active research in quantum groups and their representation theory revealed the vista of the world of \(q\)-analogues. But the missing of the concept of differentiation puts a stumbling block in the way to establish the Capelli identities by a parallel analogy. When the noncommutativity of the variables is not so strong, this difficulty is overcome at length since the notion of \(q\)-differences provides a close enough \(q\)-analogue of differentiation. In subsequent paragraphs an exposition is given of the development of these questions, including an analogue of the differential operators and the formulations of the classical Capelli identities and the Leibniz rule in the matrix form.
In the appendix, a brief supplement is given on the development during the early nineties. Mainly a simplest and fundamental quantum version of the dual pairs is constructed and the Capelli identity associated to it is stated.
From the author's epilogue which gives the future prospect of the issue discussed: \dots even in the classical case and in the cases associated to multiplicity-free actions, things have not been fully pursued with ultimate clarity. Undoubtedly, we will still find various aspects in the study of the Capelli identities, both the new and the old ones, \dots we cannot say that the value of the Capelli identity in invariant theory has vanished with it. In particular, for invariant theory based on quantum group symmetry, we cannot forsee its future at the earliest stage of the theory, and instruments like the Capelli identities may possibly by a powerful key to open up new aspects of the theory \dots. In representation theory, in spite of the huge amount of accumulations both in general theory and specific calculations, there are still some missing points, which were hard to notice. Through the study of the Capelli identities, we have come across several such points. For these reasons, we expect that the Capelli identities will continue to play fruitful roles in the future as special mathematical objects. Lie algebras; orbits; spin; Chevalley generators; Capelli identity; forms; invariants; omega process; differential operators; generating systems; invariant theory; polarization process; general linear groups; classical groups; representation theory; dual pairs; enveloping algebras; algebras of invariant differential operators; actions; highest weights; quantum groups Umeda, T.: The capelli identities, a century after Sūgaku. 46, 206-227 (1994) Representation theory for linear algebraic groups, Vector and tensor algebra, theory of invariants, Actions of groups on commutative rings; invariant theory, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), History of mathematics in the 19th century, Quantum groups (quantized enveloping algebras) and related deformations, Universal enveloping (super)algebras, Noncommutative topology, Noncommutative differential geometry, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group actions on varieties or schemes (quotients) The Capelli identity, a century after | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author gives an outline of his construction of representations of affine Lie algebras in cohomology of a certain Lagrangian subvariety in the space of framed ASD connections on the ALE spaces associated to minimal resolutions of simple surface singularities. A similar construction for finite-dimensional simple Lie algebras was discussed in an earlier paper [\textit{H. Nakajima}, Int. Math. Res. Not. 1994, No.~2, 61-74 (1994; Zbl 0832.58007)]. affine Lie algebras; resolutions of simple surface singularities Modifications; resolution of singularities (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Applications of global analysis to structures on manifolds Gauge theory on resolutions of simple singularities and affine Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This survey is a published version of author's Master Thesis. The aim of this survey is to show interesting relation between classical objects in different areas of mathematics with the common underlying core of simply laced Dynkin diagrams. The text discusses Platonic solids, binary polyhedral groups, Kleinian singularities and simply laced Lie algebras. Another obvious candidate for this list would be hereditary algebras of finite representation type. The author does not discuss the latter but refers to a survey by Idun Reiten in the Notices of the AMS. symmetry; invariant; regular element; Platonic solids; binary polyhedral groups; Kleinian singularities; simply laced Lie algebras \(n\)-dimensional polytopes, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Lie algebras of Lie groups Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras of type ADE | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The homological algebra of algebraic groups is based on the fact that the category of rational modules has enough injectives. In general however there do not exist projective modules in this category. Our main result is that an affine group scheme \(G\) has a non-zero projective module if and only if \(G/G_{\text{lr}}\) is finite, where \(G_{\text{lr}}\) is the largest normal linearly reductive subgroup scheme of \(G\). Let \(H\) be a cocommutative Hopf algebra over a field \(k\). The category of \(H\)-modules which are locally finite dimensional is naturally equivalent to the category of modules for the group scheme \(G\) whose coordinate algebra \(k[G]\) is the dual Hopf algebra \(H^0\). We consider, in the light of our main result, the question of when the category of locally finite \(H\)-modules contains a non-zero projective in the special cases: \(H=U({\mathfrak g})\), the enveloping algebra of a Lie algebra \(\mathfrak g\); and \(H=k\Gamma\), the group algebra of a group \(\Gamma\). algebraic groups; category of rational modules; projective modules; affine group schemes; cocommutative Hopf algebras; category of modules; dual Hopf algebras; category of locally finite modules; enveloping algebras; group algebras Donkin, S, On projective modules for algebraic groups, J. Lond. Math. Soc. (2), 54, 75-88, (1996) Representation theory for linear algebraic groups, Free, projective, and flat modules and ideals in associative algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group schemes, Module categories in associative algebras On projective modules for algebraic groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main objects of the paper are \(\mathbb{Z}\)-models \(\mathbf G\) of connected simply connected simple algebraic \(\mathbb{Q}\)-groups \(G\), that is smooth affine group schemes of finite type over \(\mathbb{Z}\) with general fibre \(G\) such that all special fibres \({\mathbf G}\otimes\mathbb{Z}/p\mathbb{Z}\) are reductive. The author enumerates \(\mathbb{Q}\)-groups admitting \(\mathbb{Z}\)-models and proves an analog of a theorem by \textit{G. Harder} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 4, 409-455 (1971; Zbl 0232.20088)] on the Euler-Poincaré characteristic of \({\mathbf G}(\mathbb{Z})\). If \(G(\mathbb{R})\) is compact, this gives a mass formula allowing to compute the number of different \(\mathbb{Z}\)-models for all groups of rank at most 8 (except \(E_8\)); these models are exhibited explicitly as well as two \(\mathbb{Z}\)-models of \(E_8\) obtained via its adjoint representation. connected simply connected simple algebraic \(\mathbb{Q}\)-groups; smooth affine group schemes of finite type; special fibres; \(\mathbb{Q}\)-groups admitting \(\mathbb{Z}\)-models; Euler-Poincaré characteristic; mass formula; adjoint representations B.H. Gross, Groups over \(\(\mathbb {Z}\)\). Invent. Math. 124(1-3), 263-279 (1996) Linear algebraic groups over global fields and their integers, Group schemes Groups over \(\mathbb{Z}\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a compact Lie group. The cotangent bundle \(T^*K\) of \(K\) is a symplectic manifold and the adjoint action of \(K\) on it is Hamiltonian. Therefore we can consider the reduced space at zero momentum \((T^*K)_0:=\mu^{-1}(0)/K\). Here \(\mu:T^*K\to \mathfrak k^*\) is the momentum map. From the introduction: ``Despite the huge literature on reduction of the cotangent bundle of a Lie group relative to \textit{left translation} (or right translation) the \textit{conjugation action} has received little attention.''
The complexification \(K^{\mathbb C}\) of \(K\) is a complex reductive algebraic group and there is a \((K\times K)\)-equivariant isomorphism \(T^*K\cong K^{\mathbb C}\) of real analytic varieties. So \(T^*K\) has a complex analytic structure (in fact a complex algebraic structure) which combines with its structure of a symplectic manifold to a Kähler structure. The structure of a complex algebraic variety of \(T^*K\) gives \((T^*K)_0\) the structure of a complex algebraic variety and with this structure it is isomorphic to the well-known adjoint quotient \(K^{\mathbb C}//K^{\mathbb C}\cong T^{\mathbb C}/W\) which is just an affine space in case \(K\) is simply connnected [see \textit{R.~Steinberg}, Publ. Math., Inst. Hautes Étud. Sci. 25, 281--312 (1965; Zbl 0136.30002), \S6]. Here \(T\) is a maximal torus of \(K\) and \(W\) is the Weyl group.
It is explained in the introduction and in Section~1 that the situation with the structure of symplectic Poisson manifold is much more complicated. This structure descends to \((T^*K)_0\) as a stratified symplectic Poisson structure and combines with the structure of a complex algebraic variety to a complex algebraic stratified Kähler structure on \((T^*K)_0\). The real coordinate ring \(\mathbb R[(T^*K)_0]\) contains functions that are not smooth for the complex algebraic structure, see the remark on p. 721 of the paper.
In Section 2 a general description of the stratified symplectic Poisson structure on \((T^*K)_0\) is given. This is made explicit in Section~3 for the unitary and special unitary groups and in Section~4 for several other groups.
One of the main results is that for \(K=\text{U}(n), \text{SU}(n), \text{Sp}(n), \text{SO}(2n+1,\mathbb R), G_{2(-14)}\) the real coordinate ring \(\mathbb R[T^{\mathbb C}/W]\) is generated as a Poisson algebra by the real and imaginary parts of the characters of the fundamental irreducible representations of \(K^{\mathbb C}\).
In the final section the author points out that the quantum Hilbert space for the stratified Kähler structure on \(T^{\mathbb C}/W\) is freely spanned by the holomorphic functions on \(T^{\mathbb C}/W\) that correspond to the irreducible characters of \(K^{\mathbb C}\). adjoint quotient; stratified Kähler space; Poisson manifold; Poisson algebra; Poisson cohomology; holomorphic quantization; reduction and quantization; geometric quantization; quantization on a space with singularities; normal complex analytic space; locally semialgebraic space; constrained system; invariant theory; bisymmetric functions; multisymmetric functions; quantization in the presence of singularities; costratified Hilbert space Huebschmann, J.: Stratified Kähler structures on adjoint quotients Group actions on varieties or schemes (quotients), Poisson algebras Stratified Kähler structures on adjoint quotients | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers three-dimensional toric varieties given by a fan \(\Sigma\subseteq\mathbb{R}^3\). Under the assumption that its spherical section (i.e. \(\Sigma\cap S^2\)) is homeomorphic to a disk \(D^2\) he proves that the cohomology groups \(H^{kss\bullet}(X_\Sigma,\mathbb{Z})\) are free with the following dimensions: \(h^0=1\), \(h^2=\#\Sigma^{(1)}\), and \(h^4=\#(\int\Sigma)^{(1)}\).
The assumption is particularly fulfilled for (partial) resolutions of three-dimensional affine toric varieties. This leads, as an application, to the verification of the integral MacKay correspondence for three-dimensional abelian quotient singularities. toric varieties; freeness of cohomology groups; McKay correspondence; three-dimensional abelian quotient singularities Étale and other Grothendieck topologies and (co)homologies, Toric varieties, Newton polyhedra, Okounkov bodies, \(3\)-folds, Singularities in algebraic geometry Integral cohomology of some smooth complex toric 3-folds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper surveys the representation theory of rational Cherednik algebras. We also discuss the representations of the spherical subalgebras. We describe in particular the results on category \(\mathcal O\). For type \(A\), we explain relations with the Hilbert scheme of points on \(\mathbb C^2\). We insist on the analogy with the representation theory of complex semisimple Lie algebras. rational Cherednik algebras; category \(\mathcal O\); Hilbert schemes of points; complex semisimple Lie algebras Raphaël Rouquier, Representations of rational Cherednik algebras, Infinite-dimensional aspects of representation theory and applications, Contemp. Math., vol. 392, Amer. Math. Soc., Providence, RI, 2005, pp. 103 -- 131. Hecke algebras and their representations, Simple, semisimple, reductive (super)algebras, Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Noncommutative algebraic geometry Representations of rational Cherednik algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this thesis representations of reductive groups with a multiplicity-free supersymmetric algebra are classified.
See the review of the related journal publication [\textit{T. Pecher}, J. Lie Theory 23, No. 2, 459-481 (2013; Zbl 1303.20050)]. semisimple complex Lie groups; invariant theory; skew multiplicity-free actions; supersymmetric algebras; finite-dimensional representations; skew multiplicity-free representations; exterior algebras Representation theory for linear algebraic groups, Semisimple Lie groups and their representations, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Vector and tensor algebra, theory of invariants Multiplicity-free super vector spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a \(k\)-punctured Riemann surface \(M\) of genus \(g\), a generic character variety \({\mathcal M}_\mu\) is an algebraic variety whose points parametrize representations of the fundamental group of \(M\) into \(\mathrm{GL}_n({\mathbb K})\), with images in generic semisimple \(k\)-tuple of conjugacy classes \({\mathcal C}_1,\ldots, {\mathcal C}_k\) of type \({\mathbb \mu}=(\mu^1,\ldots,\mu^k)\), at the punctures (here \(\mu^i\) denotes a partition of \(n\) counting the multiplicity of each eigenvalue of \({\mathcal C}_i\)).
This paper is the continuation of a previous one by the same authors [Duke Math. J. 160, 323--400 (2011; Zbl 1246.14063)]. There it was shown that \({\mathcal M}_\mu\) is nonsingular of pure dimension \(d\mu\), provided it is not empty. Moreover, a certain rational function \({\mathbb H}_\mu(z,w)\in{\mathbb Q}(z,w)\) was defined in terms of Macdonald symmetric functions. It was conjectured that the compactly supported mixed Hodge numbers \(\{h_c^{i,j;k}({\mathcal M}_\mu)\}_{i,j,k}\) satisfy \(h_c^{i,j;k}({\mathcal M}_\mu)=0\), unless \(i=j\), and
\[
H_c({\mathcal M}_\mu;q,t)=(t\sqrt q)^{d_\mu}{\mathbb H}_\mu(-t\sqrt q,{1\over \sqrt q}),\eqno(1)
\]
where \(H_c({\mathcal M}_\mu;q,t)=:\sum_{i,j}h_c^{i,i;j}({\mathcal M}_\mu)q^it^j\) is the compactly supported mixed Hodge polynomial. A special case of the conjecture proved in [Duke Math. J. 160, 323--400 (2011; Zbl 1246.14063)] is used here to show that a non-empty character variety \(M_\mu\) is connected.
For \(g=k=1\), set \(X={\mathbb C}^\times\times {\mathbb C}^\times\) and denote by \(X^{[n]}\) the Hilbert scheme of \(n\) points in \(X\). It is known from \textit{L. Göttsche} and \textit{W. Soergel} [Math. Ann., 296, 235--245 (1993; Zbl 0789.14002)] that the mixed Hodge polynomial of \(X^{[n]}\) is given by \(H_c(X^{[n]};1,t)=(qt^2)^n {\mathbb H}^{[n]}(-t\sqrt q,{1\over \sqrt q})\), where \({\mathbb H}^{[n]}(z,w)\) is a certain rational function in \({\mathbb Q}(z,w)\), and it is conjectured that \({\mathbb H}^{[n]}(z,w)={\mathbb H}_{(n-1,1)}(z,w)\). This conjecture together with (1) implies that \(X^{[n]}\) and \({\mathcal M}_{(n-1,1)}\) have the same mixed Hodge polynomial.
Here some cases of the conjecture are proved, namely \({\mathbb H}^{[n]}(-0,w)={\mathbb H}_{(n-1,1)}(0,w)\) and \({\mathbb H}^{[n]}(w^{-1},w)={\mathbb H}_{(n-1,1)}(w^{-1},w)\). In addition, the following relation between character varieties and quasi-modular forms is deduced:
\[
1+\sum_{n\geq 1}{\mathbb H}_{(n-1,n)}(e^{u/2},e^{-u/2})T^n={1\over u}(e^{u/2}-e^{-u/2})\exp (2\sum_{k\geq 2}G_k(T){{u^k}\over {k!}},
\]
where \(G_k(T)\) is the classical Eisenstein series for \(\mathrm{SL}(2,{\mathbb R})\). In particular, the coefficient of any power of \(u\) in the left hand side is in the ring of quasi-modular forms, generated by the \(G_k\), for \(k\geq 2\), over \({\mathbb Q}\).
The paper is also concerned with further connections between the topology of generic character varieties \({\mathcal M}_\mu\), quiver representation and multiplicities in tensor products of irreducible characters of finite general linear groups. It is proved that
\[
A_\mu(q)={\mathbb H}_\mu(0,\sqrt q),
\]
where \(A_\mu(q)\) is the value of a certain monic polynomial \(A_\mu(T)\in{\mathbb Z}[T]\) of degree \(d_\mu/2\), value which coincides with the number of absolutely indecomposable representations over \({\mathbb F}_q\) (up to isomorphism) of a \(v_\mu\)-dimensional comet-shaped quiver \(\Gamma_\mu\). character varieties; quiver representations; Hilbert schemes; representations of finite general linear groups Hausel, Tamás and Letellier, Emmanuel and Rodriguez Villegas, Fernando, Arithmetic harmonic analysis on character and quiver varieties~{II}, Advances in Mathematics, 234, 85-128, (2013) Homogeneous spaces and generalizations, Representations of finite groups of Lie type Arithmetic harmonic analysis on character and quiver varieties. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Translated from the author's introduction: Binary polyhedral groups, \(\Gamma \subset \text{SL} (2, \mathbb{C})\), are associated with Kleinian singularities, \(\mathbb{C}^2/ \Gamma\), and McKay quivers. In this work we describe an invariant-theoretic interpretation of a differential-geometric construction of P. B. Kronheimer that explains the deformation of Kleinian singularities and their simultaneous resolution by means of families of representations of McKay quivers. In the construction of the resolution a new invariant-theoretic method, the linear modification of affine algebraic quotients, is developed. We discuss the basics of this method, give some applications to the representation of oriented CDW-graphs and finally connect it to the invariant theory of McKay quivers. Connections with the theory of abstract root systems and simple Lie algebras of corresponding CDW-type \(\Delta (\Gamma)\) also turn up in the representation theory of oriented CDW-graphs. Kleinian singularities; representations of McKay quivers; linear modification; invariant theory Homogeneous spaces and generalizations, Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Linear modification of algebraic quotients, representations of the McKay quiver and Kleinian singularities | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be a finite dimensional associative \(K\)-algebra with an identity over an algebraically closed field \(K\). It is known [see \textit{Ch. Riedtmann}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 19, 275-301 (1986; Zbl 0603.16025)] that the existence of the exact sequence \(0\to N\to M\oplus Z\to Z\to 0\) of finite dimensional \(A\)-modules implies that the module \(M\) degenerates to \(N\). The author proves that the corresponding relation \(M\leq_R N\) is a partial order on the set of isomorphism classes of finite dimensional \(A\)-modules, and this relation is in fact equivalent to the existence of a short exact sequence \(0\to Z\to Z\oplus M\to N\to 0\). finite dimensional algebras; representations; degenerations of modules; Auslander-Reiten quivers; Dynkin quivers; indecomposable modules; tilted algebras; exact sequences Zwara, G, A degeneration-like order for modules, Arch. Math., 71, 437-444, (1998) Representations of associative Artinian rings, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Finite rings and finite-dimensional associative algebras, Group actions on varieties or schemes (quotients) A degeneration-like order for modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Canonical algebras had been introduced by \textit{C. M. Ringel} [Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]. Let \(k\) be an algebraically closed field, \(t\geq 2\) a natural number, \(p=(p_1,\dots,p_t)\) a sequence of natural numbers with \(p_i\geq 2\) and \(\lambda=(\lambda_3,\dots,\lambda_t)\) a sequence of pairwise different nonzero elements in \(k\). Let \({\mathcal Q}\) be the quiver
\[
\begin{matrix} & &\overset{\vec x_1}\circ &\overset {x_1}\longrightarrow &\overset {2\vec x_1}\circ &\longrightarrow &\cdots &\longrightarrow &\overset{(p_1-1)\vec x_1}\circ \\ &\overset {x_1}\nearrow &&&&&&&& \overset {x_1}\searrow \\ \overset {0}\circ &\underset {x_2}\longrightarrow &\overset{\vec x_2}\circ & \underset {x_2}\longrightarrow &\overset{2\vec x_2}\circ &\longrightarrow &\cdots &\longrightarrow &\overset{(p_2-1)\vec x_2}\circ &\underset {x_2}\longrightarrow &\overset\vec {c}\circ \\ &\underset {x_t}\searrow &&\vdots &&\vdots &&\vdots &&\underset {x_t}\nearrow \\ & &\underset {\vec x_t}\circ &\underset {x_t}\longrightarrow &\underset{2\vec x_t}\circ &\longrightarrow &\cdots &\longrightarrow &\underset {(p_t-1)\vec x_t}\circ\end{matrix}
\]
The canonical algebra \(\Lambda=\Lambda(p,\lambda)\) is the path-algebra of the quiver \({\mathcal Q}\), bound by the relations \(x^{p_i}_i-x_2^{p_2}+\lambda_i x_1^{p_1}\). If the tree \({\mathcal T}\), obtained from \({\mathcal Q}\) by deleting the vertex 0 is a Dynkin diagram, then \(\Lambda\) is a well known tame concealed algebra. If \({\mathcal T}\) is an extended Dynkin diagram, then \(\Lambda\) is a tubular algebra. Ringel [loc. cit.] gave a description of the category \(\text{mod }\Lambda\) of finite dimensional \(\Lambda\)-modules in this case. The paper under review studies the case, when \({\mathcal T}\) is a wild quiver and \(\Lambda\) then is called wild canonical.
In this case, the module category \(\text{mod }\Lambda\) decomposes into three parts, \(\text{mod }\Lambda=\text{mod}_+\Lambda\vee\text{mod}_0\Lambda\vee\text{mod}_-\Lambda\). The middle part \(\text{mod}_0\Lambda\) has as Auslander-Reiten quiver a separating family of pairwise orthogonal standard tubes. The category \(\text{mod}_+\Lambda\) has exactly one preprojective component \({\mathcal P}\), containing \(n-1\) indecomposable projectives, where \(n\) is the number of vertices of \({\mathcal Q}\). The remaining Auslander-Reiten components in \(\text{mod}_+\Lambda\) are either of type \(\mathbb{Z} A_\infty\) or have stable part \(\mathbb{Z} A_\infty\). All modules in \(\text{mod}_\geq\Lambda=\text{mod}_+\Lambda\vee\text{mod}_0\Lambda\) have projective dimension at most 1. The category \(\text{mod}_-\Lambda\) is dual to \(\text{mod}_+\Lambda \).
It was shown by \textit{W. Geigle} and \textit{H. Lenzing} [in Singularities, representations of algebras, and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)] that canonical algebras are quasitilted algebras. More precise, the hereditary category \(\text{coh} X\) of coherent sheaves over a certain weighted projective line \(X=X(p,\lambda)\) contains a tilting vector bundle \(T\) with \(\text{End}(T)=\Lambda\). The tilting object \(T\) has a decomposition \(T={\mathcal O}\oplus T'\), where \({\mathcal O}\) is a line bundle and \(T'\) is the minimal projective generator in the right perpendicular category \({\mathcal O}^\perp\). The endomorphism ring \(\text{End}(T')\) is \(\Lambda_0\), the path-algebra of the quiver \({\mathcal T}\), and \({\mathcal O}^\perp\) will be identified with \(\text{mod }\Lambda_0\).
The authors show that the subcategory \(\text{vect} X\) of \(\text{coh} X\), defined by the vector bundles over \(X\), consists of Auslander-Reiten components of type \(\mathbb{Z} A_\infty\), and they identify \(\text{mod}_\geq\Lambda\) with the torsion class \(\text{coh}_\geq X=\{{\mathcal F}\mid\text{Ext}({\mathcal O},{\mathcal F})=0\}\) in \(\text{coh} X\). Under this identification \(\text{mod}_0\Lambda\) becomes \(\text{coh}_0X\), the category of finite length sheaves, which implies the structure of \(\text{mod}_0\Lambda\).
For the study of \(\text{mod}_+\Lambda\subset\text{vect} X\), they compare the Auslander-Reiten translations \(\tau_X\) in \(\text{vect} X\) and the relative Auslander-Reiten translations \(\tau_\Lambda\) and \(\tau_{\Lambda_0}\) in \(\text{vect} X\). As result they get for a \(\Lambda\)-module \(M\) in \(\text{mod}_+\Lambda\), not in the \(\tau_\Lambda\)-orbit of a projective, that \(\tau_\Lambda^{-m-1} M=\tau^-_{\Lambda_0}\tau_\Lambda^{-m} M\) for \(m\gg 0\) and \(\tau_\Lambda^{m+1} M=\tau_X\tau_\Lambda^mM\) for \(m\gg 0\). This result not only implies the Auslander-Reiten structure of \(\text{mod}_+\Lambda\). It also shows that each Auslander-Reiten component in \(\text{mod}_+\Lambda\), different from the preprojective component, contains a right cone, consisting completely of \(\Lambda_0\)-modules and a left cone, which is a left cone of some Auslander-Reiten component in \(\text{vect} X\) simultaneously, in complete analogy to the Auslander-Reiten structure of wild tilted algebras [\textit{O. Kerner}, J. Algebra 142, No. 1, 37-57 (1991; Zbl 0737.16007)]. Especially it implies bijections between the set of stable components of \(\text{mod}_+\Lambda\), the set of regular components of \(\text{mod}\Lambda_0\) and the set of Auslander-Reiten components in \(\text{vect} X\). But in contrast to wild tilted algebras, the dimension vectors \(\dim\tau_\Lambda^m M\) for \(M\in\text{mod}_+\Lambda\) grow linearly in \(m\). categories of finite dimensional modules; hereditary categories of coherent sheaves; canonical algebras; path-algebras; quivers; tame concealed algebras; extended Dynkin diagrams; tubular algebras; wild quivers; Auslander-Reiten quivers; separating families; orthogonal standard tubes; preprojective components; indecomposable projectives; Auslander-Reiten components; quasitilted algebras; weighted projective lines; tilting vector bundles; minimal projective generators; right perpendicular categories; endomorphism rings; categories of finite length sheaves; relative Auslander-Reiten translations; wild tilted algebras; dimension vectors Lenzing, H.; de la Peña, J. A., Wild canonical algebras, Math. Z., 224, 403-425, (1997) Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representations of quivers and partially ordered sets, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representation type (finite, tame, wild, etc.) of associative algebras, Module categories in associative algebras Wild canonical algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Denote by \(\text{rep}(Q,{\mathbf d})\) the space of representations with dimension vector \(\mathbf d\) of the finite quiver \(Q\) over an algebraically closed field. There is a linear action of \(\text{Gl}({\mathbf d})\) (a product of general linear groups) on \(\text{rep}(Q,{\mathbf d})\), such that two points are in the same orbit if and only if they correspond to equivalent representations.
Assume that \(\mathbf d\) is prehomogeneous; that is, there is a Zariski open \(\text{Gl}({\mathbf d})\)-orbit in \(\text{rep}(Q,{\mathbf d})\). Let \(Z_{\mathbf d}\) be the common zero locus in \(\text{rep}(Q,{\mathbf d})\) of the algebraically independent generators of the algebra of semi-invariant polynomial functions on \(\text{rep}(Q,{\mathbf d})\), described by \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 385--395 (1991; Zbl 0779.16005)]. In other words, \(Z_{\mathbf d}\) is the nullcone for the action of the commutator subgroup \(\text{Sl}({\mathbf d})\) of \(\text{Gl}({\mathbf d})\). Consider the multiples \(n\cdot{\mathbf d}\) of the dimension vector \(\mathbf d\).
The main result of the present paper is that \(Z_{n\cdot{\mathbf d}}\) is an irreducible complete intersection for all sufficiently large \(n\). semi-invariants of quivers; complete intersections; representations of quivers; prehomogeneous dimension vectors; algebras of semi-invariant functions Christine Riedtmann and Grzegorz Zwara, On the zero set of semi-invariants for quivers, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 6, 969 -- 976 (2004) (English, with English and French summaries). Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Group actions on varieties or schemes (quotients), Group actions on affine varieties On the zero set of semi-invariants for quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group of complex \(n\times n\) unitary matrices generated by reflections acting on \(\mathbb{C}^n\). Let \(R\) be the ring of invariant polynomials, and let \(\chi\) be a multiplicative character of \(G\). Let \(\Omega\chi\) be the \(R\)-module of \(\chi\)-invariant differential forms. We define a multiplication in \(\Omega\chi\) and show that under this multiplication \(\Omega\chi\) has an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between \(\chi\)-invariant forms and logarithmic forms. finite groups generated by reflections; invariant differential forms; rings of invariant polynomials; exterior algebras A. Shepler, ''Semi-invariants of finite reflection groups,'' J. Algebra 220 (1999), 314--326. Reflection and Coxeter groups (group-theoretic aspects), Geometric invariant theory, Reflection groups, reflection geometries, Ordinary representations and characters Semi-invariants of finite reflection groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book under review is a very detailed outline of a number of ideas developed by the author with collaborators over the past two decades (see [\textit{O. A. Laudal}, Contemp. Math. 391, 249--280 (2005; Zbl 1103.14300); Acta Appl. Math. 101, No. 1--3, 191--204 (2008; Zbl 1148.14003); Geometry of time-spaces. Non-commutative algebraic geometry, applied to quantum theory. Hackensack, NJ: World Scientific (2011; Zbl 1230.14001); J. Gen. Lie Theory Appl. 5, Article ID G110104, 18 p. (2011; Zbl 1273.14009); Springer Proc. Math. Stat. 98, 93--123 (2015; Zbl 1329.83221); \textit{E. Eriksen} et al., Noncommutative deformation theory. Boca Raton, FL: CRC Press (2017; Zbl 1369.14006)]).
The book consists of 14 chapters, in addition to the bibliography, which includes 54 titles (including 14 original works by the author and co-authors), and an index. The first chapter - Introduction; it contains a brief description of philosophy, mathematical models, model space geometry, cosmology, and a reader's guide. In fact, the author discusses a hypothetical closed connection between noncommutative deformations of a thick point in an affine three-dimensional space and the basic concepts and notions of modern theoretical mathematical physics, such as time and space-time, general relativity and quantum field theory, the existence of their common ground, and many others. For example, one of his main ideas is that the Big Bang hypothesis requires the existence of a singularity at the starting point or origin of the universe. Therefore, singularity deformation theory is a very suitable tool to study this and related phenomena. Among other things, he describes a toy model for quantum theory in which the versal base space of the noncommutative deformation functor of the simplest thick point, defined by the local Artinian algebra \(k[x_1,x_2,x_3]/(x_1,x_2,x_3 )^2\), plays the key role.
(Reviewer's remark: It is curious to note that the main numerical invariant of this algebra, its three-dimensional socle, has no interpretation in the presented model.)
Then he introduces the phase space functor and describes its basic properties, studies the deformation functor of representations, blow-ups and desingularizations, Chern classes, the Dirac derivation, and the generalized de Rham complex towards the Jacobian conjecture.
The third chapter is an introduction to noncommutative algebraic geometry and to the theory of noncommutative deformations of swarms.
The next chapter is devoted to an original interpretation of the Dirac derivation and dynamical structures, including interconnection of local and global gauge groups and invariant theory. Then the author shortly discusses in the commutative case the notions of metrics, gravitation and energy, relations to Clifford algebras, the Chern-Simons classes, a generalized Yang-Mills theory, etc.
In the fifth chapter concepts of time-space and space-times are presented with the aid of the cylindrical coordinates, Kepler laws, the heat and Navier-Stokes equations.
The sixth and seventh chapters are focused on concepts of entropy, cosmology and cosmological time, including a description of the model for the Big Bang.
In the next two chapters the Universe is described as a versal base space with an explanation of such notions as density of mass, inflation, cyclical cosmology, a conformally trivial cosmological model, the role of observers and their location in this Universe, the notions of speed of photons, red-shift, etc.
The tenth and eleventh chapters are devoted to summarizing the construction of the Model, where the author offers a mathematical synthesis of general relativity, quantum field theory and Yang-Mills, he also comments the concepts of black energy and mass. The two next chapter the author studies interactions from the point of view of noncommutative deformation theory, the processes of creating new particles from old ones, the concept of entanglement, consciousness and many others, compares the Toy model and the standard one with the list of well-known structural problems of the latter discussed in this book.
The final chapter contains a number of relations to noncommutative geometry, models for quantum gravitation, the general dynamical model and ``some non-scientific thoughts about the present status of quantum theory'', a list of unsolved problems in physics, relations to classical cosmologies and the section ``So What?'' containing a number of nonformal remarks and further ideas.
To fully characterize the great significance of this book for those who are interested in modern physics and mathematics, it is perhaps very appropriate to quote the following thought: ``The mathematician who wants to be a naturalist must now assimilate a new set of physical concepts; the need for mathematical expertise is greater now than ever before'' (see [\textit{S. G. Brush}, ``Poincaré and cosmic evolution'', Phys. Today 33, No. 3, 42--49 (1980; \url{doi:10.1063/1.2913996})]). artinian algebras; thick points; noncommutative deformations; deformation functor; versal deformation; moduli suite; extensions; simple modules; phase space functor; Hochschild cohomology; Massey products; representations of associative algebras; Toy model; blow-ups; desingularizations, Hilbert schemes; Chern classes, Dirac derivation, de Rham complex; Jacobian conjecture; dynamical structure; swarms; metrics, gravitation; quantum gravitation; energy; Clifford algebras; Chern-Simons classes; Yang-Mills theory; heat equation; thermodynamics; Kepler laws; heat equation; Navier-Stokes equation; Schrödinger equation; Einstein field equation; entropy; cosmology; cosmological time; density of mass; inflation; cyclical cosmology; conformally trivial cosmological model; universe, observers; photons; red-shift; entanglement; consciousness; super symmetry bosonic fields; fermionic fields; gluons; quarks; charge; black energy; black mass Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Noncommutative algebraic geometry, Local deformation theory, Artin approximation, etc., Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Plane and space curves, Simple and semisimple modules, primitive rings and ideals in associative algebras, Representations of orders, lattices, algebras over commutative rings, Representation theory of associative rings and algebras, Rings arising from noncommutative algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory, Noncommutative geometry in quantum theory, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Quantum field theory; related classical field theories, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to relativity and gravitational theory, Relativistic cosmology Mathematical models in science | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a semisimple simply connected algebraic group over \(\mathbb{C}\), \(\mathfrak g\) its Lie algebra, \(F=\mathbb{C}((\varepsilon))\) the field of formal Laurent series, \(A=\mathbb{C}[[\varepsilon]]\) the ring of integers in \(F\). Set \(\widehat{\mathfrak g}={\mathfrak g}\otimes F\), \(\widehat{\mathfrak g}_A={\mathfrak g}\otimes A\) and \(\widehat G=G(F)\). For \(N\in\widehat{\mathfrak g}\) let \({\mathcal B}_N\subset{\mathcal B}=\widehat G/I\) (\(I\) the Iwahori subgroup) and \(X_N\subset X=\widehat G/G(A)\), resp., the set of all Iwahori subalgebras and the set of subalgebras conjugate to \(\widehat{\mathfrak g}_A\), resp. containing \(N\). Then \({\mathcal B}_N\) and \(X_N\) are locally finite unions of finite dimensional projective varieties and the dimensions of the components of \({\mathcal B}_N\) are all equal to the dimension of \(X_N\). An explicit formula for this dimension, conjectured by Kazhdan and Lusztig, is proved. semisimple simply connected algebraic groups; Lie algebras; fields of formal Laurent series; Iwahori subgroups; Iwahori subalgebras; finite dimensional projective varieties; dimensions; explicit formula R. Bezrukavnikov, ''The dimension of the fixed point set on affine flag manifolds,'' Math. Res. Lett., vol. 3, iss. 2, pp. 185-189, 1996. Linear algebraic groups over local fields and their integers, Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups The dimension of the fixed point set on affine flag manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities ``Algebraic families'' of modules and algebras play an important role in several questions of representation theory. It is often especially useful to know that some ``discrete invariants'' are constant or, at least, are semi-continuous in such families, that is they can change only in ``exceptional points'' which form a family of smaller dimension. Perhaps the best known results in this direction are those of Gabriel and Knörrer. Gabriel proved that finite representation type is an open condition for finite dimensional algebras (``fat points''), while Knörrer showed that the number of parameters for modules of prescribed rank is semicontinuous in families of commutative Cohen-Macaulay rings of Krull dimension 1 (``curve singularities''). Knörrer's theorem was used to show that the unimodal singularities of type \(T_{pq}\) are of tame Cohen-Macaulay type.
Unfortunately, the arguments of Knörrer do not work in the non-commutative case. The aim of this paper is to refine them in such a way that they could be applied to non-commutative Cohen-Macaulay algebras, too. For this purpose we introduce the notion of ``dense subrings'' which seems rather technical but, nevertheless, useful. It enables the construction of ``almost versal'' families of modules for a given algebra (cf. Theorem 3.5) and the definition of the ``number of parameters''. Just as in the commutative case, it is important that the bases of these ``almost versal'' families are projective varieties. Once having this, we are able to prove an analogue of Knörrer's theorem (cf. Theorem 4.9) and a certain variant (cf. Theorem 4.11) which turns out to be useful, for instance, to extend the tameness criterion for commutative algebras to the case of characteristic 2. The semicontinuity implies, in particular, that the set of so-called ``wild algebras'' in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, Theorem 4.9, together with the results of the authors [Compos. Math. 89, No. 3, 315-338 (1993; Zbl 0794.14010)] imply that tame is indeed an open property for curve singularities (commutative one-dimensional Cohen-Macaulay rings).
An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or elements of finite dimensional bimodules. almost versal families of modules; finite representation type; finite dimensional algebras; Cohen-Macaulay rings of Krull dimension 1; non-commutative Cohen-Macaulay algebras; projective varieties; tame algebras; curve singularities Drozd, Yu. and Greuel, G.-M.: Semi-continuity for Cohen--Macaulay modules, Math. Ann. 306 (1996), 371--389. Cohen-Macaulay modules in associative algebras, Representation type (finite, tame, wild, etc.) of associative algebras, Cohen-Macaulay modules, Singularities of curves, local rings, Formal methods and deformations in algebraic geometry Semicontinuity for representations of one-dimensional Cohen-Macaulay rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The varieties considered in this important paper are projective varieties over an arbitrary field \(F\) on which an adjoint semisimple algebraic group \(G\) acts, the action being transitive after scalar extension to a separable closure \(F_s\) of \(F\). The authors give a precise formula for the change in Schur index of arbitrary central simple \(F\)-algebras, when scalars are extended from \(F\) to the function field of such a variety. Only the case where \(G\) is of inner type is considered in detail in this paper, but the same methods also apply to groups of outer type, which the authors intend to treat in the second part.
If \(X\) is a twisted flag variety as above, then there is a parabolic subgroup \(P\) of \(G_s=G\times F_s\), uniquely determined by \(X\) up to conjugacy, such that \(X\) becomes isomorphic to \(G_s/P\) over \(F_s\). If \(G\) is of inner type, the general index reduction formula for an arbitrary central simple \(F\)-algebra \(D\) then takes the form \(\text{ind}(D\otimes F(X))=\gcd(n_\psi\text{ind}(D\otimes A(\psi)))\), where \(\psi\) runs over the characters of the center of the simply connected cover of \(G_s\), and the integers \(n_\psi\) and the central simple \(F\)-algebras \(A(\psi)\) depend only on \(\psi\). In the final sections of the paper, this formula is made explicit by a thorough description of the integers \(n_\psi\) and the algebras \(A(\psi)\) for the various types of groups \(G\) and parabolic subgroups \(P\). The authors thus recover various special cases previously obtained. The first of these special cases is due to \textit{A. Schofield} and \textit{M. Van den Bergh} [Trans. Am. Math. Soc. 333, No. 2, 729-739 (1992; Zbl 0778.12004)], who used Quillen's computation of the group \(K_0\) of Brauer-Severi varieties to obtain an index reduction formula for scalar extension to the function field of a Brauer-Severi variety. The present paper uses the same technique, relying on Panin's computation of the \(K\)-theory of twisted flag varieties. (Also submitted to MR). central simple algebras; Brauer groups; adjoint semisimple linear algebraic groups; Borel varieties; twisted flag varieties; projective varieties; change of Schur index; function fields; groups of inner type; parabolic subgroups; index reduction formula; Brauer-Severi varieties A. S. Merkurjev, I. A. Panin, A. R. Wadsworth, \textit{Index reduction formulas for twisted flag varieties}. I, \(K\)-Theory \textbf{10} (1996), no. 6, 517-596. Finite-dimensional division rings, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, \(K\)-theory of schemes Index reduction formulas for twisted flag varieties. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We prove that the Deligne-Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of \textit{X. He} [J. Algebra 320, No. 3, 1207-1219 (2008; Zbl 1195.20050)] and \textit{S. Orlik, M. Rapoport} [ibid. 320, No. 3, 1220-1234 (2008; Zbl 1222.14102)] and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures.
Let \(p\) be a prime number, let \(\mathbb{F}\) denote an algebraic closure of the finite field with \(p\) elements and let \(\mathbf G\) be a connected reductive algebraic group over \(\mathbb{F}\). We assume that \(\mathbf G\) is endowed with an isogeny \(F\colon\mathbf G\to\mathbf G\) such that \(F^\delta\) is a Frobenius endomorphism with respect to some \(\mathbb{F}_q\)-structure on \(\mathbf G\) (here, \(\delta\) is a non-zero natural number, \(q\) is a power of \(p\) and \(\mathbb{F}_q\) denotes the subfield of \(\mathbb{F}\) with \(q\) elements).
We denote by \(\mathcal B\) the variety of Borel subgroups of \(\mathbf G\) and by \(\mathcal B\times\mathcal B=\coprod_{w\in W}\mathcal O(w)\) the decomposition into orbits for the diagonal action of \(\mathbf G\). Here, \(W\) is the Weyl group of \(\mathbf G\), with set of simple reflections \(S\) corresponding to the orbits of dimension \(1+\dim\mathcal B\), and the first and last projections define an isomorphism \(\mathcal O(w)\times_{\mathcal B}\mathcal O(w')@>\sim>>\mathcal O(ww')\) when \(l(ww')=l(w) +l(w')\), where \(l\colon W\to\mathbb{Z}_{\geq 0}\) is the length function on \(W\) associated to \(S\).
Given \(w\in W\), we define the Deligne-Lusztig variety [\textit{P. Deligne} and \textit{G. Lusztig}, Ann. Math. (2) 103, 103-161 (1976; Zbl 0336.20029), Definition 1.4] associated to \(w\) by
\[
\mathbf X(w)=\mathbf X_{\mathbf G}(w)=\{\mathbf B\in\mathcal B\mid(\mathbf B,F(\mathbf B))\in\mathcal O(w)\}.
\]
By studying a class of ample sheaves on \(\mathbf X(w)\), Deligne and Lusztig proved that these varieties are affine when \(q^{1/\delta}\) is larger than the Coxeter number of \(\mathbf G\) [loc. cit., Theorem 9.7].
They proved more generally that the existence of coweights satisfying certain inequalities ensures that \(\mathbf X(w)\) is affine. Recently, Orlik-Rapoport and He studied this question. Recall that \(x,y\in W\) are \(F\)-conjugate if there exists \(a\in W\) such that \(y=a^{-1}xF(a)\). By a case-by-case analysis based on Deligne-Lusztig's criterion, they obtained the following result:
Theorem A (Orlik-Rapoport, He, cited above). If \(w\in W\) is an element of minimal length in its \(F\)-conjugacy class then \(\mathbf X(w)\) is affine.
When \(w\) is a Coxeter element, the result is due to \textit{G. Lusztig} [Invent. Math. 38, 101-159 (1976; Zbl 0366.20031), Corollary 2.8].
In this note we prove a more general affineness result and we deduce Theorem A by applying a combinatorial result on elements of minimal length in their \(F\)-conjugacy class. Before stating our results, we need some further notation. We denote by \(B^+\) the braid monoid associated to \((W,S)\). It is the monoid with presentation
\[
B^+=\langle(\underline x)_{x\in W}\mid\forall x,x'\in W,\;l(xx')=l(x)+l(x')\Rightarrow\underline{xx'}=\underline x\,\underline x'\rangle.
\]
The automorphism \(F\) of \(W\) extends to an automorphism of \(B^+\) still denoted by \(F\).
Given \(I\subset S\), let \(W_I\) denote the subgroup of \(W\) generated by \(I\) and let \(w_I\) be the longest element of \(W_I\) (the element \(w_S\) will be denoted by \(w_0\)).
The main result of this note is the following: Theorem B. Let \(I\) be an \(F\)-stable subset of \(S\) and let \(w\in W_I\) be such that there exists a positive integer \(d\) and \(a\in B^+\) with \(\underline wF(\underline w)\cdots F^{d-1}(\underline w)=\underline w_Ia\). Then \(\mathbf X(w)\) is affine.
The proof of Theorem B is by a general argument, while our deduction of Theorem A relies on combinatorial results on finite Coxeter groups which are proved by a case-by-case analysis. Deligne-Lusztig varieties; finite groups of Lie type; Weyl groups; conjugacy classes Bonnafé, C.; Rouquier, R., Affineness of Deligne-Lusztig varieties for minimal length elements, J. Algebra, 320, 3, 1200-1206, (2008) Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Linear algebraic groups over finite fields, Reflection and Coxeter groups (group-theoretic aspects) Affineness of Deligne-Lusztig varieties for minimal length elements. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We introduce and study a notion of analytic loop group with a Riemann-Hilbert factorization relevant for the representation theory of quantum affine algebras at roots of unity \(\mathcal U_{\varepsilon}(\hat{\mathfrak g})\) with non-trivial central charge. We introduce a Poisson structure and study properties of its Poisson dual group. We prove that the Hopf-Poisson structure is isomorphic to the semi-classical limit of the center of \(\mathcal U_\varepsilon(\hat{\mathfrak g})\) (it is a geometric realization of the center). Then the symplectic leaves, and corresponding equivalence classes of central characters of \(\mathcal U_\varepsilon(\hat{\mathfrak g})\), are parameterized by certain \(G\)-bundles on an elliptic curve. loop groups; representations of quantum affine algebras; quantum groups at roots of 1; Riemann-Hilbert factorization; Poisson-Lie groups; \(G\)-bundles on an elliptic curve Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Poisson algebras, Loop groups and related constructions, group-theoretic treatment Geometry of the analytic loop group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(V\) be a representation of a complex reductive group \(G\), and let \(f_1,f_2,\dots,f_n\) be homogeneous invariant functions. So \(f_i\in\mathcal O(V)\), the coordinate ring of \(V\). The addition map \(V^{\oplus k}\to V\) induces a map \(\varphi\colon\mathcal O(V)\to\mathcal O(V^{\oplus k})\). The target is multigraded and the multihomogeneous components of \(\varphi(f)\) are called polarizations of \(f\).
The authors explain why the polarizations of \(f_1,f_2,\dots,f_n\) define the nullcone of \(k\leq m\) copies of \(V\) if and only if every linear subspace \(L\) of the nullcone of \(V\) of dimension \(\leq m\) is `annihilated' by a one-parameter subgroup \(\lambda\). Here annihilation means that \(\lim_{t\to 0}\lambda(t)x=0\) for all \(x\in L\). This is then applied to many examples. There are variations with \(V^{\oplus k}\to V\) replaced with \(V^{\oplus km}\to V^{\oplus k}\) for some \(m\).
A surprising result is about the group \(\text{SL}_2\) where almost all representations \(V\) have the property that all linear subspaces of the nullcone are annihilated. This has interesting applications to the invariants on several copies. Another result concerns the \(n\)-qubits which appear in quantum computing. This is the representation of a product of \(n\) copies of \(\text{SL}_2\) on the \(n\)-fold tensor product \(\mathbb C^2\otimes\mathbb C^2\otimes\cdots\otimes\mathbb C^2\). Here they show just the opposite, namely that the polarizations never define the nullcone of several copies if \(n\geq 3\). For \(n=3\) the situation is investigated in more detail. polarizations; nullcones; qubits; Hilbert-Mumford criterion; finite-dimensional modules; complex reductive groups; homogeneous invariant functions; coordinate rings Hanspeter Kraft and Nolan R. Wallach, Polarizations and nullcone of representations of reductive groups, Symmetry and spaces, Progr. Math., vol. 278, Birkhäuser Boston, Inc., Boston, MA, 2010, pp. 153 -- 167. Representation theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Geometric invariant theory, Group actions on varieties or schemes (quotients) Polarizations and nullcone of representations of reductive groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be a smooth complex algebraic surface. The Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is a smooth variety of dimension \(2n\). \textit{E. Carlsson} studied the generating series for the intersection pairings between the total Chern class of the tangent bundle and the Chern classes of tautological bundles on \((\mathbb C^2)^{[n]}\), proving that the reduced series \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) is a quasi-modular form [Adv. Math. 229, 2888--2907 (2012; Zbl 1255.14005)]. \textit{A. Okounkov} conjectured that these reduced series are multiple \(q\)-zeta values [Funct. Anal. Appl. 48, 138--144 (2014; Zbl 1327.14026)]. \textit{Z. Qin} and \textit{F. Yu} [Int. Math. Res. Not. 2018, 321--361 (2018; Zbl 1435.14007)] proved the conjecture modulo lower weight terms via the reduced series
\[
\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q) = (q;q)_\infty^{\chi (X)} \cdot \sum_n q^n \int_{X^{[n]}} (\Pi_{i=1}^N G_{k_i} (\alpha_i,n)) c(T_{X^{[n]}})
\]
where \(0 \leq k_i \in \mathbb Z\), \(\alpha_i \in H^* (X), (q;q)_\infty = \Pi_{n=1}^\infty (1-q^n)\) and \(G_{k_i}(\alpha_i, n) \in H^* (X^{[n]})\) are classes which play a role in the study of the geometry of \(X^{[n]}\) (see work of \textit{Z. Qin} [Hilbert schemes of points and infinite dimensional Lie algebras. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1403.14003)]).
In the paper under review, the authors further study the series \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\). Defining functions \(\Theta_k^\alpha (q)\) depending on \(\alpha \in H^* (X)\) and \(k \geq 0\), they fix \(0 \leq k_1, \dots, k_N \in \mathbb Z\) and \(\alpha_1, \dots, \alpha_N \in H^* (X, \mathbb Q)\) and prove the following:
(1) If \(\langle K_X^2,\alpha_i \rangle =0\) and \(2|k_i\) for each \(i\), then the leading term \(\Pi_{i=1}^N \Theta_k^\alpha (q)\) of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is either \(0\) or a quasi-modular form of weight \(\sum (k_i+2)\).
(2) Suppose \(|\alpha_i|=4\) for each \(i\). If \(2|k_i\) for each \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is a quasi-modular form of weight \(\sum (k_i+2)\). if \(2 \not |k_i\) for some \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)=0\).
These results are proved by relating the leading term of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) for \(X\) to the leading term of \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) for \(\mathbb C^2\) studied by Carlsson [loc. cit.]. Hilbert schemes of points on a surface; quasi-modular forms; multiple zeta value; generalized partition Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Vertex operators; vertex operator algebras and related structures Hilbert schemes of points and quasi-modularity | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This book is an introduction to the theory of prehomogeneous vector spaces from the view point of Lie algebra theory. The general theory is developed and functional equations satisfied by local and global zeta functions are proved. Then the author discusses prehomogeneous vector spaces of parabolic type that were introduced by the author himself. These spaces are naturally associated to parabolic subalgebras of semisimple Lie algebras. Finally, a detailed study of prehomogeneous vector spaces of ``commutative'' parabolic type is made. prehomogeneous vector spaces of commutative parabolic type; prehomogeneous vector spaces of parabolic type; functional equations; zeta functions; parabolic subalgebras; semisimple Lie algebras Rubenthaler H., Algèbres de Lie et Espaces Préhomogènes (1992) Simple, semisimple, reductive (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras Lie algebras and prehomogeneous vector spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The representation space of a quiver \(Q\) with vector spaces \(E_i\) assigned to its vertices is the affine space \(V=\bigoplus_{a\in Q_1} {\mathrm{Hom}}(E_{t(a)},E_{h(a)})\). The group \(\mathbb{G}=\prod_{i\in Q_0} {\mathrm{GL}}(E_i)\) naturally acts on \(V\). A quiver cycle \(\Omega\) is a \(\mathbb{G}\)-stable closed irreducible subvariety in \(V\). If \(Q\) is an equioriented quiver of type \(A\), then an equivariant Grothendieck class \([\mathcal{O}_{\Omega}]\in K_{\mathbb{G}}(V)\) is known to be given as a \(\mathbb{Z}\)-linear combination of products of the stable Grothendieck polynomials applied to the standard representations of \(\mathbb{G}\). The coefficients of this linear combination are called quiver coefficients.
In the paper under review a more general notion of quiver coefficients is introduced. These coefficients can be defined for an arbitrary quiver \(Q\) without oriented loops. If \(Q\) is of Dynkin type and \(\Omega\) has rational singularities, then a formula for the quiver coefficients is given. Several conjectures are formulated. representations of quivers; quiver coefficients; Grothendieck classes; Young diagram; Dynkin type Buch, A.S., Quiver coefficients of Dynkin type, Mich. Math. J., 57, 93-120, (2008) Determinantal varieties, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], \(K\)-theory of schemes, Representations of quivers and partially ordered sets Quiver coefficients of Dynkin type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper complements the author's earlier work [Trans. Am. Math. Soc. 303, 805-827 (1987; Zbl 0628.13019)].
Let \(G=GL(1,{\mathbb{C}})^ r\) act faithfully on \(V={\mathbb{C}}^ n\) as a group of diagonal matrices. Write \({\mathcal O}(V)^ G\) (resp. \({\mathcal D}(V)^ G)\) for the G-invariant regular functions on V (resp. differential operators on V). Write \(V=\oplus^{n}_{j=1}{\mathbb{C}}x_ j\), \({\mathcal D}(X)={\mathbb{C}}[x_ 1,...,x_ n,\partial_ 1,...,\partial_ n]\) where \(\partial_ j=\partial /\partial x_ j\), and set \(L_ 0={\mathbb{C}}\), \(L_ 1=\sum_{j}({\mathbb{C}}x_ j\oplus {\mathbb{C}}\partial_ j)\), \(L_ 2=\sum_{i,j}{\mathbb{C}}(x_ i\partial_ j+x_ j\partial_ i)+\sum_{i,j}({\mathbb{C}}x_ ix_ j+{\mathbb{C}}\partial_ i\partial_ j)\). Then \(L=L_ 0\oplus L_ 1\oplus L_ 2\) is a G-stable Lie subalgebra of \({\mathcal D}(V).\)
Theorem A gives precise (and easy to check) conditions under which \({\mathcal D}(V)^ G\) is generated by \(L^ G\); in that case \({\mathcal D}(V)^ G\) becomes a homomorphic image of the enveloping algebra \(U(L^ G).\)
Theorem B. Suppose that \(L^ G_ 1=0\) (equivalently, \(L^ G\) is reductive), and that \({\mathcal D}(V)^ G\) is generated by \(L^ G\). Then \({\mathcal O}(V)^ G\) as an \(L^ G\)-module is a simple quotient of a generalized Verma module. Furthermore, if the natural map \({\mathcal D}(V)^ G\to {\mathcal D}({\mathcal O}(V)^ G)\) is surjective, then \({\mathcal D}({\mathcal O}(V)^ G)\) becomes a primitive factor ring of \(U([L^ G,L^ G]).\)
The author's earlier paper gives conditions for the surjectivity of \({\mathcal D}(V)^ G\to {\mathcal D}({\mathcal O}(V)^ G)\). With this he applies Theorems A and B to give interesting examples showing that certain primitive factors of U(sl(n,\({\mathbb{C}}))\) and U(sp(2n,\({\mathbb{C}}))\) are rings of differential operators. Similar examples, with different techniques occur in the paper [\textit{T. Levasseur}, \textit{S. P. Smith}, \textit{J. T. Stafford}, The minimal nilpotent orbit, the Joseph ideal and differential operators, J. Algebra (to appear)] and [\textit{T. Levasseur}, \textit{J. T. Stafford}, Differential operators on classical rings of invariants (preprint)]. group of diagonal matrices; G-invariant regular functions; differential operators; G-stable Lie subalgebra; enveloping algebra; simple quotient; generalized Verma module; primitive factor ring; rings of differential operators Musson, I. M.: Actions of tori on Weyl algebras. Comm. alg. 16, 139-148 (1988) Valuations, completions, formal power series and related constructions (associative rings and algebras), Universal enveloping (super)algebras, Vector and tensor algebra, theory of invariants, Simple and semisimple modules, primitive rings and ideals in associative algebras, Automorphisms and endomorphisms, Geometric invariant theory Actions of tori on Weyl algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A particular case in the superstring theory where a finite group \(G\) acts upon the target Calabi-Yau manifold \(M\) in the theory seems to attract both physicists' and mathematician's attention. Define the ``orbifold Euler characteristic'': \(\chi (M,G)= {1\over |G |} \sum_{gh=hg} \chi (M^{\langle g, h\rangle})\), where the summation runs over all the pairs \(g,h\) of commuting elements of \(G\), and \(M^{\langle g,h \rangle}\) denotes the subset of \(M\) of all the points fixed by both of \(g\) and \(h\).
Vafa's formula-conjecture. If a complex manifold \(M\) has trivial canonical bundle and if \(M/G\) has a (nonsingular) resolution of singularities \(\widetilde {M/G}\) with trivial canonical bundle, then we have \(\chi (\widetilde {M/G} ) = \chi (M,G)\).
In the special case where \(M= \mathbb{A}^n\) an \(n\)-dimensional affine space, \(\chi (M,G)\) turns out to be the number of conjugacy classes, or equivalently the number of equivalence classes of irreducible \(G\)-modules. If \(n=2\), then the formula is therefore a corollary to the classical McKay correspondence.
Let \(G\) be a finite subgroup of \(SL(2, \mathbb{C})\) and \(\text{Irr} (G)\) the set of all equivalence classes of nontrivial irreducible \(G\)-modules. Let \(X=X_G: =\text{Hilb}^G (\mathbb{A}^2)\), \(S=S_G: =\mathbb{A}^2/G\), \({\mathfrak m}\) (resp. \({\mathfrak m}_S)\) the maximal ideal of \(X\) (resp. \(S)\) at the origin and \({\mathfrak n}: ={\mathfrak m}_S {\mathcal O}_{\mathbb{A}^2}\). Let \(\pi: X\to S\) be the natural morphism and \(E\) the exceptional set of \(\pi\). Let \(\text{Irr} (E)\) be the set of irreducible components of \(E\). Any \(I\in X\) contained in \(E\) is a \(G\)-invariant ideal of \({\mathcal O}_{\mathbb{A}^2}\) which contains \({\mathfrak n}\).
Definition: \(V(I): =I/({\mathfrak m} I+{\mathfrak n})\).
For any \(\rho\), \(\rho'\), and \(\rho''\in \text{Irr} (G)\) define \(E(\rho): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho)\}\)
\(P(\rho, \rho'): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho')\}\)
\(Q(\rho, \rho', \rho''): =\{I\in \text{Hilb}^G (\mathbb{A}^2)\); \(V(I)\) contains a \(G\)-module \(V(\rho) \oplus V(\rho') \oplus V(\rho'')\}\).
Main theorem: (1) The map \(\rho \mapsto E(\rho)\) is a bijective correspondence between \(\text{Irr} (G)\) and \(\text{Irr} (E)\).
(2) \(E(\rho)\) is a smooth rational curve for any \(\rho\in \text{Irr} (G)\).
(3) \(P(\rho, \rho)= Q(\rho, \rho',\rho'') = \emptyset\) for any \(\rho,\rho', \rho''\in \text{Irr} (G)\). Hilbert schemes; orbifold Euler characteristics; irreducible components of exceptional set; superstring theory; McKay correspondence Ito, Y., Nakamura, I.: McKay correspondence and Hilbert schemes. Proc. Japan Acad. Ser. A Math. Sci., 72, 135--138 (1996) Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Global theory and resolution of singularities (algebro-geometric aspects) McKay correspondence and Hilbert schemes | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The authors give a survey about the development and the results concerning Grothendieck's conjectures relating simple singularities of surfaces and the geometry of finite dimensional complex Lie algebras. Let \(\mathfrak{g}\) be a simple complex Lie algebra of type \(A_n, D_n, E_6, E_7, E_8\) and \(G\) the corresponding simply connected simple Lie group. Let \(\mathfrak{h}\) be a Cartan subalgebra of \(\mathfrak{g}\) and \(\mathfrak{B}\) a Borel subalgebra of \(\mathfrak{g}\). Let \(B\) be the corresponding Borel subgroup of \(G\). There is a canonical map \(\gamma: \mathfrak{g}\to \mathfrak{h}/W\), \(W\) the Weyl group, defined by \(\gamma(x)=(\gamma_1(x), \ldots, \gamma_r(x))\) where \(\gamma_1, \ldots,\gamma_r\) are the homogeneous \(G\)--invariant polynomials generating \(\mathbb{C}[\mathfrak{g}]^G\). Grothendiek conjectured that \(\gamma\) has a simultaneous resolution. He also conjectured that for a subregular nilpotent element \(y\) and a transversal slice \(X\) at \(y\) to the orbit \(Gy\) the germ of the surface \(X\cap \gamma^{-1}(\gamma(0))\) at \(y\) is a simple surface singularity of the same type as the Lie algebra \(\mathfrak{g}\). simple singularities; resolution of singularities; Lie algebras; subregular nilpotent elements Lê, D. T.; Tosun, M., Simple singularities and simple Lie algebras, \textit{TWMS J. Pure Appl. Math.}, 2, 1, 97-111, (2011) Singularities in algebraic geometry, Complex surface and hypersurface singularities, Coadjoint orbits; nilpotent varieties, Deformation of singularities Simple singularities and simple Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(Q\) be an algebraic group over an algebraically closed field of characteristic zero. Let \(\mathcal{\mathfrak{q}}\) be its Lie algebra, and let \(V\) be a \(Q\)-module. Then the index of \(V\) is the non-negative integer
\[
\text{ind}\left( \mathcal{\mathfrak{q}},V\right) :=\dim V-\max_{\xi\in V^{\ast} }\left( \dim\mathcal{\mathfrak{q}}\cdot\xi\right),
\]
where \(V^{\ast}\) is the linear dual of \(V.\) It is always that case that
\[
\text{ ind}\left( \mathcal{\mathfrak{q}},V^{\ast}\right) \leq \text{ind}\left( \mathcal{\mathfrak{q} }_{v},\left( V/\mathcal{\mathfrak{q}}\cdot v\right) ^{\ast}\right) ,
\]
where
\[
\mathcal{\mathfrak{q}}_{v}=\left\{ x\in\mathcal{\mathfrak{q}\,}|\,x\cdot v=0\right\}.
\]
If we have equality, we say \(\left( Q,V\right) \) has good index behavior (GIB). In general, it is difficult to determine if a representation has GIB -- to date, no general principle is known.
Let \(G\) be a connected reductive algebraic group and let \(\mathfrak{g} =\)Lie\(\left( G\right)\). An involution \(\theta\) of \(\mathfrak{g}\) of order \(m\) provides a \(\mathbb{Z}/m\mathbb{Z}\)-grading of \(\mathfrak{g}\) into \(\mathfrak{g}=\bigoplus_{i=0}^{m-1}\mathfrak{g}_{i},\) where \(\mathfrak{g}_{i}\) is the eigenspace of \(\theta\) corresponding to the eigenvalue \(\zeta^{i}\) for \(\zeta\) a fixed primitive \(m^{\text{th}}\) root of unity. For \(G_{0}\subset G\) a connected algebraic group with Lie algebra \(\mathfrak{g}_{0}\) one has a natural action of \(G_{0}\) on \(\mathfrak{g}_{1}.\)
This work is an investigation of GIB for \(\left( G_{0},\mathfrak{g} _{1}\right) ,\) and the author focuses on two different cases, both assuming that the rank of \(\left( G_{0},\mathfrak{g}_{1}\right) \) is nonzero. In the case where \(\mathfrak{g}\) is an exceptional Lie algebra, a series of tables is given describing the finite order involutions \(\theta\) such that \(\left( G_{0},\mathfrak{g}_{1}\right) \) has GIB; the information is displayed using Kac diagrams. The other case considered is where \(\mathfrak{g=gl}_{n}\) -- in this case attention is restricted to the inner automorphisms of \(\mathfrak{g;} \) here \(\theta\) must be given by conjugation by certain diagonal matrices. Simple Lie algebras; finite order automorphisms; index of a representation W. de Graaf, O. Yakimova, Good index behaviour of {\(\theta\)}-representations, I, Algebr. Represent. Theory 15 (2012), no. 4, 613--638. Group actions on varieties or schemes (quotients), Simple, semisimple, reductive (super)algebras, Automorphisms, derivations, other operators for Lie algebras and super algebras Good index behaviour of \(\theta \)-representations. I | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0577.00010.]
Suppose G is a connected, simply connected, compact simple Lie group. Let \(\Omega\) (G) denote the space of smooth based loops in G, that is, the space of smooth maps \(S^ 1\to G\) carrying a fixed point on the circle to the identity element of the group. The topology of \(\Omega\) (G) is thoroughly understood through the work of Bott. This paper, culled from the author's 1985 Berkeley Ph.D. thesis, undertakes a study of the geometry of this space, and speculates about applications to the representation theory of Kač-Moody groups, following Graeme Segal.
A flag manifold is an orbit of G under its adjoint action. The typical orbit is diffeomorphic to G/T, where T is a maximal torus of G. Flag manifolds admit homogeneous Kähler metrics, and the author derives a general formula for the curvature of these metrics. {\S} 1 summarizes the representation theory of G, à la Borel-Weil-Bott, emphasizing the role of the intrinsic geometry of G/T.
Pressley and Segal already observed that \(\Omega\) (G) admits homogeneous Kähler metrics. In fact, \(\Omega\) (G) is a coadjoint orbit for the affine Kač-Moody group associated to G. The formula of {\S} 1 applies to compute its curvature. The Ricci curvature makes sense, and the author uses it to define a first Chern class for \(\Omega\) (G). Topologically, \(c_ 1(\Omega (G))=2n_ Gx\), where \(x\in H^ 2(\Omega (G))\) is a generator, and \(n_ G\) is the dual Coxeter number of G.
For infinite dimensional (Hilbert) manifolds the tangent bundle is always trivial, so that the usual topological definition of characteristic classes fails. However, Chern classes can be defined by endowing the manifold with a Fredholm structure. The author argues that the holonomy bundle of the Kähler connection provides this structure. There are technical obstructions to making a rigorous construction, and the author circumvents them by proving an index theorem for families of Fredholm operators parametrized by a group. Thus he can explicitly display the Fredholm structure picked out by the holonomy bundle.
{\S} 4 is completely heuristic. Following Segal, the author outlines the relationship between \(\Omega\) (G) and the representation theory of the affine Kač-Moody group. The value of \(c_ 1(\Omega (G))\) enters here, and the arguments fit with known results (e.g., the Kač character formula). The value of \(c_ 1(\Omega (G))\) enters in a completely different realm of geometry - the study of instantons on the 4-sphere, via a theorem of Atiyah and Donaldson - and again there is agreement with known results (through more heuristic arguments). The author views {\S} 4 and {\S} 5 not as rigorous mathematics, which it decidedly is not, but as a case study advocating his construction of a Fredholm structure on \(\Omega\) (G). infinite dimensional Hilbert manifolds; compact simple Lie group; smooth based loops; Kač-Moody groups; flag manifold; homogeneous Kähler metrics; curvature; representation theory; intrinsic geometry; first Chern class; dual Coxeter number; Fredholm structure; holonomy bundle; index theorem; families of Fredholm operators; affine Kač-Moody group; Kač character formula; instantons on the 4-sphere Freed, D.: Flag manifolds and infinite dimensional Kähler geometry. Infinite dimensional groups (1985) Infinite-dimensional Lie groups and their Lie algebras: general properties, Differential geometry of symmetric spaces, Fredholm structures on infinite-dimensional manifolds, Grassmannians, Schubert varieties, flag manifolds, Infinite loop spaces, Infinite-dimensional Lie (super)algebras Flag manifolds and infinite dimensional Kähler geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given an exceptional vector bundle \(E\in\text{coh }\mathbb{X}\), by a theorem of the author and \textit{H. Lenzing} [Categories perpendicular to exceptional bundles (preprint, 1993)], its left-perpendicular category \(^\perp E\) is isomorphic to a category \(\text{mod }\Sigma_E\) for some finite-dimensional hereditary algebra \(\Sigma_E\) having \(n-1\) simple modules (where \(n\) denotes the \(\mathbb{Z}\)-rank of the Grothendieck-group of \(\text{coh }\mathbb{X}\)). In general, the description of \(\Sigma_E\) is a rather difficult and largely unsolved problem. -- For all exceptional vector bundles belonging to an Auslander-Reiten-component of \(\text{coh }\mathbb{X}\) containing a line bundle, the system of indecomposable injective objects in \(^\perp E\) completes \(E\) to a tilting sheaf \({\mathcal I}_E\), whose endomorphism ring \(\text{End}({\mathcal I}_E)\) is a branch-co-extension of a tame concealed quiver \(\Gamma\) of type \(\widetilde\Delta=\widetilde\mathbb{A}_{1,1}\), \(\widetilde\mathbb{D}_4\), \(\widetilde\mathbb{E}_6\), \(\widetilde\mathbb{E}_7\), \(\widetilde\mathbb{E}_8\) by linear quivers (cf. Theorem 3.4).
As a consequence, for each exceptional vector bundle \(E\) belonging to a line-bundle-component, the quiver underlying \(\Sigma_E\) (where \(\text{mod }\Sigma_E\cong{^\perp E}\)) can be determined. -- Theorem 3.3 lists the series of quivers arising in this way. exceptional vector bundles; perpendicular categories; finite-dimensional hereditary algebras; simple modules; Grothendieck groups; Auslander-Reiten components; indecomposable injective objects; tilting sheaves; endomorphism rings; tame concealed quivers Representations of associative Artinian rings, Representations of quivers and partially ordered sets, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representation type (finite, tame, wild, etc.) of associative algebras, Module categories in associative algebras Hereditary module categories arising as categories perpendicular to exceptional vector bundles | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian \(p\)-groups \(E_r\) of rank \(r\geq 2\) by exploiting a functor from the module category of a generalized Beilinson algebra \(B(n,r)\), \(n\leq p\), to \(\text{mod\,}E_r\).
We define analogues of the above-mentioned properties in \(\text{mod\,}B(n,r)\) and give a homological characterization of the resulting subcategories via a \(\mathbb P^{r-1}\)-family of \(B(n,r)\)-modules of projective dimension 1. This enables us to apply general methods from Auslander-Reiten theory and thereby arrive at results that, in particular, contrast the findings for equal images modules of Loewy length 2 over \(E_2\) by \textit{J. F. Carlson, E. M. Friedlander} and \textit{A. Suslin} [Comment. Math. Helv. 86, No. 3, 609-657 (2011; Zbl 1229.20039)] with the case \(r>2\). Moreover, we give a generalization of the \(W\)-modules introduced by the aforementioned authors. categories of modules; generalized Beilinson algebras; group schemes; modules of constant Jordan type; \(W\)-modules; equal images property; modular representations Julia Worch, Categories of modules for elementary abelian \?-groups and generalized Beilinson algebras, J. Lond. Math. Soc. (2) 88 (2013), no. 3, 649 -- 668. Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representations of quivers and partially ordered sets, Group rings of finite groups and their modules (group-theoretic aspects), Modular representations and characters, Group schemes Categories of modules for elementary abelian \(p\)-groups and generalized Beilinson algebras. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the situation of finite dimensional modules over tame quiver algebras the degener\-ation-order coincides with the hom-order and with the ext-order. Therefore, up to common direct summands, any minimal degeneration \(N\) of a module \(M\) is induced by a short exact sequence with middleterm \(M\) and indecomposable ends \(U\) and \(V\) that add up to \(N\). We study these ``building blocs'' of degenerations and in particular the codimensions for the case where \(V\) is regular. We show by theoretical means that the classification of all the ``building blocs'' is a finite problem without affecting the codimension or the type of singularity. With the help of a computer we have analyzed completely this case: The codimensions are bounded by 2, so that the minimal singularities are known by \textit{G. Zwara} [Manuscr. Math. 123, No. 3, 237-249 (2007; Zbl 1129.14006)]. Dynkin quivers; tame quiver algebras; singularities of representations of quivers I. Wolters, On deformations of the direct sum of a regular and another indecomposable module over a tame quiver algebra, Dissertation Berg. Univ. Wuppertal, 2008, 151 pages, available from Internet: URN: urn:nbn:de:hbz:468-20090102. Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Research exposition (monographs, survey articles) pertaining to associative rings and algebras On deformations of the direct sum of a regular and another indecomposable module over a tame quiver algebra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is a textbook for the beginning graduate students. The exposition is very detailed, with a number of exercises, even with material for the final exam. There are two parts in the book: the Core Course and Selected Topics. The Core Course contains the following chapters: Semisimple Modules \& Rings and the Wedderburn Structure Theorem, The Jacobson Radical, Central Simple Algebras, The Brauer Group.
The Selected Topics part consists of: Primitive Rings and the Density Theorem, Burnside's Theorem and Representations of Finite Groups, The Global Dimension of a Ring, The Brauer Group of a Commutative Ring.
As the authors note, their approach is more homological than ring- theoretic. semisimple modules; Jacobson radical; central simple algebras; Brauer group; primitive rings; density theorem; representations of finite groups; global dimension; textbook Farb, B.; Dennis, R. K.: ''Noncommutative algebra,'', graduate texts in mathematics. (1993) Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras, Finite-dimensional division rings, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general, Simple and semisimple modules, primitive rings and ideals in associative algebras, Group rings of finite groups and their modules (group-theoretic aspects), Jacobson radical, quasimultiplication, General module theory in associative algebras, Brauer groups of schemes Noncommutative algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In these notes we present a new link between the theory of automorphic forms and geometry. For an arbitrary compact manifold one can define its elliptic genus. It is a modular form in one variable with respect to a congruence subgroup of level 2. For a compact complex manifold one can define its elliptic genus as a function in two complex variables. In the last case the elliptic genus is the holomorphic Euler characteristic of a formal power series with vector bundle coefficients. If the first Chern class \(c_1(M)\) of the complex manifold is equal to zero in \(H^2(M,\mathbb{R})\), then the elliptic genus is a weak Jacobi modular form with integral Fourier coefficients of weight 0 and index \(d/2\), where \(d= \dim_{\mathbb{C}} (M)\). The same modular form appears in physics as the partition function of \(N=2\) supersymmetric sigma model whose target space is a Calabi-Yau manifold \(M\). We note that all ``good'' partition functions appearing in physics are automorphic forms with respect to some groups. This fact reflects that physical models have some additional symmetries. If \(c_1(M)\neq 0\), then the elliptic genus of \(M\) is not an automorphic form. In these notes we define a modified Witten genus or automorphic correction of elliptic genus of an arbitrary holomorphic vector bundle over a compact complex manifold and study its properties.
We mainly present here automorphic aspects of the theory. In the proof of the theorem that the modified Witten genus is a Jacobi form we use a nice formula which relates the Jacobi theta-series, its logarithmic derivative, the quasi-modular Eisenstein series \(G_2(\tau)\) and all derivatives of Weierstrass \(\wp\)-function. To get applications to the theory of complex manifolds we study the \(\mathbb{Z}\)-structure of the graded ring of weak Jacobi forms with integral coefficients. We prove that the graded ring of Jacobi forms of weight 0 has four generators
\[
J_{0,*}^{\mathbb{Z}}= \bigoplus_{m\geq 1} J_{0,m}^{\mathbb{Z}}= \mathbb{Z}[\phi_{0,1}, \phi_{0,2}, \phi_{0,3}, \phi_{0,4}]
\]
which satisfy the single relation \(4\phi_{0,4}= \phi_{0,1}\phi_{0,3}- \phi_{0,2}^2\). The functions \(\phi_{0,1},\dots, \phi_{0,4}\) are the fundamental Jacobi forms related to Calabi-Yau manifolds of dimension \(d=2,3,4,8\). The same Jacobi forms are generating functions for the multiplicities of all positive roots of the four generalized Lorentzian Kac-Moody Lie algebras of Borcherds type constructed in [\textit{V. A. Gritsenko} and \textit{V. V. Nikulin}, Am. J. Math. 119, 181-224 (1997; Zbl 0914.11020); Mat. Sb., Nov. Ser. 187, 1601-1641 (1996; Zbl 0876.17026); Int. J. Math. 9, 153-199 (1998; Zbl 0935.11015); Int. J. Math. 9, 201-275 (1998; Zbl 0935.11016)].
The \(q^0\)-term of the Fourier expansion \((q=e^{2\pi i\tau})\) of the elliptic genus is essentially equal to the Hirzebruch \(\chi_y\)-genus of the manifold. Thus we can analyze the arithmetic properties of the \(\chi_y\)-genus of the complex manifold with \(c_1(M)= 0\) and its special values such as signature \((y=1)\) and Euler number \((y=-1)\) in terms of Jacobi modular forms. For example, we prove that the Euler number of a Calabi-Yau manifold \(M_d\) of dimension \(d\) satisfies
\[
e(M_d)\equiv 0\bmod 8 \quad\text{if}\quad d\equiv 2\bmod 8.
\]
The special values of the generators of the Jacobi ring at \(z= \frac 12, \frac 13, \frac 14\) are related to the Hauptmodules of the fields of modular functions. Using this fact we prove that
\[
\chi_y= \zeta_3(M_d) \equiv 0\bmod 9\quad\text{if}\quad d\equiv 2\bmod 6.
\]
Some other constructions (for example, \(\widehat{A}_2^{(2)}\)-genus, the second quantized elliptic genus) and other applications to the theory of vector bundles can be found in the author's course given at RIMS, Kyoto University [see also \textit{V. Gritsenko}, St. Petersbg. Math. J. 11, 781-804 (2000); translation from Algebra Anal. 11, No. 5, 100-125 (2000; Zbl 1101.14308)]. Hirzebruch genus; automorphic correction of elliptic genus; modified Witten genus; Jacobi form; Jacobi theta-series; logarithmic derivative; quasi-modular Eisenstein series; derivatives of Weierstrass \(\wp\)-function; complex manifolds; graded ring of weak Jacobi forms; Calabi-Yau manifolds; Lorentzian Kac-Moody Lie algebras of Borcherds type; elliptic genus V. Gritsenko, \textit{Complex vector bundles and Jacobi forms}, math/9906191 [INSPIRE]. Jacobi forms, Relations with algebraic geometry and topology, Almost complex manifolds, Elliptic genera, Calabi-Yau theory (complex-analytic aspects), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Calabi-Yau manifolds (algebro-geometric aspects) Complex vector bundles and Jacobi forms | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The main results of this paper are concerned with a modern version of Hilbert's fourteenth problem. The setting is as follows. Let G be an affine algebraic group over an algebraically closed field k. Let X be an affine variety over k on which G acts regularly and let k[X] be the algebra of regular functions on X. The action of G on X gives rise to an action on k[X]. Let \(k[X]^ G\) denote the subalgebra of k[X] consisting of those functions which are fixed by G. We ask: is \(k[X]^ G\) a finitely generated k-algebra?
For algebraic groups which are not reductive, the only affirmative answer known until recently was Weitzenböck's theorem [cf. \textit{R. Weitzenböck}, Acta Math. 58, 231-293 (1932; Zbl 0004.24301)]. He proved that if the additive group \({\mathbb{C}}\) acts on a vector space, then the algebra of invariant functions is finitely generated. Seshadri's proof of this theorem shows that finite generation for \({\mathbb{C}}\) is inherited from the finite generation for the reductive group \(SL_ 0({\mathbb{C}}).\)
The present paper is concerned with a class of subgroups of (arbitrary) reductive groups which also inherit finite generation. An affine algebraic group G operates on itself by right multiplication and this gives an action of G on k[G]. When G is reductive and H is a closed subgroup of G, the following conditions on H are equivalent: (i) k[G]\({}^ H\) is a finitely generated k-algebra; (ii) if X is any affine variety on which G acts, then \(k[X]^ H\) is a finitely generated k- algebra.
In this paper, two theorems are proved concerning condition (i). These theorems provide a framework for many of the known examples and lead to large classes of new examples. In stating them, we denote \(k[G]^ H\) by H'.
Theorem. Let k be an algebraically closed field, let G be a connected reductive algebraic group over k, and let H be a connected observable subgroup of G. Let L be a connected algebraic subgroup of G such that H is normal in L and \(S=L/H\) is semi-simple. Let \(U_ L\) be any maximal unipotent subgroup of L. Then H' is a finitely generated k-algebra if and only if \((U_ LH)'\) is. - Theorem. Let G be a simply connected semi- simple algebraic group and let \(B=TU\) be a Borel subgroup of G. Let H be a connected subgroup of U which is normalized by T and let \(\alpha_ 1,...,\alpha_ r\) be roots such that \(U=HU_{\alpha_ 1}...U_{\alpha_ r}.\) If \(\{\alpha_ 1,...,\alpha_ r\}\) is linearly independent over \({\mathbb{Q}}\), then H' is a finitely generated k-algebra. finite generation of algebra of invariant functions; Hilbert's fourteenth problem; algebraic group Grosshans, F., \textit{hilbert's fourteenth problem for non-reductive groups}, Math. Z., 193, 95-103, (1986) Geometric invariant theory, Group actions on varieties or schemes (quotients), Linear algebraic groups and related topics Hilbert's fourteenth problem for non-reductive groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In [Duke Math. J. 76, 365-415 (1994; Zbl 0826.17026)] \textit{H. Nakajima} has defined a new class of varieties, called quiver varieties, associated to any quiver. In the particular case of the Dynkin quiver of finite type A, these varieties are related to partial flag manifolds. For any quiver with no edge loops, he proved in [Duke Math. J. 91, 515-560 (1998; Zbl 0970.17017)] that the corresponding simply laced Kac-Moody algebra acts on the top homology groups via a convolution product. The purpose of this paper is to compute the convolution product on the equivariant \(K\)-groups of the cyclic quiver variety, generalizing the previous works [see \textit{V. Ginzburg} and \textit{E. Vasserot}, Int. Math. Res. Not. 1993, 67-85 (1993; Zbl 0785.17014) and \textit{E. Vasserot}, Transform. Groups 3, 269-299 (1998; Zbl 0969.17009)]. It is expected that equivariant \(K\)-groups should give an affinization of the quantized enveloping algebra of the corresponding Kac-Moody algebra. Similar algebras, called toroidal algebras, have already been studied in the cyclic case. Surprisingly, the convolution operators we get do not satisfy exactly the relations of the quantized toroidal algebra. We obtain a twisted version, denoted by \({\mathfrak U}_{q,t}\), of the latter. quiver varieties; Dynkin quiver of finite type A; partial flag manifolds; convolution product; equivariant \(K\)-groups; cyclic quiver variety; quantized enveloping algebra; Kac-Moody algebra; toroidal algebras Varagnolo, M., Vasserot, E.: On the \(K\)-theory of the cyclic quiver variety. Int. Math. Res. Not. no. 18, 1005-1028 (1999). arXiv:math/9902091. http://dx.doi.org/10.1155/S1073792899000525 Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Algebraic moduli problems, moduli of vector bundles, Equivariant \(K\)-theory On the \(K\)-theory of the cyclic quiver variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper is the sequel to Part I [K-Theory 10, No. 6, 517-596 (1996; Zbl 0874.16012)] by the same authors, referred to as [I]. The twisted flag varieties considered in these papers are projective varieties over an arbitrary field \(F\) on which an adjoint semisimple algebraic group \(G\) acts, the action being transitive after scalar extension to a separable closure \(F_s\) of \(F\). In [I], a general formula for the change in Schur index of an arbitrary central simple \(F\)-algebra when scalars are extended to the function field of a twisted flag variety \(X\) was given. It involves various central simple algebras \(A_G(\psi)\) attached to the elements \(\psi\) in the character group of the center of the simply connected cover of \(G_s=G\times F_s\) as well as integers \(n_{\psi,P,E}\) depending on \(\psi\), on the parabolic subgroup \(P\) of \(G_s\) such that \(X\times F_s\simeq G_s/P\) and on certain finite extensions \(E\) of \(F\).
In [I], this formula was made explicit for \(G\) simple, classical and of inner type, by a case-by-case determination of the algebras \(A_G(\psi)\) and the integers \(n_{\psi,P,E}\). The purpose of the present paper is to give similar explicit formulas in the cases not covered by [I]. The authors discuss the cases where \(G\) is a classical simple group of outer type or an exceptional simple group. They also show how to derive index reduction formulas for the flag varieties associated to a product \(G_1\times G_2\) from those for \(G_1\) and \(G_2\), and for the flag varieties of transfers (Weil restriction of scalars). Thus, index reduction formulas for all flag varieties for all adjoint semisimple groups can be obtained by combining the results in [I] with those of the present paper.
Since this paper was written, alternative proofs of some of its results, avoiding a case-by-case analysis, have been found by \textit{A. S. Merkurjev} [Doc. Math. 1, 229-243 (1996; Zbl 0854.20060), J. Ramanujan Math. Soc. 12, No. 1, 49-95 (1997; Zbl 0906.16006)]. Even if the approach in these papers is more direct, the thorough discussion of various special cases presented here is undoubtedly useful. Particularly noteworthy are the careful summary exposition of transfers given in section 7 and the description of Tits algebras for groups of type \(E_6\) and \(E_7\) in section 8.
(Also submitted to MR). central simple algebras; Brauer groups; semisimple linear algebraic groups; Borel varieties; twisted flag varieties; projective varieties; change of Schur index; function fields; groups of inner type; parabolic subgroups; index reduction formula; Brauer-Severi varieties Merkurjev, A.; Panin, A.; Wadsworth, A., \textit{index reduction formulas for twisted flag varieties II}, J. K-Theory, 14, 101-196, (1998) Finite-dimensional division rings, Group actions on varieties or schemes (quotients), Quadratic spaces; Clifford algebras, Representation theory for linear algebraic groups, \(K\)-theory of schemes Index reduction formulas for twisted flag varieties. II | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities A finite-dimensional algebra \(A\) is said to have finite representation type if there are only finitely many isomorphism types of indecomposable modules over \(A\). The Schur algebra is useful in understanding the representation theory of the general linear group; polynomial representations of degree \(d\) correspond to representations of the Schur algebra \(S(n,d)\). The Schur algebras of finite representation type were classified by \textit{K.~Erdmann} [Q. J. Math., Oxf. II. Ser. 44, No. 173, 17-41 (1993; Zbl 0832.16011)].
For \(r\in\mathbb{N}\), the infinitesimal Schur algebra \(S(n,d)_r\) is a subalgebra of \(S(n,d)\). If \(k\) is an algebraically closed field of characteristic \(p>0\), let \(G\) denote the scheme \(\text{GL}_n(k)\), and let \(T\) denote the subscheme arising from diagonal elements. Let \(G_r\) denote the kernel of the \(r\)th iterate of the Frobenius \(p\)th power endomorphism on the scheme \(G\). Then the representations of \(G_rT\) of a given degree \(d\) are equivalent to the representations of \(S(n,d)_r\).
In this paper, the authors classify the cases in which \(S(n,d)_r\) is of finite representation type. representations; general linear groups; symmetric groups; infinitesimal Schur algebras; finite representation type; indecomposable modules; polynomial representations; group schemes; Frobenius endomorphisms Doty, S; Nakano, D; Peters, K, Infinitesimal Schur algebras of finite representation type, Quart. J. Math., 48, 323-345, (1997) Representation theory for linear algebraic groups, Representations of finite symmetric groups, Representation type (finite, tame, wild, etc.) of associative algebras, Group schemes Infinitesimal Schur algebras of finite representation type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be a basic field, let \(Q\) be a quiver, and let \(\alpha\) be a representation of \(Q\) whose endomorphism ring is isomorphic to \(k\). Then \(\alpha\) is called the Schur representation of \(Q\) and its dimension-vector is called the Schur root. Let \({\mathcal M}(Q,\alpha)\) be the moduli space of representations of \(Q\) with dimension-vector \(\alpha\) [\textit{A. D. King}, Q. J. Math., Oxf. II. Ser. 45, No. 180, 515-530 (1994; Zbl 0837.16005)]. Denote by \(n\) the greatest common divisor of \(\dim_k\alpha(v)\) for all vertices \(v\) of \(Q\). It is proved that if \(1\leq n\leq 4\) then \({\mathcal M}(Q,\alpha)\) is rational. Moreover, if \(n>4\) and \(n\) divides \(420\) then the moduli space is stably rational while if \(n\) is square-free then \({\mathcal M}(Q,\alpha)\) is retract rational. finite-dimensional algebras; representations of quivers; moduli spaces; Schur representations; Schur roots; Kronecker quivers; extended Dynkin diagrams; Morita equivalences Schofield, A., Birational classification of moduli spaces of representations of quivers, \textit{Indagat. Math. (3)}, 12, 407-432, (2001) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Trace rings and invariant theory (associative rings and algebras), Representation type (finite, tame, wild, etc.) of associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Birational classification of moduli spaces of representations of quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This short paper continues the authors' earlier work [\textit{R. Farnsteiner} and \textit{D. Voigt}, Adv. Math. 155, No. 1, 1--22 (2000; Zbl 0981.16030)].
In the previous work, it was shown that, for \(k\) an algebraically closed field of characteristic \(p\geq 3\) an infinitesimal \(k\)-group \(G\) has a principal block of finite representation type if and only if \(G/M(G)\) is a semidirect product of the \(n\)-th Frobenius kernel of \(\mu_k\) with a \(V\)-uniserial normal subgroup, which is true if and only if \(H(G)\) is a Nakayama algebra, which in turn is true if and only if \(\dim(\Hom({\pmb\alpha}_{p^2},G))\leq 1\), which in turn is true if and only if \(H(G_2)\) is a Nakayama algebra, where \(G_2=\ker F^2\colon G\to G\) and \(H(G)\) is the distribution algebra of \(G\).
In this paper, the main result stated above, an analogue of Higman's theorem, is extended to the case \(p=2\). Specifically, a lemma from the previous paper is extended to the case \(p=2\), from which the main result follows. representation theory; infinitesimal group schemes; principal blocks; finite representation type; uniserial groups; Nakayama algebras Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group schemes, Representation type (finite, tame, wild, etc.) of associative algebras, Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations Representations of infinitesimal groups of characteristic 2 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be an almost simple Chevalley-Demazure group scheme of type \(A_ n\) (\(n\geq 2\)), \(D_ m\) (\(m\geq 4\)), or \(E_ 6\) and \(\Gamma\) the lattice of weights of the representation that defines \(G\). If \(G\) is universal or adjoint, denote that lattice by the respective symbols \(\Gamma_{sc}\) or \(\Gamma_{ad}\), so that \(\Gamma_{ad}\subseteq\Gamma\subseteq\Gamma_{sc}\). Let \(\sigma\) be the canonical involutive automorphism of \(\Phi\), and represent the induced involution of \(\Gamma_{sc}\) or \(\Gamma_{ad}\) by \(\sigma\) also, so that \(\Gamma\) is \(\sigma\)-stable, apart from the cases \(\Phi = D_ n\), \(n\) even, \(n\geq 4\) and \(\Gamma\) strictly between \(\Gamma_{ad}\) and \(\Gamma_{sc}\). (That strict inclusion is misprinted in the paper.) Let \(A\) be a commutative ring (with 1) with an involutive automorphism \(\sigma\) that stabilizes \(\Gamma\). Let \(G(\Phi,A) =\text{Hom}(\mathbb{Z}[G],A)\) be the group of \(A\)-valued points of \(G\), where \(\mathbb{Z}[G]\) is a Hopf algebra over \(\mathbb{Z}\). \(G(\Phi, A)\) is the Chevalley group of \(\Phi\) (or \(G\)) over \(A\). \textit{E. Abe} and the reviewer [Commun. Algebra 16, 57-74 (1988; Zbl 0647.20047)] determined the center of \(G(\Phi, A)\) for most cases. There is a corresponding automorphism, still denoted \(\sigma\), of \(\mathbb{Z}[G]\), from which an automorphism of \(G(\Phi, A)\) arises as follows. Let \(T(\Phi, A)\cong\text{Hom}(\Gamma, A^*)\) be the standard maximal torus of \(G(\Phi, A)\), where \(A^*\) is the group of invertible elements of \(A\). Let \(h(\chi)\) be the element of \(T(\Phi, A)\) that corresponds to \(\chi\in\text{Hom}(\Gamma, A^*)\), and \(\sigma(\chi) (\gamma) =\sigma(\chi (\sigma(\gamma)))\) for any \(\gamma\in\Gamma\). Then \(\sigma\) induces an automorphism of \(\text{Hom}(\mathbb{Z}[G], A) = G(\Phi, A)\) that satisfies the following. For a unipotent \(x_ \alpha (a)\) in \(G(\Phi,A)\), \(\sigma(x_ \alpha(a)) = x_{\sigma(\alpha)} (c_ \alpha\sigma(a))\), where \(c_ \alpha =\pm 1\). Also, for \(h(\chi)\in T(\Phi, A)\), \(\sigma(h(\chi)) = h(\sigma (\chi))\).
The twisted Chevalley group \(G_ \sigma (\Phi, A)\) is \(\{x\in G(\Phi,A)\mid\sigma x = x\}\). The author studies the center of this group, via the method introduced by Abe and the reviewer. His main result is as follows. Suppose the Jacobson radical \(R\) of \(A\) is trivial or that \(\Phi\) has rank \(\geq 2\). Let \(\{{\mathcal A}\}\) consist of all ordered pairs \((a,b)\) of elements of \(A\) such that \(a\overline{a} = b +\overline{b}\), where the bar indicates the image under \(\sigma\). If \(\Phi_ \sigma\) is of type \(^ 2A_{2n}\) and \(R\neq 0\), assume that there is an element \((a,b)\) in \(\{{\mathcal A}\}\) with \(a\in A^*\) and \(\{{\mathcal A}\}^*\neq\emptyset\). (Here \(*\) refers to inverse to a certain group operation defined on \(\{{\mathcal A}\}\).) If \(G\) is adjoint, then the center is trivial. If \(G\) is universal or adjoint, then the center is \(\text{Hom}(\Gamma/\Gamma_{ad}, A^*)\), and coincides with the center of the elementary subgroup of \(G(\Phi_ \sigma, A)\). The author also obtains some additional information about the center in more general cases. root systems; Weyl groups; almost simple Chevalley-Demazure group schemes; lattice of weights; involutive automorphisms; Hopf algebras; center; maximal torus; twisted Chevalley groups; Jacobson radical; elementary subgroups Linear algebraic groups over adèles and other rings and schemes, Simple groups: alternating groups and groups of Lie type, Simple, semisimple, reductive (super)algebras, Associated Lie structures for groups, Group schemes, Subgroup theorems; subgroup growth, Affine algebraic groups, hyperalgebra constructions Centers of twisted Chevalley groups over commutative rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0577.00010.]
In this article, the author gives a survey on the subject stated in the title developed by the author [Habilitationsschrift, Universität Bonn, 1984]. The author gives a definition (due to E. Looijenga) of an adjoint quotient for an arbitrary Kac-Moody Lie group G and analyses the structure of fibers. Due to Brieskorn, there is a relationship between simple singularities and simple algebraic groups. The author shows that at least to some extent there is a similar relationship between the deformation theory of simple elliptic and cusp singularities due to Looijenga and associated Kac-Moody Lie groups.
In the finite dimensional case, let G be a simply connected semi-simple algebraic group over \({\mathbb{C}}\), B be a Borel subgroup and \(T\subset B\) a maximal torus of G, N be the normalizer of T in G. Then, \(N/T=W\) is the finite Weyl group. The adjoint quotient of G is the quotient of G by its adjoint actions which is canonically isomorphic to T/W.
In the infinite-dimensional case, to obtain a reasonable adjoint quotient of G, the author uses the Tits cone attached to the root basis and its Weyl group and Looijenga's partial compactification of T/W. Its stratification into boundary components induces a partition of G which can be described in terms of the building associated with G.
Some open problems on a representation theoretic interpretation of this partition are mentioned at the end. Detailed proofs and relations to singularities may be found in the authors notes indicated above. Kac-Moody algebras; Kac-Moody group; deformation theory of singularities; adjoint quotient; Kac-Moody Lie group; semi-simple algebraic group; Borel subgroup; Weyl group; Tits cone Slodowy, P.: An adjoint quotient for certain groups attached to Kac-Moody algebras. Math. sci. Res. inst. Publ. 4, 307-333 (1985) Infinite-dimensional Lie groups and their Lie algebras: general properties, Singularities of curves, local rings, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras An adjoint quotient for certain groups attached to Kac-Moody algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theme of this thesis is the study of prehomogeneous vector spaces of parabolic type. It is divided into two parts, one is the algebraic study and is devoted mainly to the definition and the classification of prehomogeneous vector spaces of parabolic type, and the other is the analytic study which is applied to the research on local zeta functions associated with them.
In the algebraic study, the author found a natural structure of prehomogeneous vector space in a simple Lie algebra. That is; let \({\mathfrak g}\) be a simple Lie algebra, \({\mathfrak h}\) a Cartan subalgebra of \({\mathfrak g}\), and R the root system with respect to (\({\mathfrak g,h})\). We fix \(\Psi\) a base of R. Let \(\theta\) be a subset of \(\Psi\). We put \(d_ p(\theta)=\{X_{\theta}\in {\mathfrak g}; [H,X]=2pX\}\) for \(p\in Z\). Here, \(H^{\theta}\) is the element of \({\mathfrak h}_{\theta}\) defined by \(\alpha (H_{\theta})=2\) for \(\alpha \in\Psi -\theta\) and \(\alpha (H_{\theta})=0\) for \(\alpha \in\theta \). In particular, we denote \(\ell_{\theta}=d_ 0(\theta)\). Since \(\{d_ p(\theta)\}_{p\in Z}\) gives a gradation of \({\mathfrak g}\), i.e., \([d_ i(\theta),d_ j(\theta)]\subset d_{i+j}(\theta).\) The Lie algebra \(\ell_{\theta}\) acts on \(d_ p(\theta)\), i.e., \([\ell_{\theta},d_ p(\theta)]\subset d_ p(\theta)\) and this action causes a representation of the Lie group \(L_{\theta}\) in \(GL(d_ p(\theta))\) whose Lie algebra is \(\ell_{\theta}\). Then, \((L_{\theta},d_ p(\theta))\) is a prehomogeneous vector space.
In particular, when \(p=1\), the pair \((L_{\theta},d_ 1(\theta))\) is called a prehomogeneous vector space of parabolic type and furthermore, when \(Card(\psi -\theta)=1\), it is proved that \((L_{\theta},d_ 1(\theta))\) is an irreducible prehomogeneous vector space. The author decides when such prehomogeneous vector space of parabolic type is regular and classifies all of them when \({\mathfrak g}\) is simple. It is proved that ''almost all'' reduced regular irreducible prehomogeneous vector spaces are obtained as a prehomogeneous vector space of parabolic type. The author determines the real forms of the prehomogeneous vector spaces of parabolic type by making use of the real forms of Dynkin diagrams (Chapter 2), carries out the classification of regular prehomogeneous vector spaces of parabolic type which are Q-irreducible (Chapter 3) and proves that the ring of regular invariant functions on \(X_{\theta}=G\cdot ({\mathfrak h}_{\theta}+\sum_{p\geq 1}d_ p(\theta))\) is integrally closed, which is an affirmative answer for Borho's conjecture in a certain case.
In the analytic study, the author gives a relation between local zeta functions on prehomogeneous vector spaces and intertwining operators in some cases, and studies on the poles of local zeta functions. prehomogeneous vector spaces of parabolic type; algebraic study; analytic study; local zeta functions; simple Lie algebra; classification of regular prehomogeneous vector spaces; intertwining operators; poles Rubenthaler, H.: Espaces préhomogènes de type parabolique. Thèse d'état (1982) Semisimple Lie groups and their representations, Lie algebras of Lie groups, Homogeneous spaces and generalizations, Structure theory for Lie algebras and superalgebras, Graded Lie (super)algebras Espaces préhomogènes de type parabolique. (Thèse d'Etat) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0597.00009.]
In this survey paper, the author describes features of almost split sequences in algebraic geometry. In the first section, complete rational double points are considered. From work by Esnault and Knörrer, it follows that if R is a complete rational double point (viewed as a quotient ring of the ring of formal power series in 3 variables over an algebraically closed field), then the classical Artin-Verdier correspondence (between isomorphism classes of non projective indecomposable reflexive R-modules and the exceptional divisors in the desingularization of R) extends to an isomorphism between the desingularization graph of R and the underlying graph of the stable AR- quiver of R.
As a consequence, if G is a finite subgroup of S1(2,k), of order prime to char(k) and if V is the two-dimensional k[G]-module corresponding to \(G\subset S1(2,k)\), then, earlier results of the author, describing the explicit relation between the AR-quiver and certain McKay quivers, permit to prove: If R is the complete rational double point, which is the fixed point ring of the corresponding linear action of G on \(k[[ X,Y]]\), then the graph associated to the McKay quiver of V (with the trivial representation omitted) may be identified with the desingularization graph of R.
The main result of the second section, which deals with projective curves, essentially, states that the category of coherent sheaves over a connected nonsingular projective curve has almost split sequences. In fact somewhat more vaguely, if X is a connected Gorenstein projective curve, then the category of locally free sheaves on X has almost split sequences. Moreover, one may prove that if X is an irreducible projective variety, then coherent sheaves on X have almost split sequences exactly when X is a nonsingular curve.
In the last section, the author considers finite Cohen-Macaulay type algebras, i.e. having only a finite number of non-isomorphic indecomposable Cohen-Macaulay modules. Over the complex numbers, it has been shown by the author that any Cohen-Macaulay algebra R of finite Cohen-Macaulay type is an isolated singularity (and an integrally closed domain, if dim(R)\(\geq 2).\)
The hypersurfaces of finite Cohen-Macaulay type, on the other hand, are completely determined by the fact that they are just simple Arnol'd singularities (and these are explicitly known). Surprisingly, the obvious questions for complete intersection or more generally for Gorenstein singularities have been settled by Herzog, who showed that any Gorenstein \({\mathbb{C}}\)-algebra of finite Cohen-Macaulay type is necessarily a hypersurface. Considering the results concerning two-dimensional \({\mathbb{C}}\)-algebras, one may wonder wich quotient singuarities are of finite Cohen-Macaulay type, in dimension higher than two. Here, a complete answer is given by a result of the author and \textit{I. Reiten}, that states that if \(n>2\) and if G is a finite subgroup of GL(n,\({\mathbb{C}})\), not containing any pseudo-reflections, then \({\mathbb{C}}[[ X_ 1,...,X_ n]]\) G is of finite Cohen-Macaulay type exactly when \(n=3\), \(| G| =2\) and \(g(X_ i)=-X_ i\) \((i=1,2,3)\), where g is the generator of G. Note that this \({\mathbb{C}}\)-algebra is not Gorenstein hence not a hypersurface, of course. Finally, a result of the author and I. Reiten is mentioned, that explicitly describes the only rational normal scroll which is not a hypersurface and is of finite Cohen-Macaulay type (the scroll (2,1)ö). almost split sequences; complete rational double point; Artin-Verdier correspondence; stable AR-quiver; McKay quiver; desingularization graph; coherent sheaves; Gorenstein projective curve; finite Cohen-Macaulay type algebras Maurice Auslander, Almost split sequences and algebraic geometry, Representations of algebras (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 116, Cambridge Univ. Press, Cambridge, 1986, pp. 165 -- 179. Singularities in algebraic geometry, Representation theory of associative rings and algebras, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities of surfaces or higher-dimensional varieties, Rational and unirational varieties, Singularities of curves, local rings Almost split sequences and algebraic geometry | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Katz has constructed some rigid local systems of rank 7 on \(\mathbb G_m\) in finite characteristics, whose monodromy is a finite subgroup \(G\) of a complex Lie group \(M\) of type \(G_2\).
Among the five subgroups \(G\) of \(M\) which appear in his work, four occur in their natural characteristic:
\[
\mathrm{SL}_2(8) \text{ in }\operatorname{char} 2,\quad \mathrm{PU}_3(3) \text{ in } \operatorname{char} 3, \quad \mathrm{PGL}_2(7) \text{ in }\operatorname{char} 7, \quad \mathrm{PSL}_2(13) \text{ in }\operatorname{char} 13.
\]
In this paper, using the theory of Deligne-Lusztig curves, the author gives another construction of these four local systems on \(\mathbb G_m\), and finds similar local systems with finite monodromy in other complex Lie groups. Some examples of finite groups which occur in exceptional complex Lie groups \(M\), analogous to the ones above are:
\[
\begin{alignedat}{2}2
G &= \mathrm{PSL}_2(27) \text{ in }\operatorname{char}3,\quad& M&=F_4,\\
G &= \mathrm{PSU}_3(8) \text{ in }\operatorname{char}2,\quad& M&=E_7,\\
G &=\mathrm{PGL}_2(31) \text{ in }\operatorname{char}31,\quad& M&=E_8,\\
G &= \mathrm{PSL}_2(61) \text{ in }\operatorname{char}61,\quad& M&=E_8.
\end{alignedat}
\]
There are interesting families in the classical groups. For example, when \(q=p^f>2\), we have
\[
G=\mathrm{PU}_3(q) \text{ in }\operatorname{char}p,\quad M=\mathrm{Sp}_{2n},\quad 2n=q(q-1).
\]
As a bonus, the author obtains simple wild parameters for the local field \(k((1/t))\), by restricting the local systems to the decomposition group at \(t=\infty\). These parameters are representations of the local Galois group into \(M\), which have no inertial invariants on the adjoint representation and have Swan conductor equal to the rank of \(M\). rigid local systems; Deligne-Lusztig curves; finite simple subgroups of complex Lie groups; wild parameters; rigid embeddings; wild inertia group; Swan conductors; Galois group Gross, B. H., Rigid local systems on G_{\textit{m}} with finite monodromy, Adv. Math., 224, 6, 2531-2543, (2010) Linear algebraic groups over finite fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups Rigid local systems on \(\mathbb G_m\) with finite monodromy | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In previous work, the authors have shown that the Picard group of a basic finite-dimensional split algebra \(A\) over an algebraically closed field of characteristic \(0\) may essentially be computed through the normal subgroup of finite index \(H_A/H_A\cap\text{Inn}(A)\) and the corresponding finite quotient. Here, \(H_A\) consists of all \(\varphi\in\Aut(A)\), which reduce to the identity on \(B\), where \(B\) is the complement of the Jacobson radical \(J\) in the Wedderburn-Malcev decomposition of \(A\) and \(\text{Inn}(A)\) denotes the group of inner automorphisms of \(A\). While this is in theory a very effective recipe, it appears that in practice one still has to calculate ``example by example'', implying that no genuine general approach is available at the moment.
On the other hand, one may wish for such a general approach within sufficiently nice, broad and well-defined subclasses of algebras. In the present paper the authors concentrate on such a class: the case of monomial algebras \(A\) with acyclic quiver. For this class of algebras the authors' methods appear to work perfectly well, making use, moreover, of techniques basically stemming from the theory of algebraic groups. Picard groups; basic finite-dimensional split algebras; Wedderburn-Malcev decompositions; groups of inner automorphisms; monomial algebras; acyclic quivers Guil-Asensio, F.; Saorı\acute{}n, M.: The automorphism group and the Picard group of a monomial algebra. Comm. algebra 27, No. 2, 857-887 (1999) Finite rings and finite-dimensional associative algebras, Picard groups, Automorphisms and endomorphisms The automorphism group and the Picard group of a monomial algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a simple algebraic group over the algebraically closed field \(k\). A slightly strengthened version of a theorem of T. A. Springer says that (under some mild restrictions on \(G\) and \(k\)) there exists a \(G\)-equivariant isomorphism of varieties \(\varphi\colon\mathcal U\to\mathcal N\), where \(\mathcal U\) denotes the unipotent variety of \(G\) and \(\mathcal N\) denotes the nilpotent variety of \(\mathfrak g=\text{Lie\,}G\). Such \(\varphi\) is called a Springer isomorphism. Let \(B\) be a Borel subgroup of \(G\), \(U\) the unipotent radical of \(B\) and \(\mathfrak u\) the Lie algebra of \(U\). In this note we show that a Springer isomorphism \(\varphi\) induces a \(B\)-equivariant isomorphism \(\widetilde\varphi\colon U/M\to\mathfrak{u/m}\), where \(M\) is any unipotent normal subgroup of \(B\) and \(\mathfrak m=\text{Lie\,}M\). We call such a map \(\widetilde\varphi\) a relative Springer isomorphism. We also use relative Springer isomorphisms to describe the geometry of \(U\)-orbits in \(\mathfrak u\). simple algebraic groups; equivariant isomorphisms of varieties; unipotent varieties; Borel subgroups; unipotent radical; Lie algebras; Springer isomorphisms S. M. Goodwin, \textit{Relative Springer isomorphisms}, J. Algebra \textbf{290} (2005), no. 1, 266-281. Linear algebraic groups over arbitrary fields, Group actions on varieties or schemes (quotients), Solvable, nilpotent (super)algebras, Lie algebras of linear algebraic groups Relative Springer isomorphisms. | 0 |
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