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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 Given a zero-dimensional subscheme X of \({\mathbb{P}}^ 2\) one can relate the Hilbert difference functions of X to properties of the linear systems of curves of a given degree containing X. The author studies the Hilbert difference functions of a zero-dimensional subscheme X contained in a quadric Q in \({\mathbb{P}}^ 3\), and relates them to the linear systems of surfaces of a given degree containing X. zero-dimensional schemes; Hilbert difference functions; quadric; linear systems of surfaces Raciti, G.: Hilbert function and geometric properties for a closed O-dimensional subscheme of a quadric Q \subsetp3. Comm. algebra 18, No. 9, 3041-3053 (1990) Parametrization (Chow and Hilbert schemes), Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves, Schemes and morphisms Hilbert function and geometric properties for a closed zero-dimensional subscheme of a quadric \(Q\subset {\mathbb{P}}^ 3\) | 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 \(\mathcal{O}/\mathcal{J}\) denote the tautological rank \(n\) bundle over the Hilbert scheme of \(n\) points on \(\mathbb{C}^2\). The author studies the \(\mathbb{C}^{*}\)-equivariant integrals over the Hilbert scheme of points on \(\mathbb{C}^2 \) of the form
\[
F(k_1,\dots,k_N)=\sum_{n=0}^{\infty}q^{n}\int \mathrm{ch}_{k_1}(\mathcal{O}/\mathcal{J})\cdots \mathrm{ch}_{k_N}(\mathcal{O}/\mathcal{J})\cdot e(T_{\beta}\mathrm{Hilb}^n \mathbb{C}^2),
\]
where \(\mathbb{C}^{*}\) acts on \(\mathbb{C}^2\) by the formula \(z\cdot (x,y)=(zx,z^{-1}y)\), \(\mathrm{ch}\) denotes the Chern character and \(T_{\beta}\mathrm{Hilb}^n \mathbb{C}^2 \) denotes the equivariantly twisted tangent bundle to the Hilbert scheme. The main result of the paper states that \(F(k_1,\dots,k_N)\) is a quasimodular form in \(q\).
These integrals generalize the so-called Nekrasov partition function appearing in mathematical physics, which was computed explicitly by N. Nekrasov and A. Okounkov.
Although the Hilbert scheme is not compact, the integrals can be defined using the equivariant localization at isolated fixed points of the torus action. The author computes explicitly the contributions of the fixed points, what allows him to obtain the combinatorial formula for \(F(k_1,\dots,k_N)\) as an infinite sum over all partitions. He then identifies the direct sum of equivariant cohomology of \(\mathrm{Hilb}^n\mathbb{C}^2\) with the Fock space, and shows that the operators of cup products with \(\mathrm{ch}_{k_N}(\mathcal{O}/\mathcal{J})\) and \(e(T_{\beta}\mathrm{Hilb}^n \mathbb{C}^2)\) can be expressed in terms of certain vertex operators \(\Gamma_{\pm(x)}\) on this space. This approach allows him to present \(F(k_1,\dots,k_N)\) as a contour integral in 2\(N\)-dimensional space and prove the quasimodularity of this function. Hilbert schemes; moduli of sheaves; vertex operators; quasimodular forms; Nekrasov partition function DOI: 10.1016/j.aim.2011.10.003 Parametrization (Chow and Hilbert schemes), Vertex operators; vertex operator algebras and related structures, Infinite-dimensional Lie (super)algebras Vertex operators and quasimodularity of Chern numbers on the Hilbert scheme | 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 reductive group scheme of type \(A\) acting on a spherical scheme \(X\). We prove that there exists a number \(C\) such that the multiplicity \(\dim \Hom(\rho, \mathbb{C} [X(F)])\) is bounded by \(C\), for any finite field \(F\) and any irreducible representation \(\rho\) of \(G(F)\). We give an explicit bound for \(C\). We conjecture that this result is true for any reductive group scheme and when \(F\) ranges (in addition) over all local fields of characteristic 0.
Different aspects of this conjecture were studied in [\textit{P. Delorme}, Trans. Am. Math. Soc. 362, No. 2, 933--955 (2010; Zbl 1193.22014); \textit{Y. Sakellaridis} and \textit{A. Venkatesh}, Periods and harmonic analysis on spherical varieties. Paris: Société Mathématique de France (SMF) (2017; Zbl 1479.22016); \textit{T. Kobayashi} and \textit{T. Oshima}, Adv. Math. 248, 921--944 (2013; Zbl 1317.22010); \textit{B. Krötz} and \textit{H. Schlichtkrull}, Trans. Am. Math. Soc. 368, No. 4, 2749--2762 (2016; Zbl 1334.22016)]. representations of groups of Lie type Representations of finite groups of Lie type, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Linear algebraic groups over finite fields Bounds on multiplicities of spherical spaces over finite fields | 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 any quiver \(Q\) we consider spherical representations \(V\) of \(Q\) such that the isomorphism class of \(V\) is a spherical variety. We suggest an approach for classifying such representations for any \(Q\) and obtain a classification for \(Q\) being an equioriented Dynkin diagram \(A_n\). In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class. spherical representations; spherical varieties; generalized complexes; quivers; Dynkin diagrams; algebras of coinvariants Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Actions of groups and semigroups; invariant theory (associative rings and algebras), Geometric invariant theory On spherical representations of quivers and generalized complexes. | 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 introduce the notion of global equivariant hom-Lie algebra. This is a Lie algebra-like structure associated with twisted derivations. We prove several results on the structure of modules of twisted derivations and how they form global equivariant hom-Lie algebras. Particular emphasis is put on examples and results in arithmetic geometry. twisted derivations; difference equations; hom-Lie algebras; arithmetic covers of schemes; t-motives Lie algebras and Lie superalgebras, Difference algebra, Group schemes Equivariant hom-Lie algebras and twisted derivations on (arithmetic) 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 Let \({\mathfrak g}\) be a complex reductive Lie algebra with adjoint group \(G\), Cartan subalgebra \({\mathfrak h}\), and Weyl group \(W\). Then \(G\) acts naturally on the algebra \({\mathcal O}({\mathfrak g})\) of polynomial functions and the ring \({\mathcal D}({\mathfrak g})\) of polynomial coefficient differential operators on \({\mathfrak g}\). Similarly \(W\) acts on \({\mathcal O}({\mathfrak h})\) and \({\mathcal D}({\mathfrak h})\). \textit{Harish-Chandra} defined an algebra homomorphism \(\delta\) between the invariant subalgebras \({\mathcal D}({\mathfrak g})^G\) and \({\mathcal D}({\mathfrak h})^W\) [Am. J. Math. 86, 534-564 (1964; Zbl 0115.108)]. \textit{N. R. Wallach} has recently shown that this map is surjective if \({\mathfrak g}\) has no simple factor of type \(E\) [J. Am. Math. Soc. 6, 779-816 (1993; Zbl 0804.22004)]. This enables him to give an easy proof of an important theorem of Harish-Chandra on invariant eigendistributions and an elegant new approach to the Springer correspondence.
Here the authors use an abstract result on fixed rings under a finite group action of rings of differential operators to give an elementary proof of Wallach's surjectivity result that works for all \({\mathfrak g}\). They are able to simplify Wallach's arguments at several crucial points. complex reductive Lie algebras; Cartan subalgebra; Weyl group; polynomial functions; differential operators; Springer correspondence T. Levasseur and J. Stafford, Invariant differential operators and an homomorphism of Harish-Chandra, J. Amer. Math. Soc. 8 (1995), 365--372. JSTOR: Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Group actions on varieties or schemes (quotients), Rings of differential operators (associative algebraic aspects) Invariant differential operators and an homomorphism of Harish-Chandra | 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 Assume that a finite group \(G\) acts linearly on finite dimensional vector spaces \(V\) and \(M\) over a field \(\mathbb{F}\) of arbitrary characteristic. Then \(G\) also acts on the coordinate algebra \(\mathbb{F}[V]\) of \(V\), and the set of \(G\)-invariant elements is the algebra of invariants \(\mathbb{F}[V]^G\). The module of covariants \(\mathbb{F}[V]^G(M)=(\mathbb{F}[V]\otimes_{\mathbb{F}}M)^G\) of type \(M\) is defined similarly.
Let \(H<G\) be a subgroup. In the paper under review the extension of invariant algebras \(\mathbb{F}[V]^G\subset \mathbb{F}[V]^H\) is studied using modules of covariants. In particular, it is shown that if \(\mathbb{F}[V]^H\) is a free \(\mathbb{F}[V]^G\)-module, then \(G\) is generated by \(H\) and reflections of \(G\), where a \(g\in G\) is called reflection if it fixes a linear hyperplane of \(V\) point-wise. A partial case of this result for \(H=\{e\}\) is a classical theorem by Serre. modular invariant theory; algebra of invariants; module of covariants; Hilbert series Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients) Modules of covariants in 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 We cite the publisher's text and from the authors' introduction:
``This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.''
From the introduction: ``In this book, we will limit ourselves to crystals associated to finite-dimensional Lie algebras, omitting the important topic of crystals of representations of infinite-dimensional Lie algebras. Within this limited scope, we have tried to prove the essential facts using combinatorial methods. The facts one wants to prove are as follows.
Given a reductive complex Lie group \(G\), there is an associated weight lattice \(\Lambda\)
with a cone of dominant weights. Given a dominant weight \(\lambda\), there is a unique
irreducible representation of highest weight \(\lambda\). There are two operations on these that we are particularly concerned with: tensor product of representations and branching, or restriction, to Levi subgroups.
In the theory of crystal bases, one starts with the same weight lattice and cone of dominant weights. Instead of a representation, one would like to associate a special crystal to each dominant weight. If the re presentation is irreducible, the crystal should be connected. There may be many connected crystals with a given highest weight, but it turns out that there is one particular one that we call normal. We think of this as the ``crystal of the representation''. More generally, a crystal that is the disjoint union of such crystals, is to be considered normal.
The operations of tensor product and Levi branching from representation theory also make sense for crystals. The usefulness of the class of normal crystals is that the decomposition of a crystal into irreducibles with respect to these operations is again normal. Moreover, the decomposition of a representation obtained by tensoring representations or branching a representation to a Levi subgroup gives the same multiplicities as the decomposition of the tensor product or Levi branching of the corresponding normal crystals into irreducibles.
There are several ways of defining normal crystals. Kashiwara and Littelmann gave two different definitions, which then were shown to be equivalent. We give yet another definition of normal crystals, based on two key ideas: Stembridge crystals and virtual crystals (Kashiwara; Baker). For the simply-laced Cartan types, Stembridge showed how to characterize the normal crystals axiomatically. This is subject of Chapter 4. This approach does not work as well for the non-simply-laced types, but for these, there is a way of embedding certain crystals into crystals of corresponding simply-laced types. For example, to construct a normal \(\mathrm{Sp}(2r)\) crystal (for the non-simply-laced Cartan type \(C_r)\) first one constructs a \(\mathrm{GL}(2r)\) crystal (for the simply-laced Cartan type \(A_{2r-1})\). Then one finds the symplectic crystal as a ``virtual crystal'' inside the \(\mathrm{GL}(2r)\) one. This way, one may reduce many problems about crystals to the simply-laced case, including the construction of the normal crystals (see Chapter 5). \dots
A direct approach to the refined Demazure character formula seems difficult, even in the simply-laced case armed with the Stembridge axioms. Instead, what works is to construct the Demazure crystals inside the crystal \(\mathcal B_\infty\). This is an infinite crystal that contains a copy of \(\mathcal B_\lambda\) for every dominant weight \(\lambda\). We construct the Demazure crystals in \(\mathcal B_\infty\) in Chapter 12 and then deduce the properties of their counterparts in \(\mathcal B_\lambda\). After this, we are able to finish the proof of the Demazure character formula for crystals, and thereby establish the relationship with representations as shown in Chapter 13.
The infinite crystal \(\mathcal B_\infty\) is itself a remarkable combinatorial object. It is, in a
sense, the crystal of a representation, albeit an infinite-dimensional one, the Verma module with weight 0. As we discover in the proof of the refined Demazure character formula, a clear understanding of \(\mathcal B_\infty\) may be the key to the finite normal crystals. Therefore after we prove the Demazure character formula we investigate \(\mathcal B_\infty\) in more depth. Chapter 14 considers the \(*\)-involution of \(\mathcal B_\infty\), a self-map of order two with remarkable properties that was studied by Lusztig and Kashiwara.
Then in Chapter 15, we turn to another combinatorial realization of B\(\mathcal B_\infty\) that arose from Lusztig's canonical bases in the theory of quantum groups. In considering the Lusztig realization of \(\mathcal B_\infty\), we begin to see an important theme in this
subject: how combinatorial maps that arise in crystal base theory are tropicalizations of algebraic maps. The algebraic maps that admit tropicalizations are those constructed using addition, multiplication and division, but never subtraction. (These are the operations that preserve the positive real numbers, explaining why this topic touches on the theory of total positivity.) Tropicalization replaces the algebraic operations of addition, multiplication and division by piecewise linear ones, and an algebraic map corresponds to a combinatorial one given by piecewise-linear maps. Conversely, given a piecewise-linear map, we may seek an algebraic lifting, an algebraic map having the given piecewise-linear one as its tropicalization. If this is done carefully, the algebraic liftings of various maps may fit together in a way that mirrors the crystal structure
Other topics that we discuss include the crystals of tableaux of Kashiwara and Nakashima, which give explicit models for the normal crystals in the classical Cartan types A, B, C, and D (see Chapters 3 and 6). We discuss the action of the Weyl group on the crystal and the Schützenberger-Lusztig involution.
In Chapters 7 through 10, we specialize to type A to explore various aspects of tableaux theory using crystals. In particular, we discuss Lascoux and Schützenberger's theory of the plactic monoid and prove the Littlewood-Richardson rule. In the context of this discussion we emphasize analogies between the theory of crystals and representations.
Two appendices on topics in representation theory of \(\mathrm{GL}(n,\mathbb C)\) have been provided to help the reader see these analogies. Roughly, Appendix A on Schur-Weyl duality is analogous to Chapter 8 on the plactic monoid, and the material in Appendix B on the \(\mathrm{GL}(n) \times \mathrm{GL}(m)\) duality is analogous to Chapter 9 on bicrystals and the Littlewood-Richardson rule. On the crystal side, the Robinson-Schensted-Knuth (RSK) insertion algorithm in its various forms plays the role of the Schur-Weyl and \(\mathrm{GL}(n) \times \mathrm{GL}(m)\) dualities.
We have already mentioned that an important role is played in several chapters by reduced words representing the long Weyl group element. In Chapter 15 we will see that for each such reduced word \(\mathbf i\) there is a map \(v\mapsto v_{\mathbf i}\) of the crystal into \(\mathbb N^N\). We will exhibit polytopes called MV polytopes that encode the components of the vector \(v_{\mathbf i}\) in the lengths of various paths around the boundary of the polytope.
Thus reduced words for \(w_0\) are important in studying crystals. In the other direction, at least for type A, Morse and Schilling showed that crystals can be used to study the reduced words for \(w_0\) (or more generally any Weyl group element). As we have already mentioned the Schur function associated to the dominant weight or partition \(\lambda\) can be viewed as the character of the highest weight crystal of highest weight \(\lambda\) in type A. Another important class of symmetric functions are the Stanley symmetric functions (Stanley(1984)), which were introduced to study reduced expressions of symmetric group elements. Stanley symmetric functions have a positive integer expansion in terms of Schur functions. We demonstrate that these can be understood in terms of crystals by imposing a crystal structure on the combinatorial objects underlying the Stanley symmetric functions. In this
case, the insertion algorithm of Edelman and Greene (1987), which is a variant of
RSK, plays a crucial role. This is done in Chapter 10.''
This book deserves a wide readership! Kashiwara crystals; crystals of tableaux; Stembridge crystals; virtual, fundamental, normal crystals; insertion algorithms; plactic monoid; bicrystals and Littlewood-Richardson rule; crystals for Stanley symmetric functions; patterns; Weyl group action; Demazure crystals; crystals and tropical geometry; Lie algebras; representations Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Research exposition (monographs, survey articles) pertaining to combinatorics, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Combinatorial aspects of representation theory, Foundations of tropical geometry and relations with algebra, Combinatorial aspects of tropical varieties Crystal bases. Representations and combinatorics | 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 Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for \(gl(n)\), with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and extended to more general groups. These algebras are almost-graded. In this article their definition is recalled and classification and uniqueness results for almost-graded central extensions for this new class of algebras are presented. The explicit forms of the defining cocycles are given. If the finite-dimensional Lie algebra on which the Lax operator algebra is based is simple then, up to equivalence and rescaling of the central element, there is a unique non-trivial almost-graded central extension. Some results are joint work with Oleg Sheinman. Lie algebras of current type; local cocycles; central extensions; krichever- Novikov type algebras; Tyurin parameters Infinite-dimensional Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Applications of Lie algebras and superalgebras to integrable systems, Algebraic moduli problems, moduli of vector bundles, Riemann surfaces; Weierstrass points; gap sequences, Vector bundles on curves and their moduli, Relationships between algebraic curves and integrable systems, Differentials on Riemann surfaces, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Almost-graded central extensions of Lax operator 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 is an introductory survey on symplectic reflection algebras written mostly from the algebraic point of view. Here symplectic reflection algebras are introduced as deformations of orbit space singularities and this is used as motivation for most of the constructions and results of the paper. The latter include classification of symplectic singularities admitting symplectic resolutions; category \(\mathcal O\), the KZ-functor, highest weight covers and finite Hecke algebras; derived equivalences for some quiver varieties; the construction of quantizations of Hilbert schemes of points on the plane and associated geometric interpretations of representations of symplectic reflection algebras. Special cases of deformed preprojective algebras, rational Cherednik algebras and the symmetric group are discussed. symplectic reflection algebras; orbit space singularities; category \(\mathcal O\); KZ-functors; Hecke algebras; derived equivalences; Hilbert schemes; rational Cherednik algebras; deformations Gordon, I.G., Symplectic reflection algebras, (Trends in representation theory of algebras and related topics, EMS ser. congr. rep., (2008), Eur. Math. Soc. Zürich), 285-347 Associative rings and algebras arising under various constructions, Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations, Parametrization (Chow and Hilbert schemes), Group actions on varieties or schemes (quotients), Deformations of associative rings Symplectic reflection 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 We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory. finitely generated algebras; categories of finite-dimensional modules; module varieties; irreducible components; representations of quivers; deformations Crawley-Boevey, W.; Schröer, J., Irreducible components of varieties of modules, J. Reine Angew. Math., 553, 201-220, (2002) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients) Irreducible components of varieties of 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 We discuss graded subalgebras of the tensor algebra \(T(V)\) that are closed under permutations of the factors in homogeneous polynomials; these we call S-algebras. Examples are invariant algebras with respect to some subgroup of \(\text{GL}(V)\). We prove an S-version of the classical Endlichkeitssatz in the case of noncommutative invariants by using correspondences between S-algebras and certain commutative rings. We also discuss some generating functions, analogous to Hilbert series, associated to S-algebras. graded subalgebras; tensor algebras; S-algebras; invariant algebras; Endlichkeitssatz; noncommutative invariants; generating functions; Hilbert series Automorphisms and endomorphisms, Brauer groups of schemes, Vector and tensor algebra, theory of invariants, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Representations of finite symmetric groups, Combinatorial aspects of representation theory, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) S-algebras and 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 Grothendieck's anabelian conjecture on the pro-\(l\) fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This is done by estimating effectively the Galois equivariant automorphism group of the pro-\(l\) braid group on the curve. The process of the proof involves the complete determination of the groups of graded automorphisms of the graded Lie algebras associated to the weighted filtration of the braid groups on Riemann surfaces. Galois representation; anabelian geometry; braid group; pro-\(l\) fundamental groups; groups of graded automorphisms; graded Lie algebras DOI: 10.1090/S0002-9947-98-02038-8 Coverings in algebraic geometry, Braid groups; Artin groups, Separable extensions, Galois theory, Fundamental groups and their automorphisms (group-theoretic aspects) Galois rigidity of pro-\(l\) pure braid groups of algebraic curves | 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 articles of this volume (all in Japanese) will not be reviewed individually.]
Contents: \textit{Akira Fujiki}, Moduli spaces for a polarized family of compact Kähler manifolds, 1-16; \textit{Makoto Namba}, Problems on equivalence relations and automorphism groups of certain compact Riemann surfaces, 17-28; \textit{Hiroaki Terao}, Logarithmic vector fields and a generalized Coxeter equality, 29-37; \textit{Seiichiyo Kamiya}, Analytic K- theory, 38-63; \textit{Hirotaka Fujimoto}, On meromorphic maps into \(P^ N({\mathbb{C}})\), 64-82; \textit{Nobuo Sasakura}, H. Cartan's theorems A, B, with some structures, 83-92; \textit{Michitake Kita}, On Riemann-Hilbert problems in several complex variables, 93-98; \textit{Tasiaki Kóri}, A duality theorem for local cohomology on the boundary of pseudoconvex region, 99- 113; \textit{Toshio Kohno}, Determination of the rational homotopy type of open varieties by de Rham homotopy, 114-124; \textit{Mitsuyoshi Kato}, On combinatorial space forms, 125-131. Analytic varieties; Proceedings; Symposium; Kyoto; RIMS; pseudoconvex domain; analytic varieties; Moduli spaces; compact Kähler manifolds; automorphism groups of certain compact Riemann surfaces; Logarithmic vector fields; Coxeter equality; Analytic K-theory; meromorphic maps into \(P^ N({\mathbb{C}})\); H. Cartan's theorems; Riemann- Hilbert problems; duality theorem; pseudoconvex region; rational homotopy type of open varieties; de Rham homotopy; combinatorial space forms Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to algebraic topology, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings of conferences of miscellaneous specific interest, Duality theorems for analytic spaces, Complex-analytic moduli problems, Holomorphic mappings and correspondences, Compact complex surfaces, Complex Lie groups, group actions on complex spaces, Compact Riemann surfaces and uniformization Various problems on analytic varieties. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, February 4-7, 1980 | 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 presents a review of some classic and recent results on the subgroups of the Cremona group \(Cr(2)\) of birational transformations of the complex projectve plane \({\mathbb P}^2({\mathbb C})\). By a result of M. Noether since 1871, \(Cr(2)\) is generated by \(PGL(3)\) and the Cremona quadratic involution. Later in the 1980's \textit{M.Kh. Gizatullin} and \textit{V. A. Iskovskikh} give more concrete descriptions of \(Cr(2)\) by generators and relations, see [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 05, 909--970 (1982; Zbl 0509.14011); Russ. Math. Surv. 40, No. 5, 231--232 (1985); translation from Usp. Mat. Nauk 40, No. 5(245), 255--256 (1985; Zbl 0613.14012)]. However the study of the structure of \(Cr(2)\), in particular the description of the subgroups of \(Cr(2)\), is still in progress. Recently the classification of the finite subgroups of the Cremona group \(Cr(2)\) has been completed by \textit{I. V. Dolgachev} and \textit{V. A. Iskovskikh}, [see Algebra, arithmetic, and geometry. In honor of Yu. I. Manin on the occasion of his 70th birthday. Vol. I. Boston, MA: Birkhäuser. Progress in Mathematics 269, 443--548 (2009; Zbl 1219.14015)], but the complete classification of the infinite groups of \(Cr(2)\) is still unknown. This review discusses the progress in the study of the infinite subgroups \(G\) of \(Cr(2)\) especially the recent results of Cantat regarding the infinite subgroups of finite type in \(Cr(2)\), see [\textit{S. Cantat}, Ann. Math. (2) 174, No. 1, 299--340 (2011; Zbl 1233.14011)]. Cremona group; groups of finite type C. Favre, ''Le groupe de Cremona et ses sous-groupes de type fini,'' in Séminaire Bourbaki, Paris: Soc. Math. France, 2010, vol. 332, p. exp. no. 998, vii, 11-43. Birational automorphisms, Cremona group and generalizations The Cremona group and its subgroups of finite 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 an algebraically closed field and let \(\Lambda\) be a finite dimensional associative \(k\)-algebra with unit. By definition, a \(d\)-dimensional \(\Lambda\)-module is the vector space \(k^d\) endowed with a multiplication by \(\Lambda\). Making use of a purely algebraic description of the partial order \(\leq_{\text{deg}}\) [\textit{G. Zwara}, Compos. Math. 121, No. 2, 205-218 (2000; Zbl 0957.16007)], the author develops a method of decomposition of a given degeneration of \(\Lambda\)-modules into finer ones. He then applies this method to the case when \(\Lambda\) has a directed Auslander-Reiten quiver and gives a new proof of the equivalence of the orders \(\leq_{\text{deg}}\) and \(\leq_{\text{ext}}\) for such algebras. In fact, this result has been earlier obtained in a different context by \textit{K. Bongartz} [in Algebras and modules I. Workshop on representations of algebras and related topics, Trondheim, Norway, 1996. CMS Conf. Proc. 23, 1-27 (1998; Zbl 0915.16008)]. degenerations of modules; partial orders; directed Auslander-Reiten quivers; indecomposable modules; almost split sequences; vector bundles; finite-dimensional algebras Representations of associative Artinian rings, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Group actions on varieties or schemes (quotients), Representation type (finite, tame, wild, etc.) of associative algebras, Finite rings and finite-dimensional associative algebras Decompositions of degenerations | 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 define the notion of a dimension \(d\) invariant of an algebraic group \(G\) with values in a cyclic module \(M\) over a field as a natural transformation of functors \(G\to M_d\). We prove that the group \(\text{Inv}^d(G,M)\) of all invariants of dimension \(d\) is canonically isomorphic to the unramified group \(A^0(G,M_d)\). The group of cohomological invariants of dimension \(\leq 2\) is computed. We show that Rost's dimension 4 cohomological invariant of the group \(G=\text{SL}_1(A)\) for a biquaternion division algebra \(A\) is the only nontrivial invariant of \(G\) of the smallest dimension. We generalize Rost's invariant to the case of arbitrary central simple algebras of dimension 16. dimension \(d\) invariants; algebraic groups; natural transformations; unramified groups; groups of cohomological invariants; Rost's invariants; central simple algebras Merkurjev A.S.: Invariants of algebraic groups. J. Reine Angew. Math. 508, 127--156 (1999) Finite-dimensional division rings, Cohomology theory for linear algebraic groups, Higher symbols, Milnor \(K\)-theory, Group actions on varieties or schemes (quotients), Brauer groups of schemes Invariants 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 Let K be a totally real algebraic number field of degree \(n>1\). The group \(\Gamma =SL_ 2({\mathfrak O}_ K)\) acts on \({\mathbb{H}}^ n\) by the modular substitution. For a subgroup G of Aut(K/\({\mathbb{Q}})\), acting on \({\mathbb{H}}^ n\) as permutations of the coordinates, denote by \({\hat \Gamma}\) the composite of G and \(\Gamma\). The variety \(X:={\mathbb{H}}^ n/{\hat \Gamma}\) can be compactified by adding a finite number of cusps. The non-singular model \(\tilde X\) of this compactification is called Hilbert modular variety if \(G=\{1\}\) and symmetric Hilbert modular variety if \(G\neq \{1\}.\)
The author shows that \(\tilde X\) is of general type except for a finite number of fields K. In particular all Hilbert modular varieties of dimension larger than six and all symmetric Hilbert modular varieties of dimension larger than nine are of general type. - Denote by \(X_ 0\) the complement of the fixed points in X. To each (symmetric) Hilbert modular form of weight k corresponds a section of \(H^ 0(X_ 0,\Omega^{\otimes k})\). In his proof the author uses a criterion of Tai to show that for \(n>2\), except for a finite number of fields K, these sections extend to all of X. Then he discusses their extendability to \(\tilde X\) and shows, that for large N, the dimension of \(H^ 0(\tilde X,\Omega^{\otimes N})\) growths like \(N^ n\), in other words, the Kodaira dimension is equal to n and \(\tilde X\) is of general type. In the case of \(n=2\), the author refers to results of \textit{F. Hirzebruch} and \textit{D. Zagier} [in Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 43-77 (1977; Zbl 0354.14011)] in the case of Hilbert modular varieties and to the thesis of \textit{D. Bassendowski} [''Klassifikation Hilbertscher Modulflächen'', Bonn, Math. Schr. 163 (1985)] in the case of symmetric Hilbert modular varieties. Hilbert modular forms; symmetric Hilbert modular variety of general; type; Kodaira dimension S. Tsuyumine, ``On the Kodaira dimensions of Hilbert modular varieties'', Invent. Math.80 (1985) no. 2, p. 269-281 Families, moduli, classification: algebraic theory, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special surfaces On the Kodaira dimensions of Hilbert modular 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 Consider the algebraic subset \({\mathcal L}\subset\bigwedge^{2}\mathfrak{g}^\ast\otimes\mathfrak{g}= \mathbb{R}^{(n^3-n^2)/2}\) of all Lie brackets on a fixed real \(n\)-dimensional vector space \(\mathfrak{g}\). Each \(\mu\in{\mathcal L}\) defines a connected and simply connected Lie group \(G_\mu\) endowed with the left invariant Riemannian metric induced by a fixed inner product \(<.,.>\) on \(\mathfrak{g}\). The set of all left invariant Riemannian metrics on \(G_\mu\) is identified with the orbit \({\mathcal O}_\mu=\text{Gl}(n,\mathbb{R}).\mu\). By degeneration \(\mu\rightarrow\lambda\) of one Lie algebra \(\mu\in{\mathcal L}\) to another Lie algebra \(\lambda\in{\mathcal L}\) is meant that \(\lambda\) belongs to the closure of \({{\mathcal O}_\mu}\) in \(\mathbb{R}^{(n^3-n^2)/2}\).
On \({\mathcal L}\) a functional \(F\) is defined by \(\mu\rightarrow F(\mu)=\text{tr} \mathbf{R}_\mu^2\), where \(\mathbf{R}_\mu\) is a symmetric transformation defined in terms of the Ricci curvature operator of \(\mu\). Then, the limits of the flow lines of the gradient flow of \(F\) are degenerations of their starting points. This property of \(F\) is used to obtain interesting relations between the space of all left invariant metrics on \(n\)-dimensional connected and simply connected Lie groups and critical points of \(F:\)
(1) \(G_\mu\) has only one left invariant Riemannian metric up to isometry and scaling if and only if the only possible degeneration of \(\mu\) is \(0\).
(2) If \(\mu\) degenerates to \(\lambda\) and \(G_\lambda\) admits a left invariant Riemannian metric satisfying a pinched curvature condition then the same holds for \(G_\mu\).
(3) The orbit \(\text{Sl}(n,\mathbb{R}).\mu\) is closed if and only if \(G_\mu\) has left invariant Riemannian metric such that its curvature tensor is a multiple of the identity.
The closed \(\text{Sl}(n,\mathbb{R})\)-orbits of \({\mathcal L}\) are classified. Explicit 1-parameter families of mutually non-isometric Einstein solvmanifolds of dimensions 10 and 11 respectively are derived from curves of closed orbits of a representation of \(\bigwedge^2\text{Sl}(m,\mathbb{R})\otimes\text{Sl}(n,\mathbb{R})\). variety of Lie algebras; degeneration of Lie algebras; closed orbits of Lie algebras; left invariant Riemannian metrics J. Lauret, Degenerations of Lie algebras and geometry of Lie groups. \textit{Differential Geom}. \textit{Appl}. \textbf{18} (2003), 177-194. MR1958155 Zbl 1022.22019 Lie algebras of Lie groups, Differential geometry of homogeneous manifolds, Group actions on varieties or schemes (quotients), Manifolds of metrics (especially Riemannian) Degenerations of Lie algebras and geometry of Lie 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 explores the relationship between crystalline cohomology and the first author's theory of zeta-functions for function fields over finite fields. This relationship is based on the fact that, 1) the unit- root piece of the classical L-series of the Teichmüller character is the inverse characteristic polynomial of the Frobenius on the p-adic Tate-module; and 2) the reduction modulo p of this unit root piece is the reduction modulo a prime of the function field of the ''polynomial part'' of the above zeta-function.
With this tool, the authors are able to describe the variation of the p- adic Tate-modules as the prime of the function field varies. Moreover, a converse to the ''Herbrand'' criterion for function fields is established. Finally, a criterion for cyclicity of the Galois components of p-class groups of ''cyclotomic function fields'' is given. This criterion amounts to having the Frobenius have at most one non 1-unit root on the appropriate Tate-module.
In another paper, the first author has given a second criterion for cyclicity based on the values at the positive integers of his zeta- function. In particular, this second criterion implies the one given in the paper being reviewed. Moreover, because of the very simple form of the special values involved, this second criterion can be shown for all primes and some components. It is of interest that the components that arise in this manner are precisely those components that arise in the experimental evidence for a functional equation associated to these zeta- functions. Stickelberger element; Galois module structure; Gras conjecture; Drinfeld modules; Herbrand criterion; crystalline cohomology; zeta-functions for function fields over finite fields; L-series; Teichmüller character; characteristic polynomial of the Frobenius; p-adic Tate-module; p-class groups; cyclotomic function fields; 1-unit root Goss, D., Sinnott, W.: Class-groups of function fields. Duke Math. J. 52(2), 507--516 (1985). http://www.ams.org/mathscinet-getitem?mr=792185 Arithmetic theory of algebraic function fields, \(p\)-adic cohomology, crystalline cohomology, Algebraic functions and function fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Iwasawa theory Class-groups of function fields | 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 neighbourhood of a cusp on a Hilbert modular variety over a totally real field F, \([F:{\mathbb{Q}}]=n=2k\), is of the form W/G, where
\[
W=W(d)=\{z\in {\mathbb{C}}^ n| Im (z_ j)>0,\quad \prod Im (z_ j)\geq d\},\quad d>0
\]
and \(G=G(M,V)\) denotes the following group of automorphisms of \(W:z\mapsto \epsilon z+\mu \in W\) with \(\epsilon\in V\), \(\mu\in M\). Here M is a lattice in F and V a totally positive unit group in F (both of maximal rank) with \(VM=M\). The components of \(\epsilon z+\mu\) are \(\epsilon^{(j)}z_ j+\mu^{(j)}\) with the conjugate numbers \(\epsilon^{(j)}\), \(\mu^{(j)}\) of \(\epsilon\),\(\mu\). The boundary \(X=\partial W/G\) of W/G is an n-torus bundle over the (n-1)- torus \({\mathbb{R}}^{n-1}/V\) with fiber \({\mathbb{R}}^ n/M\). To the torus bundle X one can associate as follows a rational number \(\sigma\) (X), called the signature defect of X. The coordinates \(x_ 1,y_ 1,...,x_ n,y_ n (z_ j=x_ j+iy_ j)\) on \({\mathbb{C}}^ n\) induce a flat connection \(\nabla^ L\) and a Riemannian metric g on the tangent bundle of X. All Pontryagin and Stiefel-Whitney numbers of X vanish, so there is a compact oriented manifold Y with \(\partial Y=X\). Let \(L_ k(p_ 1,...,p_ k)\in H^{4k}(Y,X)\) be the Hirzebruch L-polynomial in the relative Pontryagin classes \(p_ j\) of Y. The signature defect is then defined as the difference
\[
\sigma(X):=L_ k(p_ 1,...,p_ k)[Y,X]- sign(Y).
\]
Applying the signature theorem and the Novikov additivity of the signature one sees that \(\sigma\) (X) does not depend on the particular choice of Y. In the paper under review the authors prove the equation \(\sigma(X)=L(0)\), where L(0) is the value of the L-function L(s) at \(s=0\), given for \(Re(s)>1\) by \(L(s)=\sum N(\mu)| N(\mu)|^{- 1-s}\), where \(\mu\neq 0\) runs over a system of representatives for \(M^*/V\) with the dual lattice \(M^*\) of M with respect to the trace in F. This equation has been proved in the case \(n=2\) and then in full generality conjectured by \textit{F. Hirzebruch} [Enseignement Math. 19, 183-281 (1973; Zbl 0285.14007)]. Motivated by this conjecture Atiyah, Patodi and Singer introduced about ten years ago the spectral \(\eta\)- invariant and extended with the help of this invariant the Hirzebruch signature theorem to compact oriented Riemannian manifolds with boundary [\textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer}, Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008)]. Let A be the elliptic self-adjoint operator \(A=\pm(*d-d*)\) acting on forms of even degree, \(A:\Gamma(\wedge^{ev}T^*X)\to \Gamma(\wedge^{ev}T^*X)\). Denote by \(\lambda\) the eigenvalues of A. The \(\eta\)-invariant \(\eta_ A(0)\) of A is defined as the value at \(s=0\) of the meromorphic function \(\eta_ A(s)\) given for \(Re(s)>>0\) by \(\eta_ A(s)=\sum \lambda | \lambda |^{-1-s}\). Note that \(\eta_ A(0)\) depends on the chosen connection on TX. In the case of the Levi-Cività connection on TX (with respect to the metric g) the equation \(\eta_ A(0)=\sigma(X)\) has been proved in the cited paper. Here the authors prove first that \(\eta_ A(0)\) does not change by replacing the Levi-Cività connection by the flat connection \(\nabla^ L\). In the second step the authors prove the equality \(\eta_ A(0)=L(0)\), where \(\eta_ A\) is defined with respect to the flat connection \(\nabla^ L\). This part constitutes in fact the bulk of the paper. The basic idea here is to unwind the operator A along the fibers of X. In this way one gets for \(Re(s)>>0\) the formula
\[
\eta_ A(s)=\sum N(\mu)| N(\mu)|^{-1-s/n}\eta(h(\mu),s),\quad 0\neq \mu \in M^*/V
\]
with \(h=h(\mu)=| N(\mu)|^{-1/n}\) and the \(\eta\)-function \(\eta\) (h,s) of certain family of operators \(B_ h:L^ 2({\mathbb{R}}^{n-1};\wedge^{ev}{\mathbb{R}}^{2n- 1})\circlearrowright\). This gives the representation
\[
\eta_ A(s)=\eta(\alpha,s)L(s/n)+\gamma(s),
\]
\[
\gamma(s)=\sum N(\mu)| N(\mu)|^{-1-s/n}(\eta(h(\mu),s)-\eta(\alpha,s))
\]
with a suitable constant \(\alpha>h(\mu)\) for all \(\mu\). The crucial point in the proof is to show \(\gamma(0)=0\) and \(\eta(\alpha,0)=1\). Here the authors have to cope with substantial difficulties. To this end they use the heat- equation method and the Feynman-Kac representation of the heat kernel to estimate the integral representation of \(\eta\) (\(h(\mu)\),s)-\(\eta\) (\(\alpha\),s). The essential point, however, seems to be algebraic in nature, namely the vanishing of certain traces (cf. Lemma 10.3). eta invariants; signature defects of cusps; special values of L- functions; cusp on Hilbert modular variety; lattice in totally real field; Hirzebruch L-polynomial; Hirzebruch signature theorem; flat connection; Feynman-Kac representation of the heat kernel Atiyah, MF; Donnelly, H; Singer, IM, Eta invariants, signature defects of cusps, and values of \(L\)-functions, Ann. Math., 118, 131-177, (1983) Heat and other parabolic equation methods for PDEs on manifolds, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special surfaces, Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Global ground fields in algebraic geometry, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Characteristic classes and numbers in differential topology, Quaternion and other division algebras: arithmetic, zeta functions, Totally real fields, Connections (general theory) Eta invariants, signature defects of cusps, and values of 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 Let \(k\) be an algebraically closed field. A finite braided tensor category \(\mathscr{D}\) over \(k\) is \textit{non-degenerate} if the full subcategory \(\mathscr{D}'\) of the objects in \(\mathscr{D}\) that centralise every other object (with respect to the braiding) is equivalent to \(\operatorname{Vec}_k\) (via the usual functor \(\operatorname{Vec}_k \to \mathscr{D}\)). A \emph{minimal extension} of a finite Tannakian category \(\mathscr{E}\) over \(k\) is a non-degenerate finite braided tensor category \(\mathscr{D}\) over \(k\), containing \(\mathscr{E}\) as a maximal symmetric tensor subcategory. In the case when \(k\) is of characteristic zero, the minimal extensions of a given \(\mathscr{E}\) are classified in [\textit{V. Drinfeld} et al., ``Group-theoretical properties of nilpotent modular categories'', Preprint, \url{arXiv:0704.0195}; Sel. Math., New Ser. 16, No. 1, 1--119 (2010; Zbl 1201.18005)], and [\textit{T. Lan} et al., Commun. Math. Phys. 351, No. 2, 709--739 (2017; Zbl 1361.81184)], and the paper under review extends this classification to arbitrary characteristic.
From now on, \(k\) is of positive characteristic. Let \(G\) be a finite group scheme over \(k\). Denote its coordinate algebra by \(\mathcal{O}(G)\), and the category of finitely generated \(\mathcal{O}(G)\)-modules by \(\operatorname{Coh}(G)\). Given a normalised 3-cocycle \(\omega\in H^3(G,\mathbb{G}_{\mathrm{m}})\), the tensor category \(\operatorname{Coh}(G,\omega)\) is given by \(\operatorname{Coh}(G)\) as its underlying abelian category, but with associativity constraint twisted by \(\omega\). It is then a fact (Lemma 2.5), that the centre \(\mathscr{Z}(\operatorname{Coh}(G,\omega))\) of \(\operatorname{Coh}(G,\omega)\) is canonically a minimal extension of \(\operatorname{Rep}(G)\). For the definition of the centre of a monoidal category, see, e.g., Definition 7.13.1 in [\textit{P. Etingof} et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001)].
The main contributions of the paper are laid out in three steps. Firstly, the author proves in Theorem 1.1 that every minimal extension of \(\mathscr{E}=\operatorname{Rep}(G)\) is obtained as \(\mathscr{Z}(\operatorname{Coh}(G,\omega))\) for some \(\omega\in H^3(G,\mathbb{G}_{\mathrm{m}})\), in the way indicated above. Secondly, it is shown in Corollary~1.2 that the construction in Theorem~1.1 induces a natural bijection. To state this correspondence, fix a non-degenerate finite braided tensor category \(\mathscr{D}\). There is a natural equivalence relation on the set of triples \((G,\omega,F)\), where \(G\) is a finite \(k\)-group scheme, \(\omega\in H^3(G,\mathbb{G}_{\mathrm{m}})\) and \(F \colon \mathscr{D} \to \mathscr{Z}(\operatorname{Coh}(G,\omega))\) is a braided tensor equivalence. Corollary 1.2 then says that the set of equivalence classes of such triples is naturally in bijection with the set of maximal symmetric tensor subcategories of \(\mathscr{D}\). Lastly, the set of equivalence classes of minimal extensions of \(\mathscr{E}\) up to equivalence naturally form a group, and the author shows in Corollary 1.3 that this group is isomorphic to \(H^3(G,\mathbb{G}_{\mathrm{m}})\).
The proof of Theorem 1.1 splits roughly into two steps. First, a braided tensor category \(\mathscr{C}\) is constructed, together with a braided tensor equivalence \(F\colon \mathscr{D} \to \mathscr{Z}(\mathscr{C})\), exhibiting \(\mathscr{D}\) as the centre of this \(\mathscr{C}\). The category \(\mathscr{C}\) is defined as the \(G\)-de-equivariantisation of \(\mathscr{D}\), namely, as the category of \(\mathcal{O}(G)\)-modules in \(\mathscr{D}\). The functor \(F\) is induced by the free module functor (Corollary 3.4). The second step, is then to construct a \textit{quasi}-tensor equivalence \(\tilde{\Phi}\colon \mathscr{C} \to \operatorname{Coh}(G)\) (Corollary 3.10). The transportation of the associativity constraint from \(\mathscr{C}\) to \(\operatorname{Coh}(G)\) along \(\tilde{\Phi}\) then corresponds to the sought after element cocycle \(\omega \in H^3(G, \mathbb{G}_{\mathrm{m}})\). finite tensor categories; non-degenerate braided tensor categories; Tannakian categories; finite group schemes; restricted Lie algebras Tannakian categories, Braided monoidal categories and ribbon categories, Group schemes, Lie algebras and Lie superalgebras Minimal extensions of Tannakian categories 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 [For part I, see the preceding review Zbl 0890.12005.]
Part II is concerned with applications of classical invariant theory to statistical physics and to theta functions. The main theorem in chapter 2 is stated as follows: For a partition function \(\xi(s)= \sum^\infty_{l=1} \gamma_l s^{dl}\) satisfying \(\gamma_t\geq 0\) \((l\geq 1)\) and \(\alpha>0\), the \(2n\)-apolar of \(\xi (s)\)
\[
A_{2n} \bigl(\xi(s), \xi(s)\bigr) =s^{2n} \sum^{2n}_{l=0} (-1)^k {2n \choose k} \left({d \over ds} \right)^{2n-k} \xi(s) \left({d \over ds} \right)^k \xi(s)
\]
has the expansion
\[
A_{2n} \bigl(\xi(s), \xi(s)\bigr) =\sum^\infty_{l=2} \beta_{n,l} s^{-\alpha l}
\]
such that \(\beta_{n,l} \geq 0\) \((l\geq 2)\). This means, for a given partition function \(\xi(s)\) with nonnegative relative probabilities, we construct a sequence of partition functions \(A_{2n} (\xi(s), \xi(s))_{n\geq 1}\) with the same properties, which may be considered a sequence of symbolic higher derivatives of \(\xi(s)\).
The main theorem in chapter 3 is stated as follows: For given theta functions \(\varphi_1(z)\) and \(\varphi_2(z)\) of level \(n_1\) and \(n_2\), respectively, in \(g\) variables \(z=(z_1, z_2, \dots, z_g)\), the \(r=(r_1,r_2, \dots, r_g)\)-apolar
\[
A_r \bigl(\varphi_1(z), \varphi_2(z)\bigr) =\sum_{0\leq j\leq r-j} {(-1)^{|j|} \over n_1^{|j|} n_2^{|r-j|}} {r\choose j} \left({\partial \over \partial z} \right)^j \varphi_1(z) \left({\partial \over\partial z} \right)^{r-j} \varphi_2(z)
\]
is a theta function of level \(n_1+n_2\), and
\[
\begin{multlined} \left({\partial\over \partial z} \right)^h \varphi_1(z) \left({\partial \over\partial z} \right)^k \varphi_2(z)= \sum_{0\leq j\leq h+k} \sum_l(-1)^{|l|} n_1^{|l|} n_2^{|j-l|} {h\choose l} {k\choose j-l} \\ {(-1)^{|h|} (n_1 n_2)^{|h+k-j |} \over (n_1+n_2)^{|h+k|}} \left({\partial \over\partial z} \right)^j A_{h+k-j} \bigl(\varphi_1(z), \varphi_2(z)\bigr). \end{multlined}
\]
{}. differential algebra; decomposition of differential polynomials; apolars; invariant theory; theta functions; statistical partition function Morikawa, H, On differential polynomials II, Nagoya Math. J, 148, 73-112, (1997) Differential algebra, Theta functions and abelian varieties, Exact enumeration problems, generating functions, Equilibrium statistical mechanics On differential polynomials. 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 In this Bourbaki seminar exposé, the author discusses the proof given by \textit{M. Haiman} [J. Am. Math. Soc. 14, 941--1006 (2001; Zbl 1009.14001)] of the Macdonald's positivity conjecture obtained via the solution of the \(n!\)-conjecture of \textit{A. Garsia} and \textit{M. Haiman} [Proc. Nat. Acad. Sci. U.S.A. 90, 3607--3610 (1993; Zbl 0831.05062)].
In 1988 I. G. Macdonald defined symmetric functions \({\tilde H}_{\lambda}(X)\) with coefficients in \({\mathbb Q}(q,t)\) (\(\lambda \) a partition of a positive integer \(n\)) unifying the theory of Hall-Littlewood and Jack polynomials and conjectured that the coefficients of their expansions through Schur functors are polynomials in \(q,t\) with coefficients positive integers. In order to prove this positivity conjecture, one should construct, for each partition \(\lambda \), a bigraded representation of the symmetric group \(S_n\) whose bigraded Frobenius character is \({\tilde H}_{\lambda}(X)\).
In 1991, Garsia and Haiman, inspired by similar constructions for \(q\)-Kostka polynomials [cf. \textit{A. Garsia} and \textit{C. Procesi}, Adv. Math. 94, 82--138 (1992; Zbl 0797.20012)], proposed such a construction and reduced the proof of the positivity conjecture to the proof of two facts. One of these facts is the \(n!\)-conjecture : if \(R = {\mathbb C}[x_1,\dots ,x_n;y_1\) \(,\dots ,y_n]\) and if \(D_{\lambda}\) is the determinant of the \(n\times n\) matrix whose \(k\)th-column is the transpose of \((x_ky_k,\dots ,x_ky_k^{{\lambda}_1},\dots , x_k^iy_k,\dots ,x_k^iy_k^{{\lambda}_i},\dots )\) then the \({\mathbb C}\)-vector subspace of \(R\) spanned by all the partial derivatives of \(D_{\lambda}\) has dimension \(n!\).
First, the author explains the way Haiman reduces the proof of the \(n!\) conjecture to the following statement about the Hilbert scheme \(H_n\) of 0-dimensional subschemes of length \(n\) of \({\mathbb C}^2\): the reduced fibre product \(X_n\) of the canonical morphisms \(H_n\rightarrow ({\mathbb C}^2)^n/S_n\) and \(({\mathbb C}^2)^n\rightarrow ({\mathbb C}^2)^n/S_n\) is Cohen-Macaulay, Gorenstein. Then, he shows how one reduces the proof of this fact to the following statement: consider the polygraph \(Z(n,\ell )\subset ({\mathbb C}^2)^n\times ({\mathbb C}^2)^{\ell}\) defined as the union of the graphs of all linear maps \(({\mathbb C}^2)^n\rightarrow ({\mathbb C}^2)^{\ell}\) having as components canonical projections \(({\mathbb C}^2)^n\rightarrow {\mathbb C}^2\). Then the coordinate ring \(R(n,\ell )\) of \(Z(n,\ell )\) is a free module on the polynomial ring \({\mathbb C}[y_1,\dots ,y_n]\) (where \(x_i,y_i\), \(i=1,\dots ,n\) are the coordinates on \(({\mathbb C}^2)^n\)). The proof of this last fact is referred to M. Haiman's paper (where it occupies more then 30 pages).
In a final paragraph, the author mentions that this method allowed \textit{M. Haiman} [Invent. Math. 149, 371--407 (2002; Zbl 1053.14005)] to solve other conjectures on diagonal harmonics. The proof of these (combinatorial) conjectures uses even deeper geometrical results like the Lefschetz fixed point formula of Atiyah-Bott and \textit{T. Bridgeland's, A. King's} and \textit{M. Reid's} [J. Am. Math. Soc. 14, 535--554 (2001; Zbl 0966.14028)] interpretation of the McKay correspondence as an equivalence of derived categories. Hilbert schemes of points; symmetric functions; representations of the symmetric group Parametrization (Chow and Hilbert schemes), Symmetric functions and generalizations, Representations of finite symmetric groups On the \(n!\)-conjecture | 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 field, \(n\geq 1\) an integer and \(R\) a parabolic subgroup of \(\text{GL}_n(K)\). Then \(R\) acts on its unipotent radical \(R_u\) and on any unipotent normal subgroup \(U\) by conjugation. Each parabolic subgroup \(R\) is the group of automorphisms of a finite dimensional \(K\)-algebra \(\Lambda(d)\), \(d\in\mathbb{N}^t\), which is Morita equivalent to the path algebra \(\Lambda=K\mathbb{A}_t\) of a directed Dynkin quiver \(\mathbb{A}_t\) with \(t\) vertices. The actions of \(R\) on \(U\) are described by means of the category \(\text{Mat}(B)\) of matrices in the sense of Yu. Drozd over a subbimodule \(B\) of the radical of \(\Lambda\) viewed as a \(\Lambda\)-\(\Lambda\)-bimodule. In this way, for each \(d\in\mathbb{N}^t\) there is a parabolic group \(R(d)\) of invertible elements in \(\Lambda(d)\) and a unipotent subgroup \(U(d)\) of \(R(d)\). One of the main results of the paper asserts that there is a quasi-hereditary algebra \(\mathcal A\) such that the orbits of \(R(d)\) on \(U(d)\) correspond bijectively to the isomorphism classes of \(\Delta\)-filtered \(\mathcal A\)-modules of \(\Delta\)-dimension vector \(d\). The bijection is induced by a morphism of algebraic varieties. In particular, it preserves degenerations and families. A quiver with relations of the quasi-hereditary algebra \(\mathcal A\) is determined. Dynkin quivers; parabolic subgroups; unipotent radicals; categories of matrices; quasi-hereditary algebras; orbits; directed algebras; filtered modules Brüstle, T.; Hille, L.: Actions of parabolic subgroups in gln on unipotent normal subgroups and quasi-hereditary algebras. Coll. math. 83, 281-294 (2000) Representations of quivers and partially ordered sets, Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc., Group actions on varieties or schemes (quotients), Module categories in associative algebras, Endomorphism rings; matrix rings Actions of parabolic subgroups in \(\text{GL}_n\) on unipotent normal subgroups and quasi-hereditary 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 See the review in Zbl 0786.14029. Lie algebras; Mumford-Tate groups; Tate modules; dimension of an abelian variety Arithmetic ground fields for abelian varieties, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Abelian varieties and 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 See also Prog. Math. 160, 177-203 (1998; Zbl 0928.11023).
We give a review of the recent results concerning Siegel modular forms with respect to the paramodular groups of genus 2 and their applications to algebraic geometry and physics. Some facts mentioned below have not been published before, like
Rationality Theorem A. The moduli space of \((1,2)\)-, \((1,3)\)- or \((1,4)\)-polarized Abelian surfaces is rational.
A large portion of the results presented here were obtained by the author together with \textit{V. Nikulin} and with \textit{K. Hulek} during his stay at RIMS and were published in RIMS-preprints. polarized Abelian surfaces; arithmetic lifting; Lorentzian Kac-Moody algebras; Siegel modular forms; paramodular groups of genus 2; moduli space Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Relationship to Lie algebras and finite simple groups, Modular and Shimura varieties, Algebraic moduli of abelian varieties, classification, \(K3\) surfaces and Enriques surfaces, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Heights, Complex multiplication and moduli of abelian varieties Precious Siegel modular forms of genus two | 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 0562.00002.]
Let \(\Lambda\) be a connected hereditary finite dimensional algebra over a field k, then the category reg(\(\Lambda)\) of regular \(\Lambda\)-modules in parametrized by just one parameter curve \(X=X(\Lambda)\). In this paper, the author gives a formal definition of this curve X, by specifying its associated category coh(X) of coherent sheaves and proves in this way that X is a noncommutative projective curve (with singularities). In fact, for each \(F\in coh(X)\) one shows that both \(\Gamma\) (X,F) and \(H^ 1(X,F)\) are finite dimensional left \(\Lambda\)-modules and that the pair of these modules completely determines F, i.e. the categories coh(X) and mod(\(\Lambda)\) (of finite dimensional left \(\Lambda\)-modules) completely determine each other. If one assumes the field k to be algebraically closed then the situation is extremely simple, since one then proves that (up to isomorphism) the category coh(X) is completely determined by the singularity type of X, which is given by a Dynkin diagram \(A_{p,q}\), \(D_ n\), \(E_ 6\), \(E_ 7\), \(E_ 8\), which determines X. connected hereditary finite dimensional algebra; regular \(\Lambda \) - modules; coherent sheaves; noncommutative projective curve; singularity type; Dynkin diagram Representation theory of associative rings and algebras, Finite rings and finite-dimensional associative algebras, Singularities of curves, local rings Curve singularities arising from the representation theory of tame hereditary 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 An intriguing correspondence between four-qubit systems and simple singularity of type \(D_4\) is established. We first consider the algebraic variety \(X\) of separable states within the projective Hilbert space \(\mathbb {P}(\mathcal {H})=\mathbb {P}^{15}\). Then, cutting \(X\) with a specific hyperplane \(H\), we prove that the \(X\)-hypersurface, defined from the section \(X \cap H \subset X\), has an isolated singularity of type \(D_4\); it is also shown that this is the `worst-possible' isolated singularity that one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version, by proving that the equation of the dual variety of \(X\), which is nothing but the Cayley hyperdeterminant of type \(2 {\times} 2 {\times} 2 {\times} 2\), can be expressed in terms of the stochastic local operation and classical communication-invariant polynomials as the discriminant of the miniversal deformation of the \(D_4\)-singularity. As a consequence of this correspondence one obtains a finer-grained classification of entanglement classes of four-qubit systems. quantum information theory; entangled states; simple singularities of hypersurfaces; hyperdeterminants; Dynkin diagrams F. Holweck, J.-G. Luque and M. Planat, Singularity of type \(D_4\) arising from four qubit systems, preprint (2013), arXiv:1312.0639. Quantum information, communication, networks (quantum-theoretic aspects), Quantum coherence, entanglement, quantum correlations, Quantum state spaces, operational and probabilistic concepts, Complex surface and hypersurface singularities, Deformations of complex singularities; vanishing cycles, Multilinear algebra, tensor calculus, Homogeneous spaces and generalizations, Vector and tensor algebra, theory of invariants Singularity of type \(D_4\) arising from four-qubit systems | 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 commutative semisimple algebra over a field \(k\), and let \((V,B)\) be a finitely generated symplectic module over \(A\). The paper studies the action of the symplectic group \(\text{Sp}_A (V,B)\) on each set of \(d\)-dimensional \(k\)-subspaces of \(V\) with ``\(B'\)-rank'' \(r\), where \(B' = f \circ B\) is a \(k\)-symplectic form induced by a nonzero \(k\)-linear map \(f : A \to k\). For \(\dim_k A = 2\) or 3, are determined all the cases of \((d,r)\) whose number of orbits is finite. It is assumed here that \(k\) is infinite and that \(V\) is a faithful \(A\)-module. symplectic forms; finite dimensional commutative semisimple algebras; finitely generated symplectic modules; actions; symplectic groups; number of orbits; faithful modules Representation theory for linear algebraic groups, Exterior algebra, Grassmann algebras, Classical groups (algebro-geometric aspects) On the finiteness of certain double coset 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 Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities (of Hilbert varieties) arising from totally real cubic fields. As an example, the implementation is used to compute values of partial zeta functions associated to these cusps following Shintani's approach.
The resolutions derived here are based on the work of \textit{F. Ehlers} [Math. Ann. 127-156 (1975; Zbl 0301.14003)] and \textit{E. Thomas} and \textit{A. T. Vasquez} [Math. Ann. 247, 1-20 (1980; Zbl 0403.14005)]. fundamental domain; resolutions of cusp singularities; algorithms; totally real cubic fields; values of partial zeta-functions; Hilbert varieties Grundman, HG, Explicit resolutions of cubic cusp singularities, Math. Comp., 69, 815-825, (2000) Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Algebraic number theory computations, Global theory and resolution of singularities (algebro-geometric aspects), Modular and Shimura varieties, Cubic and quartic extensions, Zeta functions and \(L\)-functions of number fields Explicit resolutions of cubic cusp 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 main result of the paper gives the reduced Gröbner basis with respect to an admissible term of order type \(\omega\) of the intersection of a descending chain of polynomial ideals. It is also shown that the result does not hold for any admissible order. Using this and previous results of \textit{P. Gianni, B. Trager} and \textit{G. Zacharias} [J. Symb. Comput. 6, No. 2/3, 149-167 (1988; Zbl 0667.13008)], one can compute a generating set for arbitrary intersections. The authors apply their main theorem to Lagrange interpolation on algebraic sets, obtaining the ideal \(I\) of an algebraic set as the intersection of zero-dimensional ideals whose affine Hilbert functions converge towards the affine Hilbert function of \(I\). reduced Gröbner basis; admissible term order of order type \(\omega\); Lagrange interpolation on algebraic sets; ideal of an algebraic set; affine Hilbert functions Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Relevant commutative algebra, Abstract approximation theory (approximation in normed linear spaces and other abstract spaces), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Intersections of sequences of ideals generated by polynomials | 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 note one studies the hyperplane singularities of holomorphic functions of three complex variables of the form \(f=x^ 2g(y_ 1,y_ 2)\), where g has an isolated singularity. It is proved that the Milnor fibre has the homotopy type of an Eilenberg-MacLane space; the fundamental group and the homology groups are computed. homotopy type of Milnor fibre; hyperplane singularities; homology groups Complex surface and hypersurface singularities, Milnor fibration; relations with knot theory, Local complex singularities, Singularities of surfaces or higher-dimensional varieties On infinitely determined hyperplane 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 For each Sophie Germain prime \(g\geq 5\), we construct an absolutely simple polarized abelian variety of dimension \(g\) over a finite field, whose automorphism group is a cyclic group of order \(4g+2\). We also provide a description on the asymptotic behavior of prime numbers that are related to our construction. absolutely simple polarized abelian varieties over finite fields; automorphism groups; distribution of primes Abelian varieties of dimension \(> 1\), Varieties over finite and local fields, Isogeny, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Distribution of primes Absolutely simple polarized abelian varieties of odd Sophie Germain prime dimension over finite fields with maximal automorphism 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 \(E\cong (\mathbb Z/p)^r\) (\(r\geq 2\)) be an elementary Abelian \(p\)-group and let \(k\) be an algebraically closed field of characteristic \(p\). A finite dimensional \(kE\)-module \(M\) is said to have constant Jordan type if the restriction of \(M\) to every cyclic shifted subgroup of \(kE\) has the same Jordan canonical form. I begin by discussing theorems and conjectures which restrict the possible Jordan canonical form. Then I indicate methods of producing algebraic vector bundles on projective space from modules of constant Jordan type. I describe realisability and non-realisability theorems for such vector bundles, in terms of Chern classes and Frobenius twists. Finally, I discuss the closely related question: can a module of small dimension have interesting rank variety? The case \(p\) odd behaves throughout these discussions somewhat differently to the case \(p=2\). modular representations; elementary Abelian groups; modules of constant Jordan type; vector bundles; rank varieties; Chern classes; Frobenius twists; endotrivial modules Benson, D.: Modules for elementary abelian \(p\)-groups. In: Proceedings of the International Congress of Mathematicians (ICM 2010), pp. 113-124 (2010) Modular representations and characters, Group rings of finite groups and their modules (group-theoretic aspects), Vector bundles on curves and their moduli, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Modules for elementary Abelian \(p\)-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 Suppose \(k\) is a perfect of positive characteristic \(p\). Let \(W(k)\) be its ring of Witt vectors and let \(K_0\) be the field of fractions of \(W(k)\). Let \(K\) be a finite totally ramified extension of \(K_0\) and we fix a uniformizer \(\pi \in K\). Let \(\mathcal{O}_K\) be a \(p\)-adic discrete valuation ring with perfect residue field \(k\). Put \(\mathfrak{S} = W(k)[[u]]\). Let \(\mathrm{BT}^{\varphi}_{/\mathfrak{S}}\) be the category of \(\varphi\)-module of height \(\leq 1\), whose objects are pairs \((\mathfrak{M},\varphi_{\mathfrak{M}})\) where \(\mathfrak{M}\) is a finite free \(\mathfrak{S}\)-module and \(\varphi_{\mathfrak{M}} : \mathfrak{M} \to \mathfrak{M}\) is a \(\varphi\)-semi-linear endomorphism such that \(E(u)\) annihilates \(\mathrm{coker}(1 \otimes \varphi_{\mathfrak{M}} : \varphi^*\mathfrak{M} \to \mathfrak{M})\) where \(E(u) \in W(k)[u]\) is an Eisenstein polynomial such that \(E(\pi) = 0\) and \(E(0)=p\). Let \((\mathrm{Mod}/\mathfrak{S})^{\leq 1}\) be the category of torsion \(\varphi\)-modules of height \(\leq 1\), whose objects are pairs \((\mathfrak{M}, \varphi_{\mathfrak{M}})\) where \(\mathfrak{M}\) is a finitely generated \(p^{\infty}\)-torsion \(\mathfrak{S}\)-module with non-zero \(u\)-torsion and \(\varphi_{\mathfrak{M}} : \mathfrak{M} \to \mathfrak{M}\) satisfies the same condition as in \(\mathrm{BT}^{\varphi}_{/\mathfrak{S}}\).
\textit{C. Breuil} (see [Ann. Math. (2) 152, No. 2, 489--549 (2000; Zbl 1042.14018)]) conjectured that there exist exact anti-equivalences of categories between the category of \(p\)-divisible groups over \(\mathcal{O}_K\) and \(\mathrm{BT}^{\varphi}_{/\mathfrak{S}}\) and between category of \(p\)-power order finite flat group schemes over \(\mathcal{O}_K\) and \((\mathrm{Mod}/\mathfrak{S})^{\leq 1}\) satisfying some extra conditions.
\textit{M. Kisin} [Progr. Math. 253, 457--496 (2006; Zbl 1184.11052)] proved the conjecture when \(p>2\) and later proved the conjecture for the Cartier duals of connected \(p\)-divisible groups without restriction on \(p\). The subject paper proves the conjecture without connectedness assumptions when \(p=2\). The same result is obtained by \textit{E. Lau} [``Displayed equations for Galois representations'', \url{arxiv:1012.4436}; ``A relation between Dieudonne displays and crystalline Dieudonne theory, \url{arxiv:1006.2720}] by extending Zink's theory of windows and displays for arbitrary \(p\)-divisible groups with no restriction on \(p\) (over more general bases than discrete valuation rings). \textit{T. Liu} [Compos. Math. 144, No. 1, 61--88 (2008; Zbl 1133.14020)] also proved the same result using the \((\varphi, \hat{G})\)-modules. The subject paper's approach is more related Kisin's original approach. classification of finite flat groups schemes; Kisin theory Kim, W., The classification of \textit{p}-divisible groups over 2-adic discrete valuation rings, Math. Res. Lett., 19, 1, 121-141, (2012) Group schemes, \(p\)-adic cohomology, crystalline cohomology, Galois theory The classification of \(p\)-divisible groups over 2-adic discrete valuation 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 purpose of this expository article is to give a detailed survey of recent developments in the area of Hopf Galois extensions and crossed products of Hopf algebras, which are generalizations of the Clifford theory for groups, the Blattner theory of representations of Lie algebras and the theory of finite group schemes. An important part of the paper concerns the theory of Hopf modules, which can be viewed as a formalization of descent theory. Proofs of some new or not easily accessible results are given. The last section is a report on joint work with S. Montgomery on crossed products. survey; Hopf Galois extensions; crossed products of Hopf algebras; Clifford theory; finite group schemes; Hopf modules H. J. Schneider, Hopf-Galois extensions, crossed products and Clifford theory, in: Advances in Hopf Algebras, Lecture Notes in Pure and Appl. Math., 158 (1994), pp. 267--298. Smash products of general Hopf actions, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Twisted and skew group rings, crossed products, Group schemes Hopf Galois extensions, crossed products, and Clifford 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 \(B\) be a quaternion algebra with center a totally real number field \(K\) of degree \(m\). Let \(n\) be the number of infinite places \(\varphi:K \hookrightarrow \mathbb{R}\) such that \(B\otimes_\varphi \mathbb{R}\) is isomorphic to \(M_2(\mathbb{R})\) and set \(G(\mathbb{R}) = SL(2,\mathbb{R})^n \times SU(2)^{m-n}\). Let \({\mathcal H}\) be the Poincaré upper half plane \(\{z\in\mathbb{C} \mid \text{Im} (z)>0\}\); the quotient of \(G(\mathbb{R})\) by a maximal compact subgroup is isomorphic to \({\mathcal H}^n\). Let \(\rho\) be a ``polymer'' representation of \(G(\mathbb{R})\); let \(L\) be a lattice in \(\mathbb{R}^{2n}\) and let \(\Gamma\) be a cocompact torsion-free discrete subgroup of \(G(\mathbb{R})\) such that \(\rho (\Gamma) L\subset L\). The quotient \(X= \Gamma \backslash {\mathcal H}^n\) is a Hilbert modular variety and one constructs a projective variety \(Y\) with a morphism \(Y\to X\) whose fibers are abelian varieties (a ``Kuga fiber variety''). The author determines the total Chern class of \(Y\). quaternion algebras; total Chern class of Kuga fiber variety; Hilbert modular variety Algebraic moduli of abelian varieties, classification, Modular and Shimura varieties, Totally real fields Characteristic classes of Kuga fiber varieties of quaternion 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 \(R\) be an integral domain and \(K\) its quotient field. The main purpose of the first part of the book is to study the ring \(\text{Int} (R) = \{f \in K [X]\); \(f(R) \subset R\}\). It is well known (G. Pólya, 1915) that, when \(R = \mathbb{Z}\), \(\text{Int} (R)\) is a free \(R\)-module generated by a family \(\{h_n\}_{n \in \mathbb{N}}\) of polynomials with \(\deg h_n = n\) (these are in fact the binomial polynomials). An open problem is the determination of all rings \(R\) with this property.
The author discusses this problem, in particular in the case when \(R = I_K\) is the ring of integers of a number field \(K\). Another interesting question is about the algebraic properties of \(\text{Int} (R)\): noetherianity, Skolem property, maximal and prime ideals, Krull dimension, Prüfer property, etc. This first part also deals with the values of the successive derivatives of polynomials or of rational functions.
The second part is devoted to the study of fully invariant subsets of a field by polynomial mappings: if \(f \in \mathbb{Q} [X]\) and \(S \subset \mathbb{Q}\) satisfy \(f(S) = S\), then either \(S\) is finite or \(\deg f = 1\). The aim of study is to determine the fields with this property or its analogue in the case of several variables. In particular, is this property stable by purely transcendental extension (yes) or by finite extension? The last chapter deals with polynomial cycles: by a theorem of I. N. Baker (1960) every polynomial of degree \(\geq 2\) in \(\mathbb{C} [X]\) has cycles of every order with at most one exception. The author considers this question in algebraic number fields.
This nice, short (130 pages) but dense book makes a sound review of the question. As often as possible, concise proofs are given. More technical results or related questions are described and references are given; the text is well supplemented by many exercises given at the end of each chapter. The appendix states a list of 21 open problems and the book contains 11 pages of bibliographical references (from 1895 to 1994). It is interesting to have such a synthesis on questions which are often studied but scattered in the literature. binomial polynomials; values of the successive derivatives of polynomials or rational functions; polynomial functions; integral domain; ring of integers of a number field; fully invariant subsets; polynomial mappings; several variables; transcendental extension; finite extension; polynomial cycles; many exercises; open problems; bibliographical references Narkiewicz, Władysław, Polynomial mappings, Lecture Notes in Mathematics 1600, viii+130 pp., (1995), Springer-Verlag, Berlin Research exposition (monographs, survey articles) pertaining to number theory, Polynomials in number theory, Polynomials in general fields (irreducibility, etc.), Research exposition (monographs, survey articles) pertaining to commutative algebra, Polynomials over commutative rings, Polynomials over finite fields, Polynomial rings and ideals; rings of integer-valued polynomials, Rational and birational maps, Polynomials (irreducibility, etc.) Polynomial mappings | 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 Galois modules of finite commutative group schemes of period p over a ring of Witt vectors are studied. Necessary and sufficient (for \(p>2)\) conditions are given for such a module to arise from a group scheme. Answers are obtained (in the negative) to a question of Shafarevich on the existence of abelian varieties with good reduction everywhere over the rationals and to a question of Tate on the existence of nontrivial p- divisible groups over the integers, for \(3\leq p\leq 17\). Galois modules of finite commutative group schemes; ring of Witt vectors; abelian varieties with good reduction everywhere over the; rationals; nontrivial p-divisible groups over the integers; abelian varieties with good reduction everywhere over the rationals Group schemes, Formal groups, \(p\)-divisible groups, Witt vectors and related rings Group schemes of period p over the ring of Witt vectors | 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 Consider the following zeta functions (firstly studied by F. J. Grunewald, D. Segal and G. C. Smith) for finitely generated abstract or profinite groups \(G\): \(\zeta^*_G(s)=\sum_{H\in\chi^*(G)}|G:H|^{-s}\) where \(*\in\{\leq,\triangleleft,\wedge\}\) and \(\chi^\leq(G)=\{H\mid H\) is a subgroup of finite index in \(G\}\), \(\chi^\triangleleft(G)=\{H\mid H\) is a normal subgroup of finite index in \(G\}\), \(\chi^\wedge(G)=\{H\mid H\) is a subgroup of finite index in \(G\) and \(\widehat H\cong\widehat G\}\), here \(\widehat G\) denotes the profinite completion of \(G\). The ``local factors'' associated with these zeta functions for each prime \(p\) are given by: \(\zeta^*_{G,p}(s)=\sum_{H\in\chi^*_p(G)}|G:H|^{-s}\) where \(\chi^*_p(G)=\{H\in\chi^*(G)\mid H\) has \(p\)-power index in \(G\}\).
For the class of finitely generated torsion-free nilpotent groups these zeta-functions can be decomposed as an Euler product of the local factors: \(\zeta^*_G(s)=\prod_p\zeta^*_{G,p}(s)\). In the paper under review contributions to the following questions are made: Question 1: For a fixed group \(G\), how does \(\zeta^*_{G,p}(s)\) behave as \(p\) varies over all primes \(p\)? Question 2: For a fixed prime \(p\), how does \(\zeta^*_{G,p}(s)\) behave as \(G\) varies over \(p\)-adic analytic groups of fixed dimension \(d\) (or nilpotent groups of a fixed Hirsch length \(d\))? Question 3: For a fixed finitely generated nilpotent group \(G\) do there exist finitely many rational functions \(W^*_1(Y,X),\dots,W^*_n(Y,X)\) of two variables over \(\mathbb{Q}\) such that for each prime \(p\) there is an \(i\) for which \(\zeta^*_p(s)=W^*_i(p,p^{-s})\)? Partial answers to questions 1 and 2 are given in the reviewed article. The proof uses the method of Denef and van den Dries for the calculation of \(p\)-adic integrals over semi-algebraic, respectively over sub-analytic sets and results of Macintyre, which allows the calculation of this \(p\)-adic integrals uniformly for all primes \(p\). zeta functions; profinite groups; subgroups of finite index; profinite completions; torsion-free nilpotent groups; Euler products; \(p\)-adic analytic groups; \(p\)-adic integrals M. du Sautoy,Zeta functions of groups and rings: uniformity, Israel Journal of Mathematics86 (1994), 1--23. Nilpotent groups, Zeta functions and \(L\)-functions, Non-Archimedean analysis, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Analysis on \(p\)-adic Lie groups, Limits, profinite groups, Other Dirichlet series and zeta functions, Quantifier elimination, model completeness, and related topics Zeta functions of groups and rings: Uniformity | 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 Quaternionic Shimura surfaces are quotients of the product of two copies of the upper half plane by irreducible cocompact arithmetic groups. In the present paper we are interested in (smooth) quaternionic Shimura surfaces admitting an automorphism with one-dimensional fixed locus; such automorphisms are involutions. We propose a new construction of surfaces of general type with \(q = p_g = 0\) as quotients of quaternionic Shimura surfaces by such involutions. These quotients have finite fundamental group. Shimura surfaces; surface automorphisms; quotients by finite groups; surfaces of general type Automorphisms of surfaces and higher-dimensional varieties, Modular and Shimura varieties, Surfaces of general type Minimal surfaces of general type with \(p_g = q = 0\) arising from Shimura 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 A loop group LG is the group of maps of the circle \(S^ 1\) into some topological group G (with group law coming from pointwise multiplication in G). The book under review is devoted to the study of structure and representations of LG in the case when G is either a compact or a complex Lie group.
Groups similar to LG enter mathematics in several points. Their Lie algebras form a class of Kac-Moody Lie algebras, the so-called affine algebras. In the last few years these algebras were extensively studied, and their deep relations with various branches of mathematics (combinatorics, finite groups) and physics (quantum field theory, especially string models) were displayed [see, for example, Vertex operators in mathematics and physics, \textit{J. Lepowsky}, \textit{S. Mandelstam} and \textit{I. M. Singer} (eds.) (Publ., Math. Sci. Res. Inst. 3) (1985; Zbl 0549.00013)].
The present book differs from the other sources in that it mostly adopts analytic and geometric, rather that algebraic and combinatoric, approaches.
The first part of the book studies the group LG itself. After the introduction (Chapter I) and a survey of the results about finite- dimensional representations of Lie groups (Chapter 2) the authors give in Chapter 3 general facts about infinite-dimensional Lie groups and consider LG from this viewpoint. In Chapter 4 they study one of the most important properites of loop groups, namely the existence of natural central extensions by the circle group T; these extensions are sometimes more important than loop groups themselves. In this chapter the extensions are constructed and studied by differential geometric methods. Chapter 5 contains a brief survey of Kac-Moody Lie algebras as Lie algebras of loop groups.
Chapter 6 is one of the main chapters in the first part of the book. In this chapter LG is represented as a group of operators in an appropriate Hilbert space, namely in the space \(H=L^ 2(S^ 1,V)\) of \(L^ 2\)- functions on the circle with values in some finite-dimensional representation of G; LG acts in H pointwise. The idea (coming from quantum field theory) is to consider the polarization of H, i.e. the decomposition \(H=H_+\oplus H_-\) where \(H_+\) (resp. \(H_-)\) is the space of functions with vanishing negative (resp. positive) Fourier coefficients. Properties of operators from LG with respect to this decomposition form a very interesting and important part of the theory.
Another very important concept in the first part of the book is the notion of the Grassmannian Gr(H) of a polarized Hilbert space H, introduced in Chapter 7. The authors study the canonical (determinant) line bundle on Gr(H), Schubert cell decomposition of Gr(H), etc.
Chapter 8 introduces the fundamental homogeneous space X of LG that is defined as LG/G where \(G\subset LG\) is considered as the subgroup of constant loops. Two main properties of X are as follows. First, X can be considered as a (infinite-dimensional) complex manifold via the identification \(X=LG_{{\mathbb{C}}}/L_+G_{{\mathbb{C}}}\) where \(G_{{\mathbb{C}}}\) is the complexification of G and \(L_+G_{{\mathbb{C}}}\) consists of boundary values of analytic mappings of the disk \(| z| <1\) into \(G_{{\mathbb{C}}}\). Second, X can be canonically imbedded into Gr(H), thus inheriting from Gr(H) the stratification by Schubert cells and other nice features.
The remaining Chapters 9-14 deal with the representation theory of loop groups. We will not describe the content of these chapters and restrict ourselves to giving their titles: Ch. 9: Representation theory. Ch. 10: The fundamental representation. Ch. 11: The Borel-Weil theory. Ch. 12: The spin representation. Ch. 13: ''Blips'' or ''vertex'' operators. Ch. 14: The Kac character formula and the Bernstein-Gel'fand-Gel'fand resolution. determinant line bundle; vertex operators; loop group; Kac-Moody Lie algebras; affine algebras; infinite-dimensional Lie groups; central extensions; circle group; Grassmannian; polarized Hilbert space; Schubert cell decomposition; homogeneous space; complex manifold; Borel-Weil theory; spin representation; Kac character formula; Bernstein-Gel'fand- Gel'fand resolution A. Pressley and G. Segal, \textit{Loop Groups}, Clarendon Press, Oxford (1986). Infinite-dimensional Lie groups and their Lie algebras: general properties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Research exposition (monographs, survey articles) pertaining to topological groups, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), Grassmannians, Schubert varieties, flag manifolds, Harmonic analysis on homogeneous spaces, Homogeneous complex manifolds Loop 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 author studies an abelian variety \(A\) defined over \(\overline \mathbb{Q}\) and characterizes the abelian variety of CM type by its property of periods. The obtained result brings the criterion for algebraicity of the values of the Siegel modular function at algebraic points and of other various modular functions.
This result can be regarded as the extension of the well known algebraicity criterion (so-called Schneider's theorem) for the value of the elliptic modular function. See also the paper of the author and \textit{J. Wolfart} [J. Reine Angew. Math. 463, 1-25 (1995; Zbl 0827.11043)]. transcendency; abelian variety of CM type; periods; values of the Siegel modular function at algebraic points; modular functions; Schneider's theorem; elliptic modular function Shiga, H.: On the transcendency of the values of the modular function at algebraic points. Soc. math. France astérisque 209, 293-305 (1992) Transcendence theory of other special functions, Complex multiplication and moduli of abelian varieties, Abelian varieties of dimension \(> 1\), Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Algebraic theory of abelian varieties On the transcendency of the values of the modular function at algebraic points | 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 \(F\) be a field, \(p\) a prime number or \(p=0\), \(\text{Set}\) the category of sets, and \(\text{Fields}/F\) the category of field extensions of \(F\) and field \(F\)-homomorphisms. Let \(\mathcal F\colon\text{Fields}/F\to\text{Sets}\) be a functor (an ``algebraic structure'') and \(K,E\in\text{Fields}/F\). An element \(\alpha\in\mathcal F(E)\) is said to be \(p\)-defined over \(K\) (and \(K\) is called a field of \(p\)-definition of \(\alpha\)), if there exist a finite field extension \(E'/E\) of degree prime to \(p\) (so \(E'=E\) in case \(p=0\)), a field \(F\)-homomorphism \(K\to E'\), and an element \(\beta\in \mathcal F(K)\), such that the image of \(\alpha\) under the map \(\mathcal F(E)\to\mathcal F(E')\) coincides with the image of \(\beta\) under the map \(\mathcal F(K)\to\mathcal F(E')\). The essential \(p\)-dimension of \(\alpha\), denoted by \(\text{ed}_p^{\mathcal F}(\alpha)\), is defined as the least transcendence degree \(\text{tr.deg}_F(K)\), when \(K\) runs across the fields of \(p\)-definition of \(\alpha\). The essential \(p\)-dimension of the functor \(\mathcal F\) is the supremum \(\text{ed}_p(\mathcal F)\) of the \(p\)-dimensions \(\text{ed}_p^{\mathcal F}(\alpha)\), taken over fields \(E\in\text{Fields}/F\) and all \(\alpha\in\mathcal F(E)\). The number \(\text{ed}_0(\mathcal F)\) is called the essential dimension of \(\mathcal F\), and is also denoted by \(\text{ed}(\mathcal F)\).
When \(G\) is an algebraic group over \(F\), the essential dimension of \(G\) is the essential \(p\)-dimension of the functor \(\mathcal F_G\colon\text{Fields}/F\to\text{Sets}\) taking a field \(E\) to the set of isomorphism classes of principal homogeneous \(G\)-spaces (\(G\)-torsors) over \(\text{Spec}(E)\). In particular, if \(G=\text{PGL}(n)\) over \(F\), then \(\mathcal F_G\) is isomorphic to the functor taking \(E\) to the set of isomorphism classes of central simple \(E\)-algebras of degree \(n\). It is known [see \textit{Z. Reichstein} and \textit{B. Youssin}, Can. J. Math. 52, No. 5, 1018-1056 (2000; Zbl 1044.14023)] that \(\text{ed}_p(\text{PGL}_F(p))=2\) and \(\text{ed}_F(\text{PGL}_F(n))=\text{ed}_p(\text{PGL}_F(p^r))\), for each prime \(p\), where \(r\) is the maximal integer for which \(p^r\) divides \(n\).
The main result of the paper under review proves that \(\text{ed}_p(\text{PGL}_F(p^2))=p^2+1\), provided that \(p\) is different from the characteristic of \(F\). As shown by the author, when \(p=2\neq\text{char}(F)\), this implies Rost's theorem, which states that \(\text{ed}(\text{PGL}_F(4))=\text{ed}_2(\text{PGL}(4))=5\). The proof of the main theorem contains several results of independent interest. Some of them concern central simple algebras, Brauer groups and transfers of Brauer-Severi varieties. Others provide information about the algebraic torus of norm \(1\) elements of a bicyclic field extension of degree \(p^2\) as well as on its quotient group of \(R\)-equivalence classes. essential \(p\)-dimension; central simple algebras; Brauer groups; Severi-Brauer varieties; \(R\)-equivalence; Chow groups; character groups of algebraic tori Alexander S. Merkurjev, Essential \?-dimension of \?\?\?(\?²), J. Amer. Math. Soc. 23 (2010), no. 3, 693 -- 712. Linear algebraic groups over finite fields, Brauer groups (algebraic aspects), Finite-dimensional division rings, Group actions on varieties or schemes (quotients) Essential \(p\)-dimension of \(\mathrm{PGL}(p^2)\). | 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 The classification theory of holomorphic and/or algebraic vector bundles on complex projective algebraic varieties has been one of the most challenging topics, and one of the most active areas of research in algebraic geometry over the past thirty years. By now, a powerful conceptual and methodical framework for tackling moduli problems in the theory of vector bundles has been developed, and the corresponding classification theory has reached a quite satisfactory state for vector bundles on curves (or compact Riemann surfaces), algebraic surfaces, and projective spaces. In the last decade, the ubiquity and significance of moduli spaces of vector bundles (and their intrinsic geometry) in mathematical physics, particularly in gauge theory and in conformal quantum field theory, has become increasingly apparent. Despite the great progress achieved in the field, so far, and compared to the wide interest that the subject has gained in mathematics and physics, there has been no adequate reflection of this development in the current textbook literature, at least not until very recently. Certainly, moduli theory of vector bundles, together with its huge and rather involved technical framework, is still beyond the scope of any normal textbook on algebraic geometry, so that the available treatises on this subject are merely specific research monographs assuming appropriate prerequisites, or survey articles providing an overview of the development (without detailed proofs).
The present work is one of the first (and very few) textbooks dealing with topics in the theory of algebraic vector bundles and their moduli spaces. The text consists of two major parts, each of which grew out of courses that the author had taught on several occasions between 1991 and 1993. Part I is a set of lectures on vector bundles on algebraic curves, held by the author at the University of Paris VII in 1991. Designed for students who already had a profound background knowledge in algebraic geometry, the material presented here covers the (now rather classical) moduli theory of (semi-)stable vector bundles of given rank and degree over smooth projective curves, basically so along the original approach (via geometric invariant theory) due to A. Grothendieck, D. Mumford, and C. S. Seshadri. The presentation provided here is much more elementary, selective and textbook-like than \textit{C. S. Seshadri}'s advanced monograph ``Fibrés vectoriels sur les courbes algébriques'' [Astérisque 96 (Paris 1982; Zbl 0517.14008)]. For the convenience of the audience (or the reader, respectively), the first two sections review briefly the basic facts and techniques from general algebraic geometry, as they are frequently used in the sequel. This includes, mostly without proofs, generalities on vector bundles, operations on vector bundles, coherent sheaves and their cohomology, GAGA correspondences, and Riemann-Roch theory for vector bundles. Section 3 deals with the topological aspects of vector bundles and their characteristic classes, whereas chapter 4 provides the technical framework for constructing and investigating Hilbert schemes. Chapter 5 introduces to families of vector bundles over curves, to the concept of boundedness for families, and to the crucial notion of Mumford-Takemoto (semi-)stability for coherent sheaves and vector bundles. The following chapter 6 provides the reader with some fundamental methods and facts from geometric invariant theory, thus preparing him for the construction of moduli spaces of (semi-)stable vector bundles as geometric quotients of subschemes of Hilbert schemes modulo reductive groups. The moduli spaces \(M(r,d)\) of semi-stable rank-\(r\) vector bundles of degree \(d\) over smooth projective curves are then explicitly constructed in chapter 7. In contrast to the classical Mumford-Seshadri approach, the author follows here the method of \textit{C. T. Simpson} [Moduli of representations of the fundamental group of a smooth projective variety.'' I and II, Publ. Math., Inst. Hautes Étud. Sci. 79, 47--129 (1994; Zbl 0891.14005) and 80, 5--79 (1995; Zbl 0891.14006)] which has the advantage of being extendable to semi-stable sheaves on singular curves as well as to higher-dimensional projective varieties. Chapter 8, the concluding chapter of part I, describes more closely the geometry of the moduli spaces \(M(r,d)\). The main results are the theorems of existence, irreducibility and smoothness of the open locus of stable vector bundles, for which there is barely any other reference in the literature. Actually, the author has adapted a method of proof that he and J.-M. Drézet had developed, some years ago, to study stable bundles on the projective plane, and which is fully explained in part II of the present book.
Part II of the text is much more advanced, in its content and presentation, than the fairly down-to-earth exposition in the first part. It deals with semi-stable sheaves on the projective plane and their moduli spaces, a topic to which the author himself has contributed a great deal of methods and results. This part grew out of a course which the author had taught at the workshop ``Vector bundles on surfaces'' (Nice 1993). Chapter 9 contains some basic material on semi-stable sheaves on smooth projective surfaces, including the concept of semi-stability in the sense of Gieseker-Maruyama (i.e., with respect to reduced Hilbert polynomials) and the existence of the corresponding Harder-Narasimhan filtrations for torsion-free coherent sheaves. Chapters 10 and 11 turn to restrictions of semi-stable sheaves to curves in a surface. Using the Shatz stratification of a complete family of torsion-free sheaves as a fundamental ingredient, here as well as in the sequel, the author proves a particular variant of \textit{H. Flenner}'s general restriction theorem [cf. Comment. Math. Helv. 59, 635--650 (1984; Zbl 0599.14015)], thus obtaining a suitable generalization of the celebrated Grauert-Mülich theorem for surfaces. Chapter 12 discusses F. Bogomolov's result on the non-negativity of the discriminant of a semi-stable sheaf on a surface and, in addition, an elegant proof of C. Simpson's estimate for the dimension of the space of global sections of a semi-stable sheaf in the special case of a surface as a base space. Bounded families of coherent sheaves on a surface are the topic of chapter 13. The proof of boundedness for families of semi-stable sheaves with given Hilbert polynomial, which is in full generality due to M. Maruyama, is here given in two steps. First it is shown that boundedness holds for the projective plane, essentially so by using Beilinson monads, and then the case of an arbitrary smooth projective base surface is reduced to that particular case. The construction of the coarse moduli spaces for semi-stable sheaves (over a polarized projective surface) with given Hilbert polynomial is outlined in chapter 14. Again the author follows C. Simpson's recent approach (cited above), thereby referring to the methodically similar construction in the curve case (chapter 7). Chapter 15 focuses on the infinitesimal study of the Shatz stratification, culminating in a smoothness criterion for the single strata in the case of the projective plane and, subsequently, in computing the codimension of the Harder-Narasimhan-Shatz strata in this case. Conditions for the existence of semi-stable sheaves with given Chern classes on \(\mathbb P^2 (\mathbb C)\) are derived in chapter 16, while the irreducibility of the corresponding moduli spaces is proved in the following chapter 17. These two chapters provide a lucid account on the results obtained by \textit{J.-M. Drézet} and \textit{J. Le Potier} in Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 193--243 (1985; Zbl 0586.14007). The concluding chapter 18 is devoted to the proof of J.-M. Drézet's theorem on the structure of the Picard groups of the moduli spaces \(M_{\mathbb P^2} (r,c_1, c_2)\). The proof presented here is different from Drézet's original proof (1988); it is based upon a weak version of the Brill-Noether theorem for vector bundles on the projective plane, which recently has been communicated by \textit{L. Göttsche} and \textit{A. Hirschowitz} [EUROPROJ Annual Conference (Catania 1993)].
All in all, the present book provides a profound and lucid introduction to some central topics in the theory of algebraic sheaves and their classifying spaces. It is, in a certain sense, complementary to another very recent text on this subject, namely to the monograph ``The geometry of moduli spaces of sheaves'' by \textit{D. Huybrechts} and \textit{H. Lehn} [Aspects of Mathematics, E 31. Braunschweig: Vieweg (1997; see the preceding review Zbl 0872.14002)]. The latter treatise is perhaps more advanced, comprehensive and up-to-date, however does not treat the case of vector bundles over curves as thoroughly, and does not particularly emphasize the specific case of sheaves on the projective plane. In regard of this, the book under review is perfectly suited for complementary reading to the other one, and conversely. At any rate, two up-to-date textbooks on algebraic vector bundles are finally available, and the mathematical community will certainly appreciate that! vector bundles; moduli spaces; geometric invariant theory; Hilbert schemes; Harder-Narasimhan filtrations; coarse moduli spaces; Shatz stratification; Picard groups C. Li and X. Zhao, \textit{The MMP for deformations of Hilbert schemes of points on the projective plane}, Algebraic Geometry, to appear. Algebraic moduli problems, moduli of vector bundles, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli Lectures on 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 The authors find the \(\widetilde D_5\)-singularity in the intersection of nilpotent variety of \(\mathfrak s\mathfrak l(2,{\mathbb C}) \oplus \mathfrak s\mathfrak l(2,{\mathbb C})\) with a four-dimensional \` good\'\ slice, depending on two parameters. They also find the semi-universal deformation in these terms.
In an Appendix some facts about pencils of quadrics are collected. simple elliptic singularities; \(\tilde{D}_5\)-singularities; Lie algebras Singularities in algebraic geometry, Linear algebraic groups over the reals, the complexes, the quaternions, Local complex singularities A new construction of \(\tilde{D}_5\)-singularities and generalization of Slodowy slices | 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 extends a structure theorem of \textit{F. Catanese} and \textit{F.-O. Schreyer} [in: Algebraic geometry. A volume in memory of Paolo Francia. 79--116 (2002; Zbl 1053.14048)] for good birational canonical projections into \({\mathbb P}^3\) of surfaces of general type to the case of projections into 3-dimensional weighted projective spaces. In order to achieve this extension, he has to generalize, first, to the weighted case the main technical result used by Catanese and Schreyer, namely Beilinson's description of the derived category of coherent sheaves on projective spaces.
Let \(\mathbb K\) be a field, \(n\in {\mathbb N}\), \(w=(w_0,\dots ,w_n)\in {\mathbb N}^{n+1}_+\), \(| w| =w_0+\cdots +w_n\), and \({\mathbb P}(w)\) the \(n\)-dimensional weighted projective space obtained as the quotient of \({\mathbb K}^{n+1}\setminus \{0\}\) by the action of the multiplicative group \(\text{G}_{\text{m}}={\mathbb K}^{\ast}\) given by \(t\cdot x = (t^{w_0}x_0,\dots ,t^{w_n}x_n)\). If \(\pi : {\mathbb K}^{n+1}\setminus \{0\} \rightarrow {\mathbb P}(w)\) is the canonical surjection then \({\pi}_{\ast}{\mathcal O}_{{\mathbb K}^{n+1}\setminus \{0\}}\) is a sheaf of \(\mathbb Z\)-graded \({\mathbb K}\)-algebras. The structure sheaf of the scheme \({\mathbb P}(w)\) is \({\mathcal O}_{\mathbb P} := ({\pi}_{\ast}{\mathcal O})_0\). Since the category \(\text{Coh}{\mathbb P}\) of coherent \({\mathcal O}_{\mathbb P}\)-modules presents several pathologies, the author works, instead, with the graded scheme \(\overline{\mathbb P}(w)=({\mathbb P}(w),{\mathcal O}_ {\overline{\mathbb P}})\), where \({\mathcal O}_{\overline{\mathbb P}}= {\pi}_{\ast}{\mathcal O}\), and with the category \(\overline{\text{Coh}}\;\overline{\mathbb P}\) of coherent sheaves of graded \({\mathcal O}_{\overline{\mathbb P}}\)-modules.
In particular, if \(\mathcal F\) is an object of \(\overline{\text{Coh}}\;\overline{\mathbb P}\) and \(d\in {\mathbb Z}\) then \({\mathcal F}(d)\) is the object of \(\overline{\text{Coh}}\;\overline{\mathbb P}\) coinciding with \(\mathcal F\) as sheaves but with the grading modified such that \({\mathcal F}(d)_i={\mathcal F}_{d+i}\). One also defines \(\text{H}^i(\overline{\mathbb P}, {\mathcal F})\) to be the cohomology group \(\text{H}^i({\mathbb P},{\mathcal F}_0)\) which is isomorphic to \(\text{Ext}^i_{\overline{\mathbb P}} ({\mathcal O}_{\overline{\mathbb P}},{\mathcal F})\).
Since every object of \(\overline{\text{Coh}}\;\overline{\mathbb P}\) has a finite resolution with finite direct sums of objects of the form \({\mathcal O}_{\overline{\mathbb P}}(a)\), \(a\in {\mathbb Z}\), and since the Koszul complex \({\mathcal K}^{\bullet}\) associated to the canonical epimorphism \(\bigoplus {\mathcal O}_{\overline{\mathbb P}}(-w_i)\rightarrow {\mathcal O}_{\overline{\mathbb P}}\) is exact, it follows that the sheaves \({\mathcal O}_{\overline{\mathbb P}}(j)\), \(-| w| <j\leq 0\), generate the derived category \({\text{D}}^b(\overline{\text{Coh}}\;\overline{\mathbb P})\) as a triangulated category. If \({\Omega}^j_{\overline{\mathbb P}}\) is the kernel of the Koszul differential \({\mathcal K}^{-j}\rightarrow {\mathcal K}^{-j+1}\), one deduces that the sheaves \({\Omega}^j_{\overline{\mathbb P}}(j)\), \(0\leq j\leq n\), and \({\mathcal O}_{\overline{\mathbb P}}(j)\), \(n-| w| <j<0\), generate \({\text{D}}^b(\overline{\text{Coh}}\;\overline{\mathbb P})\). The first main result obtained by the author states that if \({\overline{\mathcal Y}}^{\bullet}\) is a minimal (in a technical sense) finite complex of finite direct sums of objects of the above form isomorphic, in the derived category, to an object \(\mathcal F\) of \(\overline{\text{Coh}}\;\overline{\mathbb P}\) then the multiplicity in \({\overline{\mathcal Y}}^i\) of \({\Omega}^j_{\overline{\mathbb P}}(j)\) is the dimension of \(\text{H}^{i+j}(\overline{\mathbb P},{\mathcal F}(-j))\) and the multiplicity of \({\mathcal O}_{\overline{\mathbb P}}(j)\) (\(n-| w| <j<0\)) is the dimension of \(\text{H}^i(\overline{\mathbb P},{\mathcal F}\otimes {\mathcal N}^{\bullet}_{(j)})\), where \({\mathcal N}^{\bullet}_{(j)}\) is a certain subcomplex of \({\mathcal K}^{\bullet}(-j)\).
Now, let \(S_0\) be a minimal surface of general type, \(R={\bigoplus}_{i\geq 0}\text{H}^0(S_0,{\omega}^i_{S_0})\) its canonical ring, \(w\in {\mathbb N}^4_{+}\) with \(\text{gcd}(w_0,\dots ,w_3)=1\) and \({\sigma}_i\in \text{H}^0(S_0,{\omega}^{w_i}_{S_0})\), \(i=0,\dots ,3\), defining a morphism \({\phi}_0 : S_0\rightarrow {\mathbb P}(w)\), birational onto its image. Applying the above Beilinson type theorem to \({\mathcal F}=({\bigoplus}_{i\in {\mathbb Z}}{\phi}_{0\ast}{\omega}^i_{S_0})(2)\), the author obtains his main result asserting that there exists an exact sequence :
\[
0\rightarrow ({\mathcal O}_{\overline{\mathbb P}}\oplus {\mathcal E})^{\vee}(-1-| w| ) \overset {\alpha}\longrightarrow {\mathcal O}_{\overline{\mathbb P}}\oplus {\mathcal E}\rightarrow {\bigoplus}_{i\in {\mathbb Z}}{\phi}_{0\ast}{\omega}^i_{S_0} \rightarrow 0
\]
where \(\mathcal E\) is a direct sum of sheaves of the form \({\Omega}^j_{\overline{\mathbb P}}(j-2)\), \(j=0,1,2\), or \({\mathcal O}_{\overline{\mathbb P}}(j-2)\), \(3-| w| <j<0\), and \(\alpha ={\alpha}^{\vee}(-1-| w| )\). The multiplicities in \(\mathcal E\) of the sheaves \({\Omega}^j_{\overline{\mathbb P}}(j-2)\) depend only on the invariants of \(S_0\). Moreover, \(\alpha \) satisfies a certain ``rank condition'' which allows one to recuperate \(S_0\) from \(\alpha \).
Finally, the author determines explicitly the above resolution for \(S_0\) a double cover of the Jacobian \(A\) of a smooth projective curve \(C\) of genus 2 branched over a smooth divisor \(B\in | 2{\Theta}| \) (\(S_0\) has invariants \(p_g=q=2\), \(K^2=4\)) and for \(w=(1,1,2,3)\). These surfaces are interesting because their bicanonical map is not birational. In particular, the author shows that the multiplicities in \(\mathcal E\) of the sheaves \({\mathcal O}_{\overline{\mathbb P}}(j-2)\) depend on the projection \({\phi}_0\) (not only on the invariants of \(S_0\)). The determination of these multiplicities requires a deep knowledge of the ring of theta functions \(R(A,{\Theta})={\bigoplus}_{n\geq 0}\text{H}^0(A,{\mathcal O}_A(n{\Theta}))\). surfaces of general type; graded schemes; derived categories; ring of theta functions A. Canonaco, The Beilinson complex and canonical rings of irregular surfaces, Mem. Amer. Math. Soc. 183 (2006). Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special varieties, Graded rings, Derived categories, triangulated categories, Theta functions and curves; Schottky problem, Surfaces of general type The Beilinson complex and canonical rings of irregular 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 See the preview in Zbl 0724.14014. defects of cusp singularities; totally real cubic number field; plurigenera of Hilbert modular varieties; non-rational Hilbert modular threefold; arithmetic genus Grundman, H.G.: Defects of cusp singularities and the classification of Hilbert modular threefolds. Math. Ann.292, 1-12 (1992) Modular and Shimura varieties, \(3\)-folds, Global ground fields in algebraic geometry, Singularities in algebraic geometry Defects of cusp singularities and the classification of Hilbert modular threefolds. | 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 Hilbert modular group; resolution of singularities at cusps F.Hirzebruch, The Hilbert Modular group, resolution of the singularities at the cusps and related problems. Sém. Bourbaki, exp. 396 (1971). Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Modifications; resolution of singularities (complex-analytic aspects), Characteristic classes and numbers in differential topology, Special surfaces The Hilbert modular group, resolution of the singularities at the cusps and related problems | 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 0588.00014.]
A la donnée d'une algèbre de Lie semi-simple complexe L, ou de facon équivalente son système de racines \(\Phi\), et d'un réseau \(\Gamma\) contenant le groupe des poids radiciels \(P_ r\) et contenu dans le groupe des poids P de \(\Phi\), on sait associer un schéma en groupes affine G sur \({\mathbb{Z}}\) qui jouit des propriétés suivantes: \((1)\quad G({\mathbb{C}})\) est un groupe de Lie connexe, semi-simple complexe et Lie(G(\({\mathbb{C}}))\simeq L\); \((2)\quad pour\) chaque corps k, G(k) est un groupe algébrique semi-simple déployé sur k, du même type que G(\({\mathbb{C}})\). Un tel schéma est appelé schéma de Chevalley- Demazure; il est unique selon M. Demazure.
En particulier, pour chaque anneau commutatif unitaire R, on obtient un groupe G(R) (le groupe des points rationnels sur R). Pour toute racine \(\alpha\), on a un groupe additif a un paramètre \(e_{\alpha}: {\mathbb{G}}_{\alpha}(R)\to G(R)\), \(t\mapsto e_{\alpha}(t)\). On note \(U_{\alpha}(R)\) l'image de ce morphisme et E(R) le sous-groupe de G(R) engendré par les \(U_{\alpha}(R)\) pour tout \(\alpha\in \Phi\), E(R) est appelé le sous-groupe élémentaire de G(R).
Le but de cet article est de démontrer avec les techniques inaugurées par L. N. Vaserstein que pour un système \(\Phi\) de racines réduit sans composantes de type \(A_ 1\) le sous-groupe élémentaire E(R) est normal dans \(G(R)=G_{\Phi,\Gamma}(R)\). group of rational points; semi-simple groups; root systems; Chevalley groups; Chevalley-Demazure group schemes [101] Vavilov N., ''Intermediate subgroups in Chevalley groups'', Groups of Lie Type and their Geometries (Como, 1993), Cambridge Univ. Press, Cambridge, 1995, 233--280 Linear algebraic groups over adèles and other rings and schemes, Group schemes, Simple, semisimple, reductive (super)algebras, Linear algebraic groups over arbitrary fields, Subgroup theorems; subgroup growth, Classical groups (algebro-geometric aspects) Normalité des groupes élémentaires dans les groupes de Chevally sur un anneau. (Normality of the elementary groups in the Chevalley groups over a ring) | 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 \({\mathfrak g}=\bigoplus_{j\in\mathbb{Z}_m}{\mathfrak g}_j\) be a \(\mathbb{Z}_m\)-graded simple Lie algebra satisfying \([{\mathfrak g}_i,{\mathfrak g}_j]\subset{\mathfrak g}_{i+j}\). Let \(G\) be a connected reductive group with Lie algebra \({\mathfrak g}_0\). The natural \({\mathfrak g}_0\)-action on \({\mathfrak g}_1\) induces a \(G\)-module structure on \({\mathfrak g}_1\), called a \(\Theta\)-representation. The representation of the derived subgroup of \(G\) on \({\mathfrak g}_1\) is called the reduction of the \(\Theta\)-representation. The main result of the present paper is to show that every equivariant polynomial automorphism of a \(\Theta\)-representation and of the reduction of an irreducible \(\Theta\)-representation is a multiple of the identity. graded simple Lie algebras; connected reductive groups; equivariant polynomial automorphisms; irreducible representations A. Kurth:Equivariant Polynomial Automorphisms. Ph.D. Thesis Basel (1996). Representation theory for linear algebraic groups, Rational and birational maps, Group actions on varieties or schemes (quotients), Graded Lie (super)algebras Equivariant polynomial automorphisms of \(\Theta\)-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 diophantine problems with realizing a finite group \(G\) as the Galois group of a regular extension \(L/\mathbb{Q}(x)\) of \(\mathbb{Q}\): \(\overline{\mathbb{Q}}\cap L= \mathbb{Q}\). It illustrates, for example, why you can't dictate where to put the branch points (ramified places) of \(L/\mathbb{Q}(x)\). There is an if and only if condition that \(G\) is the group of a Galois regular extension of \(\mathbb{R}(x)\) with only real branch points. It is that involutions generate \(G\) (Theorem 1.1 (a)). If you can't realize \(G\) with real branch points, you certainly can't realize it with rational branch points.
A more sophisticated topic produces a group theory test for finding \(L/\mathbb{R}(x)\) with real branch points, regular but not necessarily Galois, with Galois closure over \(\mathbb{C}\) the (geometric) group \(G\). This direction concludes with a profinite topic. Given \(G\) passing this test, is there a totally nonsplit extension of \(G\) that doesn't pass it? Answers use the Universal Frattini Cover (or minimal projective cover) of \(G\) from [\textit{M. Fried} and \textit{M. Jarden}, Field arithmetic (Ergebnisse 11) (1986; Zbl 0625.12001); Chapter 20].
Of course, the real algebraic numbers are close to the algebraic closure. The field of totally real numbers \(\mathbb{Q}^{tr}\) -- algebraic numbers all of whose conjugates are real -- gives closer comparison with regular realization of groups over \(\mathbb{Q}\). Theorem 5.7 uses [\textit{M. Fried} and \textit{H. Völklein}, Ann. Math., II. Ser. 135, 1-13 (1992; Zbl 0765.12002)] and [\textit{F. Pop}, Fields of totally \(\Sigma\)-adic numbers (Preprint 1991)] to show each finite group is the Galois group of a regular extension of \(\mathbb{Q}^{\text{tr}}(x)\). Theorem 5.7 follows trivially if we know \(G(\overline{\mathbb{Q}}/\mathbb{Q}^{\text{tr}})\) -- generated by the conjugate of complex conjugation -- is freely (in a profinite sense) generated by involutions. \textit{M. Fried}, \textit{D. Haran} and \textit{H. Völklein} [C. R. Acad. Sci., Paris, Sér. I 317, No. 11, 995-999 (1993)] have shown this. Pop has recently announced \(p\)-adic analogs of this based on techniques of Harbater.
The second part of the paper, descends fields of definition from \(\mathbb{R}\) to \(\mathbb{Q}\). Hurwitz family techniques completely reduce the regular version of the inverse Galois problem to finding \(\mathbb{Q}\)-rational point on varieties [\textit{M. Fried} and \textit{H. Völklein}, Math. Ann. 290, 771-800 (1991; Zbl 0763.12004)]. One result considers realizing the symmetric group \(S_ m\) as the group of a Galois regular extension of \(\mathbb{Q}(x)\), satisfying two further conditions. These are that the extension has no more than four branch points, and it has some totally real residue class field specializations. Theorem 4.11 shows such extensions exist for \(m=3-10\). This topic demonstrates the diophantine difficulty of realizing groups by covers having quotients of \(G(\mathbb{Q}^{\text{tr}}/\mathbb{Q})\) as specializations.
Finally, the paper considers the regular realization problem on the dihedral groups \(\{D_ p\), \(p\) a prime\}. Suppose \(p>7\). Theorem 5.1 shows if \(D_ p\) is the group of a Galois regular extension of \(\mathbb{Q}(x)\), the extension has at least 6 branch points. This interprets rational points on modular curves as coming from realization of certain dihedral group covers. It then applies Mazur's Theorem uniformly bounding \(\mathbb{Q}\) points on the modular curves \(\{C_ 1(p)\), \(p\) a prime\}. \textit{B. Mazur} and \textit{S. Kamienny} [Rational torsion of prime order in elliptic curves over number fields (Preprint 6/92)] suggest no bound on the number of branch points allows realization of more than finitely many \(D_ p\). Indeed, the Galois problem is harder than universally bounding elliptic curve torsion points over all extensions of \(\mathbb{Q}\) of a given degree. coverings of \(\mathbb{P}^ 1\); Galois groups; Hurwitz monodromy group; finite groups; Galois group of regular extension of \(\mathbb{Q}\); universal Frattini cover; rational point on varieties; regular realization problem on dihedral groups; real branch points; totally nonsplit extension; field of totally real numbers; fields of definition; inverse Galois problem; symmetric group; modular curves Dèbes, Pierre; Fried, Michael D., Nonrigid constructions in Galois theory, Pacific J. math., 163, 1, 81-122, (1994) Inverse Galois theory, Rational points, Varieties over global fields, Coverings of curves, fundamental group, Symmetric groups Nonrigid constructions in Galois 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 [Topology 3, 131--139 (1964; Zbl 0136.43102)], \textit{L. Auslander} conjectured that any crystallographic group (a discrete subgroup in the affine group \(\mathrm{GL}_n(\mathbf R)\cdot R^n\) which acts properly discontinuously and such that the quotient space is compact) contains a solvable subgroup of finite index. In fact, he gave a proof of this statement, but soon a flaw was discovered in his proof. So this statement is now known as Auslander's conjecture. In [Adv. Math. 25, 178--187 (1977; Zbl 0364.55001)], \textit{J. W. Milnor} asked whether this statement is actually true for any affine subgroup which acts properly discontinuously. The answer to this question turned out to be negative: \textit{G. A. Margulis} [Sov. Math., Dokl. 28, 435--439 (1983; Zbl 0578.57012); translation from Dokl. Akad. Nauk SSSR 272, 785--788 (1983)] constructed a nonabelian free group of affine transformations with Zariski-dense linear part in the Lie group \(\mathrm{SO}(2,1)\), acting properly discontinuously on \(\mathbf R ^3\). Later this counterexample was generalized.
In the paper under review, the author constructs a wide class of counterexamples to Milnor's question. The main result is the follows:
Let \(G\) be a noncompact semisimple real Lie group. Consider the ``affine group'' -- the semidirect product \(G\cdot g\) for the adjoint action of \(G\) on its Lie algebra \(g\). Then there is a discrete subgroup \(\Gamma \subset G \cdot g\) whose linear part is Zariski-dense in \(G\) and which is free, nonabelian and acts properly discontinuously on the affine space corresponding to \(g\).
The general strategy of the proof comes from Margulis's original paper. But some new ideas are presented. discrete subgroups of Lie groups; affine group; Auslander conjecture; Milnor conjecture; flat affine manifold; Margulis invariant; quasi-translation; free group; Schottky group Smilga, I.: Proper affine actions on semisimple Lie algebras. arXiv:1406.5906 Discrete subgroups of Lie groups, Linear algebraic groups over finite fields, Linear algebraic groups over the reals, the complexes, the quaternions, Classical groups (algebro-geometric aspects), Other geometric groups, including crystallographic groups, Simple, semisimple, reductive (super)algebras, Automorphisms, derivations, other operators for Lie algebras and super algebras Proper affine actions on semisimple 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 We study the spectrum of integral group rings of finitely generated Abelian groups \(G\) from the scheme-theoretic viewpoint. We prove that the (closed) singular points of \(\text{Spec\,}\mathbb{Z}[G]\), the (closed) intersection points of the irreducible components of \(\text{Spec\,}\mathbb{Z}[G]\) and the (closed) points over the prime divisors of \(|t(G)|\) coincide. We also determine the formal completion of \(\text{Spec\,}\mathbb{Z}[G]\) at a singular point. affine schemes; singularities; Kähler differentials; spectra of integral group rings; finitely generated Abelian groups; singular points Group rings, Group rings of infinite groups and their modules (group-theoretic aspects), Singularities in algebraic geometry, Ideals in associative algebras, Modules of differentials On spectra of Abelian group 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 Let K be a field of characteristic 0, \(f_ 1,...,f_{\ell}\in K(x)\) be nonconstant rational functions. Write \(f_ i=h_{i1}/h_{i2}\) where \(h_{i1},h_{i2}\in K[x]\) are coprime. Denote by \(V(f_ 1(X_ 1)+...+f_ 1(X_ 1))\) the algebraic set in affine \(\ell\)-space \({\mathbb{A}}_ K^{\ell}\) defined by the polynomial \(\prod^{\ell}_{i=1}h_{i2}(X_ i)\sum^{\ell}_{i=1}f_ i(X_ i).\) The main theorem states: If \(\ell \geq 3\) then \(V(f_ 1(X_ 1)+...+f_{\ell}(X_{\ell}))\) is irreducible. This was proved by \textit{A. Schinzel} - even for arbitrary K, but only for polynomials - [cf. Pac. J. Math. 118, 531-563 (1985; Zbl 0571.12011)]. The author gives a short proof of the main theorem using Galois-theoretic methods and results of one of his earlier papers [Ill. J. Math. 17, 128-146 (1973; Zbl 0266.14013)]. He also indicates how these results can help to shorten arguments of Schinzel in the case of positive characteristic.
Now let \(\ell =2\) and \(K={\mathbb{C}}\). For positive integers n,m let \({\mathcal R}(n,m)\) denote the set of ordered pairs (h,g) of rational functions in \({\mathbb{C}}(x)\) of respective degrees n and m. To exclude trivial cases the author introduces the concept of newly reducible pairs; if h and g are polynomials then this implies that they have the same degree. - The existence of a newly reducible pair \((h,g)\in {\mathcal R}(n,m)\) is equivalent to a pure group theoretic existence problem (theorem 2.3); as a consequence one gets, that for each \(n>3\) there exists m and newly reducible \((h,g)\in {\mathcal R}(n,m)\). The existence problem for polynomials is open; if h in addition has the property that there are no proper subfields between \({\mathbb{C}}(x)\) and \({\mathbb{C}}(h(x))\) then the existence of newly reducible pairs of polynomials in \({\mathcal R}(n,m)\) implies \(n\in \{7,11,13,15,21,31\}\) [by using the classification of finite simple groups; see the author, ``Rigidity and applications of the classification of finite simple groups to monodromy'', Part II (Preprint)] and for every such n there exists (h,g) (cf. loc. cit.).
The (n,m)-problem might helps to solve the existence problem mentioned above: Call a pair \((h',g')\) of polynomials in \({\mathcal R}(n,m)\) hereditarily irreducible if \(V(h'(h_ 1)-g'(g_ 1))\) is irreducible for each pair of nonconstant polynomials \((h_ 1,g_ 1)\). The (n,m) problem is the following: Does there exist an hereditarily irreducible \((h',g\}\) in \({\mathcal R}(n,m)?\) For \(n=m=2\) the answer is negative. The (2,3)-problem has an affirmative solution if there exists a group with certain properties. In the description of this group a parameter \(k=\gcd (n\cdot \deg (g_ 1),m\cdot \deg (h_ 1))/1cm(n,m)\) occurs. The author shows that in case \(k=1,2\) there is no group with the required properties. The cases \(k\geq 3\) remain open. irreducibility of sets defined by multivariate polynomials; classification of finite simple groups Fried M D. Irreducibility results for separated variables equations. J Pure Appl Algebra, 1987, 48: 9--22 Polynomials over commutative rings, Varieties and morphisms, Finite simple groups and their classification, Relevant commutative algebra Irreducibility results for separated variables equations | 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 note is a supplement to our paper [Ann. Math., II. Ser. 118, 131-177 (1983; Zbl 0531.58048)]. Hirzebruch conjectured that the values at zero of the Shimizu L-functions are realized as the signature defects of cusps associated to Hilbert modular varieties. In the paper cited above we claimed to have established the Hirzebruch conjecture but, as was pointed out to us by W. Müller, we only dealt with the ''split'' case. In fact our method of proof extends with essentially no change to the non-split case. eta invariants; signature defects of cusps; special values of; L- functions; cusp on Hilbert modular variety; lattice in totally; real field; Hirzebruch L-polynomial; Hirzebruch; signature theorem; flat connection; Feynman-Kac; representation of the heat kernel M. F. Atiyah, H. Donnelly, and I. M. Singer, Eta invariants, signature defects of cusps, and values of \?-functions, Ann. of Math. (2) 118 (1983), no. 1, 131 -- 177. , https://doi.org/10.2307/2006957 M. F. Atiyah, H. Donnelly, and I. M. Singer, Signature defects of cusps and values of \?-functions: the nonsplit case. Addendum to: ''Eta invariants, signature defects of cusps, and values of \?-functions'', Ann. of Math. (2) 119 (1984), no. 3, 635 -- 637. Heat and other parabolic equation methods for PDEs on manifolds, Characteristic classes and numbers in differential topology, Connections (general theory), Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Quaternion and other division algebras: arithmetic, zeta functions, Totally real fields, Singularities in algebraic geometry, Global ground fields in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Special surfaces Signature defects of cusps and values of L-functions: The nonsplit 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 The values of the modular function \(j(\tau)\) at imaginary quadratic arguments in the upper half plane are known as singular moduli. They are algebraic integers, and their differences turn out to be highly divisible numbers. We determine the prime factorization of the absolute norm of \(j(\tau_ 1)-j(\tau_ 2)\), where \(\tau_ 1\) and \(\tau_ 2\) are arguments of discriminants \(d_ 1\) and \(d_ 2\), and show that all primes \(\ell\) which divide this norm must divide a positive integer of the form \((D-x^ 2)/4\), where \(D=d_ 1d_ 2\). We also factor values of the modular polynomial \(\Phi_ m(x,y)\) at singular moduli; in this case the primes dividing \(\Phi_ m(j(\tau_ 1),j(\tau_ 2))\) must divide a positive integer of the form \((m^ 2D-x^ 2)/4.\)
Two methods of proof are given. The first is algebraic, and exploits the connection between the arithmetic of maximal orders in quaternion algebras of prime discriminant \(\ell\) over \({\mathbb{Q}}\) and the geometry of supersingular elliptic curves in characteristic \(\ell\). The second is analytic, and is based on the calculation of the Fourier coefficients of the restriction to the diagonal of an Eisenstein series for the Hilbert modular group of \({\mathbb{Q}}(\sqrt{D})\). Both methods may be viewed as the special case \(N=1\) of the theory of local heights for Heegner points on \(X_ 0(N)\). modular function \(j(\tau)\); singular moduli; prime factorization of the absolute norm; modular polynomial; arithmetic of maximal orders in quaternion algebras; geometry of supersingular elliptic curves; Fourier coefficients; Eisenstein series; Hilbert modular group; local heights; Heegner points Gross, B. H.; Zagier, D. B., \textit{on singular moduli}, J. Reine Angew. Math., 355, 191-220, (1985) Modular and automorphic functions, Algebraic moduli of abelian varieties, classification, Special algebraic curves and curves of low genus, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Quaternion and other division algebras: arithmetic, zeta functions On singular moduli | 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 relate a certain category of sheaves of \(k\)-vector spaces on a complex affine Schubert variety to modules over the \(k\)-Lie algebra (for \(\text{char\,}k>0\)) or to modules over the small quantum group (for \(k=0\)) associated to the Langlands dual root datum. As an application we give a new proof of Lusztig's conjecture on quantum characters and on modular characters for almost all characteristics. Moreover, we relate the geometric and representation-theoretic sides to sheaves on the underlying moment graph, which allows us to extend the known instances of Lusztig's modular conjecture in two directions: We give an upper bound on the exceptional characteristics and verify its multiplicity-one case for all relevant primes.
One of the fundamental problems in representation theory is the calculation of the simple characters of a given group. This problem often turns out to be difficult and there is an abundance of situations in which a solution is out of reach. In the case of algebraic groups over fields of positive characteristic we have a partial, but not yet a full answer.
In 1979, George Lusztig conjectured a formula for the simple characters of a reductive algebraic group defined over a field of characteristic greater than the associated Coxeter number; [cf. \textit{G. Lusztig}, Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)]. Lusztig outlined in 1990 a program that led, in a combined effort of several authors, to a proof of the conjecture for almost all characteristics. This means that for a given root system \(R\) there exists a number \(N=N(R)\) such that the conjecture holds for all algebraic groups associated to the root system \(R\) if the underlying field is of characteristic greater than \(N\). This number, however, is unknown in all but low rank cases.
One of the essential steps in Lusztig's program was the construction of a functor between the category of intersection cohomology sheaves with complex coefficients on an affine flag manifold and the category of representations of a quantum group (this combines results of \textit{M. Kashiwara} and \textit{T. Tanisaki} [Duke Math. J. 77, No. 1, 21-62 (1995; Zbl 0829.17020)], and \textit{D. Kazhdan} and \textit{G. Lusztig} [J. Am. Math. Soc. 6, No. 4, 905-947, 949-1011 (1993; Zbl 0786.17017); ibid. 7, No. 2, 335-381, 383-453 (1994; Zbl 0802.17007, Zbl 0802.17008)]). This led to a proof of the quantum (i.e. characteristic 0) analog of the conjecture. \textit{H. H. Andersen, J. C. Jantzen} and \textit{W. Soergel} then showed that the characteristic zero case implies the characteristic \(p\) case for almost all \(p\) [cf. Representations of quantum groups at a \(p\)-th root of unity and of semisimple groups in characteristic \(p\): independence of \(p\). Astérisque 220 (1994; Zbl 0802.17009)].
One of the principal functors utilized in Lusztig's program was the affine version of the Beilinson-Bernstein localization functor. It amounts to realizing an affine Kac-Moody algebra inside the space of global differential operators on an affine flag manifold. A characteristic \(p\) version of this functor is a fundamental ingredient in Bezrukavnikov's program for modular representation theory [cf. \textit{R. Bezrukavnikov, I. Mirković} and \textit{D. Rumynin}, Ann. Math. (2) 167, No. 3, 945-991 (2008; Zbl 1220.17009)], and recently Frenkel and Gaitsgory used the Beilinson-Bernstein localization idea in order to study the critical level representations of an affine Kac-Moody algebra [cf. \textit{P. Fiebig}, Duke Math. J. 153, No. 3, 551-571 (2010; Zbl 1207.20040)].
There is, however, an alternative approach that links the geometry of an algebraic variety to representation theory. It was originally developed in the case of finite-dimensional complex simple Lie algebras by \textit{W. Soergel} [J. Am. Math. Soc. 3, No. 2, 421-445 (1990; Zbl 0747.17008)]. The idea was to give a ``combinatorial description'' of both the topological and the representation-theoretic categories in terms of the underlying root system using Jantzen's translation functors. This approach gives a new proof of the Kazhdan-Lusztig conjecture, but it is also important in its own right: when taken together with the Beilinson-Bernstein localization it establishes the celebrated Koszul duality for simple finite-dimensional complex Lie algebras [cf. \textit{W. Soergel}, loc. cit., and \textit{A. Beilinson, V. Ginzburg, W. Soergel}, J. Am. Math. Soc. 9, No. 2, 473-527 (1996; Zbl 0864.17006)].
In this paper we develop the combinatorial approach for quantum and modular representations. We relate a certain category of sheaves of \(k\)-vector spaces on an affine flag manifold to representations of the \(k\)-Lie algebra or the quantum group associated to Langlands' dual root datum (the occurrence of Langlands' duality is typical for this type of approach). As a corollary we obtain Lusztig's conjecture for quantum groups and for modular representations for large enough characteristics.
The main tool that we use is the theory of sheaves on moment graphs, which originally appeared in the work on the localization theorem for equivariant sheaves on topological spaces by \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25-83 (1998; Zbl 0897.22009)] and \textit{T. Braden} and \textit{R. MacPherson} [Math. Ann. 321, No. 3, 533-551 (2001; Zbl 1077.14522)]. In particular, we state a conjecture in terms of moment graphs that implies Lusztig's quantum and modular conjectures for all relevant characteristics.
Although there is no general proof of this moment graph conjecture yet, some important instances are known: The smooth locus of a moment graph is determined by \textit{P. Fiebig} [loc. cit.], which yields the multiplicity-one case of Lusztig's conjecture in full generality. Moreover, by developing a Lefschetz theory on a moment graph we obtain in [\textit{P. Fiebig}, J. Reine Angew. Math. 673, 1-31 (2012; Zbl 1266.20059)] an upper bound on the exceptional primes, i.e. an upper bound for the number \(N\) referred to above. Although this bound is huge (in particular, much greater than the Coxeter number), it can be calculated by an explicit formula in terms of the underlying root system. Kazhdan-Lusztig polynomials; irreducible characters; highest weight modules; simple Lie algebras; quantized enveloping algebras; reductive algebraic groups; positive characteristic; root systems; intersection cohomology sheaves; Schubert varieties; character formulae; Coxeter numbers; Lusztig conjecture; affine flag manifolds; affine Kac-Moody algebras; moment graphs Fiebig, Peter, Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture, J. Amer. Math. Soc., 0894-0347, 24, 1, 133\textendash 181 pp., (2011) Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Modular representations and characters, Sheaf cohomology in algebraic topology Sheaves on affine Schubert varieties, modular representations, and Lusztig's conjecture. | 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 \(p\) be a prime integer. For any integers \(1\leqslant s\leqslant r\), \(\text{Alg}_{p^r,p^s}\) denotes the class of central simple algebras of degree \(p^r\) and exponent dividing \(p^s\). For any \(s<r\), we find a lower bound for the essential \(p\)-dimension of \(\text{Alg}_{p^r,p^s}\). Furthermore, we compute an upper bound for \(\text{Alg}_{8,2}\) over a field of characteristic 2. As a result, we show \(\text{ed}_2(\text{Alg}_{4,2})=\text{ed}(\text{Alg}_{4,2})=3\) and \(3\leqslant\text{ed}(\text{Alg}_{8,2})\leqslant 10\) over a field of characteristic 2. categories of field extensions; essential \(p\)-dimension; transcendence degrees; central simple algebras; essential dimension; Brauer groups; cyclic algebras Baek, S., Essential dimension of simple algebras in positive characteristic, C. R. Acad. Sci. Paris Sér. I Math., 349, 375-378, (2011) Finite-dimensional division rings, Brauer groups (algebraic aspects), Group actions on varieties or schemes (quotients) Essential dimension of simple algebras 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 Affine Hecke algebras \(H_{v_ 0}\) arise naturally in the representation theory of semisimple p-adic groups. Representations of \(H_{v_ 0}\) correspond to representations with Iwahori-fixed vectors, which are important to understand for applications to number theory.
A classification of simple modules for \(H_{v_ 0}\) is obtained by \textit{D. Kazhdan}, \textit{G. Lusztig} [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] in the special case that the c(s) in the defining relations \((T_ s+1)(T_ s-v_ 0^{2c(s)})=0\) are independent of s. The approach, based on equivariant K-theory, is not suited to the general case.
The author [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026)], defined a graded analogue \(\bar H_{r_ 0}\) of \(H_{v_ 0}\), for which the representation theory in the general case may be studied using equivariant homology and intersection cohomology. In the present paper, the author connects the representation theory of the \(H_{v_ 0}\) to that of \(\bar H_{r_ 0}\). The center \({\mathcal Z}\) of \(H_{v_ 0}\) is determined, generalizing unpublished work of Bernstein from the special case. A simple \(H_{v_ 0}\)-module M determines a unique maximal ideal T of \({\mathcal Z}\) such that \(TM=0\). The completion \(\hat {\mathcal Z}\) of the center is taken with respect to the T-adic topology. Then the completion of the affine Hecke algebra is defined by \(\hat H_{v_ 0}=H_{v_ 0}\otimes_{{\mathcal Z}}\hat {\mathcal Z}.\)
Graded algebras \(\bar H_{r_ 0}\) are defined via powers of a maximal ideal of a certain commutative subalgebra \({\mathcal O}\) of \(H_{v_ 0}\). The centers are determined and again completions are defined. The first reduction theorem is that the completion of an affine Hecke algebra with respect to a maximal ideal of the center is isomorphic to the ring of \(n\times n\) matrices over the completion of a smaller affine Hecke algebra. The second reduction is that a certain natural homomorphism from an affine Hecke algebra into a suitable completion of its graded version becomes an isomorphism when the first algebra is completed. The classification of simple \(H_{v_ 0}\)-modules is then reduced essentially to the classification of simple \(\bar H_{r_ 0}\)-modules, in the case the parameter \(v_ 0\in {\mathbb{C}}^ x\) has infinite order. representation theory of semisimple p-adic groups; simple modules; equivariant homology; intersection cohomology; affine Hecke algebra; Graded algebras; completions Lusztig, G, \textit{affine Hecke algebras and their graded version}, J. Amer. Math. Soc., 2, 599-635, (1989) Representations of Lie and linear algebraic groups over local fields, Analysis on \(p\)-adic Lie groups, Graded rings and modules (associative rings and algebras), Representation theory for linear algebraic groups, Local ground fields in algebraic geometry, \(p\)-adic theory, local fields, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Affine Hecke algebras and their graded version | 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 elliptic plane curve \(Q\), the authors consider the problem of constructing holomorphic self-maps \(f\) of \(\mathbb P^2\) that leave \(Q\) invariant. The criterion for a self-map of \(Q\) to extend to \(\mathbb P^2\) is stated. The authors look at the singular points of \(Q\).
In contrast with the smooth case, most singular elliptic curves do not admit nontrivial self-maps. The obstructions given by the singular points of \(Q\) are discussed. Two invariants are defined in terms of Weierstrass' \(\sigma\) and \(\zeta\) functions, and an invariance criterion for the elliptic plane curves with ordinary singularities is stated.
The authors prove that there do not exist self-maps of \(\mathbb P^2\), for which \(Q\) is critical and invariant, and the backward orbit of any point of \(Q\) is dense in the Julia set of \(f\). The case of a smooth cubic \(Q\) is discussed. The classic tangent process on \(Q\) provides examples of self-maps that leave \(Q\) invariant. If it is required \(f\) to leave invariant a line of lines, \(Q\) must be isomorphic to the Fermat cubic. The case when \(f\) has minimal degree 2 is also discussed.
When an elliptic plane curve has enough symmetries, the invariants associated to its singular points can be calculated easily. The simplest case is the dual of a smooth cubic. Special families of elliptic quartics with two singular points are considered. Computer-generated pictures illustrate tangent processes on such curves. elliptic plane curve; holomorphic self-maps; obstructions; invariants at singular points; ordinary singularities; backward orbit; invariant critical components; Julia set; invariant smooth cubics; elementary maps; dual of smooth cubic; Fermat cubic; tangent process; symmetries; Weierstrass' \(\sigma\) and \(\zeta\) functions; elliptic quartics with two singular points; Cassini quartic; quartics with a cusp and a node; mixed quartic; invariant cuspidal quartic [BD02]A. Bonifant and M. Dabija, \textit{Self-maps of }P 2\textit{with invariant elliptic curves}, in: Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), Contemp. Math. 311, Amer. Math. Soc., Providence, RI, 2002, 1--25. Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables, Projective techniques in algebraic geometry, Elliptic curves Self-maps of \(\mathbb{P}^2\) with invariant elliptic curves | 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 study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations. Braid groups; Yang-Baxter groups; dynamical Yang-Baxter relations; classical Yang-Baxter relations; Kohno-Drinfeld algebras; 3-term relations algebras; Gaudin elements; Jucys-Murphy elements; small quantum cohomology; \(K\)-theory of flag varieties; Pieri rules; chromatic number; Tutte and Betti polynomials; reduced polynomials; Chan-Robbins-Yuen polytope; \(k\)-dissections of a convex \((n+k+1)\)-gon; Lagrange inversion formula; Richardson permutations; multiparameter deformations of Fuss-Catalan and Schröder polynomials; poly-Bernoulli numbers; Stirling numbers; Euler numbers; Brauer algebras; VSASM; CSTCPP; Birman-Ko-Lee monoid; Kronecker elliptic sigma functions Kirillov, Anatol N., On some quadratic algebras {I}~{\(\frac{1}{2}\)}: combinatorics of {D}unkl and {G}audin elements, {S}chubert, {G}rothendieck, {F}uss--{C}atalan, universal {T}utte and reduced polynomials, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 12, 002, 172~pages, (2016) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Yang-Baxter equations, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On some quadratic algebras. I \(\frac{1}{2}\): Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte and reduced polynomials | 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 real semisimple Lie group. This paper discusses the ``cohomological induction'' construction of representations of \(G\). The author sketches the most important results about the duality of cohomologically induced modules and the existence of invariant hermitian forms on modules in the ``middle degree''. Then he discusses the ``positivity'' conditions which imply the vanishing of all cohomologically induced modules except the one in the ``middle degree'' and irreducibility and unitarity of the latter. The final section reviews the author's work on various generalizations of these results and their relationship to still mysterious ``unipotent representations''. A number of illuminating examples of such representations is discussed. infinite dimensional representations; unipotent representations; real semisimple Lie groups; construction of representations; duality of cohomologically induced modules; invariant hermitian forms Jr., D. A. Vogan: Unipotent representations and cohomological induction. Contemporary math. 154, 47-70 (1993) Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Semisimple Lie groups and their representations, Applications of Lie groups to the sciences; explicit representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vanishing theorems in algebraic geometry, Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)), Local cohomology of analytic spaces, Hyperbolic and Kobayashi hyperbolic manifolds Unipotent representations and cohomological induction | 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 \(G\) denote a semisimple algebraic group over a field of characteristic \(p>0\). Then \textit{R. Bezrukavnikov, I. Mirković}, and \textit{D. A. Rumynin} [Ann. Math. (2) (to appear)] have established a derived localization theorem for the sheaf \(\mathcal D\) of algebras of crystalline differential operators on the flag variety \(X=G/B\).
Let now \(\overline{\mathcal D}=\mathcal End_{X^{(1)}}(\mathcal O_X)\) where \(X^{(1)}\) is the twist by the Frobenius morphism \(F\) on \(X\). Then \(\overline{\mathcal D}\) is a central reduction of \(\mathcal D\). The authors prove in this paper a derived localization theorem for \(\overline{\mathcal D}\) in the case where \(G=\text{Sp}_4\). They do so by checking via explicit calculations of the socle series for baby Verma modules that \(F_*\mathcal O_X\) is tilting (this requires \(p\geq 5\)). This generalizes earlier work of the first author with \textit{Y. Hashimoto} and \textit{D. Rumynin} [Contemp. Math. 413, 43-62 (2006; Zbl 1121.14041)] where the same result is proved for \(G=\text{SL}_3\). crystalline differential operators; localization theorems; symplectic groups; flag varieties; Frobenius morphisms; invertible sheaves; simple Lie algebras; derived equivalences; enveloping algebras Kaneda, M.; Ye, J., Equivariant localization of \(\overline{D}\)-modules on the flag variety of the symplectic group of degree 4, J. Algebra, 309, 236-281, (2007) Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Universal enveloping (super)algebras, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Group actions on varieties or schemes (quotients), Sheaves of differential operators and their modules, \(D\)-modules, Representation theory for linear algebraic groups Equivariant localization of \(\overline D\)-modules on the flag variety of the symplectic group of degree 4. | 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 \(R\) be any ring. There are two group structures on formal series in one variable over \(R\); the first one is \(G=1+xR[[x]]\) with the product, the second one is \(H=x+x^2R[[x]]\), with the composition. Moreover, \(H\) acts on \(G\) giving a semidirect product. These three groups are affine, so have a Hopf algebra of coordinates. This is generalized to locally finite categories. If \(C\) is such a category, it admits a large algebra, with a Cauchy-type product, and the set \(G\) of formal series in this large algebra such that the constant term is \(1\) is a group for the product. Under technical conditions, a group \(H\) of substitutions also exists. It is proved that both of these groups are affine, so have a coordinate Hopf algebra, which are free commutative algebras. Moreover, the action of \(H\) over \(G\), inducing a semidirect product, induces, at the level of their coordinate Hopf algebras, a smash coproduct. locally finite categories; coordinate Hopf algebras of affine groups; formal series; smash coproducts Hopf algebras and their applications, Connections of Hopf algebras with combinatorics, Affine algebraic groups, hyperalgebra constructions, Connections of group theory with homological algebra and category theory, Valuations, completions, formal power series and related constructions (associative rings and algebras) Two interacting coordinate Hopf algebras of affine groups of formal series on a category. | 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 Part II of the book, whose first Part has been reviewed in [Histoires hédonistes de groupes et géométries. Tome 1. Paris: Calvage et Mounet (2013; Zbl 1275.51001)]. Certainly that review is also in vogue in respect to Part II. In Part II, one can find a work-out of some topics of Part I, as well as some new topics. As such we mention, among others, Hecke algebras, Bruhat decomposition, Dynkin diagrams, Clifford chains, classical Lie groups, finite subgroups of \(\text{SO}(3)\) and \(\text{SO}(\mathbb{R})\) and its representations, McKay correspondence and characters of subgroups of \(\text{SU}_2(\mathbb{C})\).
Both the two parts give a beautiful view into group theory and geometry. actions of groups; linear algebra; topological groups; endomorphisms; Grassmannians; echelon matrices; groups preserving a bilinear form; quaternion fields; algebraic combinatorics; Lie groups; Platonic solids; topics from the projective plane; orthogonal groups; unitary groups; symplectic groups; Young tableaux; algebraic geometry; algebraic curves; surfaces configurations; special varieties; graphes; projective line; conics; representation theory; McKay correspondance Ph. Caldero, J. Germoni, \textit{Histoires Hédonistes de Groupes et de Géométries [Hedonistic Histories of Groups and Geometries].} Vol. 2, Calvage et Mounet, Paris, 2015. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, Affine analytic geometry, Projective analytic geometry, Geometry of classical groups, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), Euclidean analytic geometry, Analytic geometry with other transformation groups, General theory of linear incidence geometry and projective geometries, Representations of finite symmetric groups, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Curves in algebraic geometry, Singularities of curves, local rings Hedonistic histories of groups and geometries. Vol. 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 The author constructs examples of algebraic surfaces which are desingularizations of singular \(({\mathbb Z}/2\mathbb{Z})^2\)-covers of \({\mathbb P}^1\times{\mathbb P}^1\). One series of examples are simple canonical surfaces with \(p_g=4\) (the minimal possible value for a canonical surface) and \(K^2=11,...,28\), that disproves a prediction of Enriques about \(24\) as the maximum value of \(K^2\) in this case. Among other examples one finds surfaces with \(p_g=q=1\) and \(K^2=4,5\), and an infinite series of surfaces whose canonical map is composed of a pencil of curves of genus \(2\) and \(3\), with non-constant moduli. simple canonical algebraic surfaces; bidouble Galois covers; resolution of singularities; moduli of surfaces of general type F. Catanese, Singular bidouble covers and the construction of interesting algebraic surfaces, in Algebraic Geometry: Hirzebruch 70, eds. P. Pragacz \textit{et al.}, Proc. Algebraic Geometry Conference in honor of F. Hirzebruch's 70th birthday, \textit{Contempory Mathematics}, Vol. 241 (Springer, 1999), pp. 97-120. Coverings in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Special surfaces, Singularities of surfaces or higher-dimensional varieties, Surfaces of general type, Families, moduli, classification: algebraic theory Singular bidouble covers and the construction of interesting algebraic 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 ``In this paper we consider certain abelian varieties defined over \(\mathbb{Q}\) which arise as simple factors of the Jacobian variety \(J_ 0(N)\) of the modular curve \(X_ 0(N)\). These abelian varieties with `extra twist' were investigated fully by \textit{K. A. Ribet} [Math. Ann. 253, 43-62 (1980; Zbl 0421.14008)] and \textit{F. Momose} [J. Fac. Sci., Univ. Tokyo, Sect. IA 28, 89-109 (1981; Zbl 0482.10023)]. Here we shall be interested in the case where the variety \(A\) is two-dimensional and the extra twist is given by the character attached to a quadratic field \(k\); the \(L\)-series of \(A\) then turns out to be the Mellin transform \(L(F,s)\) of a cuspidal automorphic form \(F\) of weight 2 over \(k\), with rational integer coefficients. Such cusp forms, for imaginary quadratic \(k\), have been studied by the author \((\dots)\) and others \((\dots)\), seeking a Weil-Taniyama-type correspondence between cusp forms of weight 2 and elliptic curves over \(k\). Our main result \((\dots)\) is that if \(A\) remains simple over \(k\) then \(L(F,s)\) is \textit{not} the \(L\)-series of an elliptic curve defined over \(k\).''
Numerical examples are given in the range \(N\leq 300\). Actually, 12 relevant abelian varieties \(A\) are listed, 2 of which (one with CM, one without) fail to split over the twisting field \(k\). The final section is devoted to a discussion of the Weil-Taniyama conjecture over imaginary quadratic fields in relation to the author's results. modular curve; abelian varieties with twist; \(X_ 0\); simple factors of the Jacobian variety; correspondence between cusp forms of weight 2 and elliptic curves; Taniyama-Weil conjecture J.E. Cremona, Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields, J. London Math. Soc. (2), 45 (1992), 404-416. MR 93h:11056 Arithmetic ground fields for abelian varieties, Modular and Shimura varieties, Abelian varieties of dimension \(> 1\), Elliptic curves over global fields, Holomorphic modular forms of integral weight, Quadratic extensions Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields | 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, the author discusses some generalizations of the two-dimensional tree level conformal field theory to the framework of moduli spaces of locally trivialized holomorphic \(G\)-bundles over the Riemann sphere.
His approach is based upon \textit{G. Segal}'s definition for two-dimensional conformal field theory (Preprint, 1989), and his generalizing constructions use the theory of operads and their algebras. More precisely, the author studies the moduli space \({\mathcal E} (m)\) of locally trivialized holomorphic rank-\(m\) vector bundles over the Riemann sphere as an analytic partial operad, their relations with infinite loop groups, and their link to infinite Grassmannians. This leads to the classification and construction of all modular functors of dimension one over the moduli spaces \({\mathcal E} (m)\), including their determinant bundles. The main theorem of the present work is that the category of integrable algebras over one-dimensional modular functors on \({\mathcal E} (m), \; m >0\), is isomorphic to the category of the so-called integrable generalized affine vertex operator algebras, which is introduced here, too. Conformal field theory; vector bundles; moduli of vector bundles; loop groups; operads; modular functors; vertex operator algebras; infinite Grassmannians. Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Vertex operators; vertex operator algebras and related structures, Moduli problems for differential geometric structures, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Groups and algebras in quantum theory and relations with integrable systems, Loop groups and related constructions, group-theoretic treatment Some generalizations of genus zero two-dimensional conformal field 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 Assume \(G\) is a connected reductive algebraic group over an algebraically closed field \(K\) of characteristic zero. The author classifies the finite dimensional \(G\)-modules \(V\) of the reductive group \(G\) with spherical orbits. It is shown that any module with this property can be realized as a spherical module after an extension of the group by a central torus.
A \(G\)-action on an irreducible variety \(X\) is called an action with spherical orbits if there is an open subset \(X_0\subseteq X\) such that the orbit \(Gx\) is spherical for any \(x\in X_0\). Let \(V\) be a finite dimensional \(G\)-module and \(V=V_1\oplus\cdots\oplus V_k\) a decomposition into simple \(G\)-submodules. Denoted by \(T\) the \(k\)-dimensional algebraic torus acting by dilatations on any \(V_i\). Then \(\overline G=TG\) is a linear group acting on \(V\). The author shows that the \(\overline G\)-module \(V\) is spherical, and then lists all ``minimal'' indecomposable linear actions with spherical orbits. A characterization of arbitrary \(G\)-modules with spherical orbits is obtained. In addition, it is shown that for any \(G\)-module with spherical orbits the algebras of \(U\)- and \(G^s\)-invariants are free, where \(U\) (\(G^s\)) is the maximal connected unipotent (semisimple) subgroup of \(G\). At the end, the author obtains a classification of \(G\)-actions with spherical orbits on projective spaces \(\mathbb{P}(V)\). algebras of invariants; Borel subgroups; central tori; group actions; homogeneous spaces; maximal semisimple subgroups; maximal unipotent subgroups; simple modules; reductive groups; representations; spherical actions; spherical modules; spherical orbits; spherical varieties; weights Representation theory for linear algebraic groups, Homogeneous spaces and generalizations, Group actions on affine varieties, Group actions on varieties or schemes (quotients), Geometric invariant theory, Linear algebraic groups over arbitrary fields A classification of reductive linear 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 this important paper the author generalizes to the noncommutative case of relatively free algebras the classical theorem of Shephard-Todd and Chevalley from (commutative) invariant theory: the algebra of invariants of a finite linear group is isomorphic to the polynomial algebra if and only if the group is generated by pseudo-reflections.
Let \(F_n(I)\) be a relatively free algebra over a field \(K\) of characteristic 0. This is the free algebra \(K\langle x_1,\dots,x_n\rangle\) modulo the T-ideal \(I\) of all polynomial identities in \(n\) variables of a PI-algebra \(R\). Then the general linear group \(\text{GL}_n(K)\) acts as a group of automorphisms on \(F_n(I)\). The main result is that the algebra of invariants \(F_n(I)^G\) of a finite linear group \(G\neq\langle 1\rangle\) is isomorphic to the relatively free algebra \(F_n(I)\) if and only if \(G\) is generated by pseudo-reflections and \(I\) contains the polynomial identity \([[x_2,x_1],x_1]\). Hence the theorem of Shephard-Todd and Chevalley can be extended only to algebras satisfying all polynomial identities of the Grassmann (or exterior) algebra. Before it was known that the main result of \textit{R. M. Guralnick} [Linear Algebra Appl. 72, 85-92 (1985; Zbl 0579.20039)] implies that the algebras of invariants of relatively free algebras are never relatively free if the T-ideal contains only matrix polynomial identities. The complete answer obtained in the paper under review is based on various deep arguments from combinatorics, theory of PI-algebras, representations of the general linear group and the description of the pseudo-reflection groups. Hilbert series; groups generated by pseudo-reflections; Grassmann algebras; relatively free algebras; theorem of Shephard-Todd and Chevalley; algebras of invariants; polynomial algebras; T-ideals; PI-algebras; general linear groups; groups of automorphisms; matrix polynomial identities Domokos, M., Relatively free invariant algebras of finite reflection groups, Trans. Amer. Math. Soc., 348, 2217-2233, (1996) Automorphisms and endomorphisms, \(T\)-ideals, identities, varieties of associative rings and algebras, Vector and tensor algebra, theory of invariants, Exterior algebra, Grassmann algebras, Actions of groups on commutative rings; invariant theory, Reflection and Coxeter groups (group-theoretic aspects), Reflection groups, reflection geometries, Classical groups (algebro-geometric aspects) Relatively free invariant algebras 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 Let \(X\) be a closed hypersurface of a smooth variety \(Z\) over a field \(k\) of characteristic zero and \(E=\{ H_1, \ldots, H_m \}\) a collection of smooth hypersurfaces of \(Z\) with normal crossings. The pair \((X,E)\) has \textit{simple normal crossings} (or is snc) at a point \(a \in Z\) if there is a regular system of parameters \((x_1, \ldots, x_p,y_1, \ldots, y_q)\) of \({\mathcal O}_{Z,a}\) such that each irreducible component of \(X\) at \(a\) defined by \(x_i\) and each \(H \in E\) by \(y_j\), for suitable indices \(i, j\). If \((X,E)\) is snc at each point of \(Z\), we say that it is a snc pair. The main result of the paper under review is the following theorem.
Consider a pair \((X,E)\) as above. Then there is sequence
\[
(1) \quad Z=Z_0 \leftarrow Z_1 \leftarrow \cdots {\leftarrow} Z_t
\]
of blowing-ups, having smooth centers \(C_i \subset Z_i\), such that (letting \({\sigma}_i\) denote the \(i\)-th blowing-up, \(X_{i+1}\) the strict transform of \(X_i\) and \(E_{i+1}\) the collection of strict transforms of the hypersurfaces in \(E_i\) together with the exceptional divisor of \(\sigma _i\)) then \(C_i\) has snc with respect to \(E_i\) for all \(i\), \(C_i\) contains no snc singularity of \((X_i,E_i)\) (hence \(\sigma_{j+1}\) is an isomorphism over the set of snc of \((X_i,E_i)\)) and \((X_t,E_t)\) has only snc singularities. The association of the ``partial'' desingularization sequence (1) to \((X,E)\) is functorial with respect to smooth morphisms that preserve the number of irreducible components of \(X\) at each point \(a \in X\). Each \(C_i\) is necessarily contained in \(X_i\).
Two of the authors had obtained somewhat weaker results before. For instance, in [\textit{E. Bierstone} and \textit{P. D. Milman}, Adv. Math. 231, No. 5, 3022--3053 (2012; Zbl 1257.14002)] they found similar results but where, in the notation above, the composite morphism \({\sigma}_t \ldots {\sigma}_1\) is an isomorphism over the set of snc points of \((X,E)\), although the more precise statement of the new theorem on each morphism \(\sigma _i\) is not available.
As in the mentioned previous paper, the present algorithm to obtain the sequence (1) involves the use of inv, the fundamental desingularization function introduced in [\textit{E. Bierstone} and \textit{P. D. Milman}, Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)]. With the aid of inv and another numerical function \({\mu}_{H,k}\) (involving orders of certrain ideals along suitable exceptional hypersurfaces, previously introduced in [\textit{E. Bierstone} and \textit{P. D. Milman}, Publ. Res. Inst. Math. Sci. 44, No. 2, 609--639 (2008; Zbl 1151.14012)]), the authors characterize the snc points of each pair \((X_i,E_i)\) that appears in the sequence (1). Using centers where inv reaches a maximum, the authors reach a situation where they have to deal exclusively with the numbers \(\mu\). At this point, they start using another technique, a process they call ``cleaning''.
They give examples examples illustrating how the present method works, as well as one showing that it is different from that of the mentioned Adv. Math paper. resolution of singularities; simple normal crossings; desingularization invariant; cleaning; partial resolution algorithm Bierstone, E.; Silva, S.; Milman, P. D.; Vera Pacheco, F., Desingularization by blowings-up avoiding simple normal crossings, Proceedings of the American Mathematical Society, 142, 4099-4111, (2014) Global theory and resolution of singularities (algebro-geometric aspects), Singularities of surfaces or higher-dimensional varieties, Modifications; resolution of singularities (complex-analytic aspects), Singularities in algebraic geometry, Local complex singularities, Invariants of analytic local rings Desingularization by blowings-up avoiding 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 The very interesting paper under review can be seen as a finite type version of a celebrated result in representation theory, due to \textit{E. Date} et al. [J. Phys. Soc. Japan 50, 3806--3812 (1981; Zbl 0571.35099)]. The latter supplied a precise description of the ring \(\mathbf{C}[x_1,x_2,\ldots]\) in infinitely many indeterminates as a representation of the Lie algebra of the complex valued matrices of infinite size with finitely many non-zero diagonals.
The DJKM representation is based on the fact that the so-called \textit{Fermionic Fock Space} can be seen as the fundamental representation of the infinite Lie algebra \(gl_\infty(\mathbb{C})\). From a representation theoretical point of view, the \textit{Fermionic Fock space} can be roughly thought of an infinite wedge power of the vector space \(\mathbb{C}[X^{-1}, X]\) of Laurent polynomials in the indeterminate \(X\). The natural question the authors asked themselves is how the DJKM picture can be detected already for polynomial rings in finitely many indeterminates. To explain what and how they do let us walk a few steps backward, to render more precisely the feeling of their result.
To begin with, le \(K[X]\) be the ring of polynomials in one indeterminate over a field \(K\) of characteristic zero. Let \(B_r:=K[x_1,\ldots,x_r]\) be the \(K\)-algebra of polynomials in the indeterminates \(\mathbf{x}:=(x_1,\ldots,x_r)\). It is easy to convince oneself that \(B_r\) is isomorphic to the \(r\)-th exterior power \(\bigwedge^rK[X]\). Although a number of mathematicians like to explain this fact as a special case of some sophisticated \textit{Geometric Satake Correspondence}, the naive reason is that both spaces possess a basis parametrized by all the partitions of length at most \(r\). It is also easy to see that, as the \(r\)-th exterior power of \(K^n\) is a representation of the Lie algebra \(gl_n(K)\) of the \(K\)-valued \(n\times n\) square matrices, then \(\bigwedge^rK[X]\) turns into a representation of the Lie algebra \(gl_\infty(K)\) of the \(K\)-valued matrices \((a_{ij})_{i,j\geq 0}\) whose entries are all zero but finitely many. The isomorphism \(B_r\rightarrow \bigwedge^rK[X]\) then makes \(B_r\) itself into a representation of \(gl_\infty(K)\).
The main result of the paper under review, Theorem 4.11, consists in determining what the authors call the \textit{generating formal power series} \(\mathcal{E}(z,w^{-1}, t_1,\ldots, t_r)\), which describes \(B_r\) as a representation of the Lie algebra \(gl_\infty(K)\). This means the following. Recall that the elementary matrices \(E_{i,j}\) with all entries zero but \(1\) in position \((i,j)\) form a basis of \(gl_\infty(K)\) and that, interpreting \(B_r\) as the ring of symmetric polynomials in \(r\) indeterminates, it possesse a basis of Schur determinants \(\Delta_{\lambda}\) constructed out of the complete symmetric polynomials, parametrized by partitions of length at most \(r\). If \(\mathbf{t}_r:=(t_1,\ldots,t_r)\) is an \(r\)-tuple of formal variables, denote by \(s_{\lambda}(\mathbf{t}_r)\) the Schur polynomials in the indeterminates \(\mathbf{t}_r\), like at p. 40 of the book by \textit{I. G. Macdonald} [Symmetric functions and Hall polynomials. With contributions by A. V. Zelevinsky. Reprint of the 1998 2nd edition. Oxford: Oxford University Press (2015; Zbl 1332.05002)].
The determination of the generating formal power series of the aforementioned Theorem 4.11 heavily relies on the techniques introduced in the 2005 reviewer's paper [\textit{L. Gatto}, Asian J. Math. 9, No. 3, 315--322 (2005; Zbl 1099.14045)] and substantially improves results by \textit{L. Gatto} and \textit{P. Salehyan} [Commun. Algebra 48, No. 1, 274--290 (2020; Zbl 1442.14156)].
How does this relate with DJKM work? This is widely discussed in the last section of the paper under review. In a nutshell, taking a suitable limit \(r\to\infty\), one obtains an isomorphism from the ring \(B:=B_\infty\) to the charge zero vector subspace \(\mathcal{F}_0\) of the \textit{fermionic Fock space} \(\mathcal{F}:=\bigwedge^{\infty/2}K[X^{-1}, X]\). There is a vector space isomorphism of \(\mathcal{F}_0\) with the ring \(B\) of polynomials in infinitely many indeterminates \((x_1,x_2,\ldots)\) (the \textit{bosonic Fock space}), because both spaces possess a basis parametrized by partitions. The literature (see, e.g., [\textit{V. G. Kac} et al., Highest weight representations of infinite dimensional Lie algebras. World Scientific (2013)]) often refers to this isomorphism as the \textit{boson-fermion correspondence.} As a consequence, \(B\) is made into a representation of the Lie algebra \(gl(\infty)\), induced by the natural one living on the infinite wedge power. Moreover the vertex operators occurring in the DJKM representation are nothing but an infinite dimensional version of what the authors call \textit{Schubert derivations}, borrowing the own reviewer terminology, which, as the name suggest, are devices useful to cope with Schubert Calculus. The paper concludes itself with an essential but comprehensive reference list. Hasse-Schmidt derivations; vertex operators on exterior algebras; representation of Lie algebras of matrices; bosonic and fermionic representations by Date-Jimbo-Kashiwara-Miwa; symmetric functions Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Vertex operators; vertex operator algebras and related structures, Exterior algebra, Grassmann algebras, Symmetric functions and generalizations On the vertex operator representation of Lie algebras of matrices | 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 Cayley-Hamilton algebra \(A\), let \(\text{trep}_n(A)\) denote the variety of trace preserving \(n\)-dimensional representations of \(A\), and let \(\text{triss}_n(A)\) be its quotient modulo the \(\text{GL}_n\)-action. If \(\text{BS}_n(A)\) denotes the trace preserving Brauer-Severi variety, there is a fibration \(\pi\colon\text{BS}_n(A)\to\text{triss}_n(A)\).
The authors consider smooth orders \(A\), i.e.~orders in a central simple algebra for which \(\text{trep}_n(A)\) is a smooth variety. For such Cayley-Hamilton algebras \(A\), the étale local structure of the maps \(\text{BS}_n(A)\to\text{trep}_n(A)\to\text{triss}_n(A)\) can be described in terms of marked quivers. Using this method, the authors describe the flat locus of Brauer-Severi schemes of smooth orders. Furthermore, they analyse the singularities of \(\text{triss}_n(A)\) where the flat locus does not coincide with the Azumaya locus, and describe the fibers near such singularities. This generalizes classical results of M.~Artin. Brauer-Severi schemes; simple representations; moduli spaces; flat loci; maximal orders; central simple algebras; smooth curves; marked quivers Bocklandt R., Symens S., Van den Weyer G.: The flat locus of Brauer-Severi fibrations of smooth orders. J. Algebra 297(1), 101--124 (2006) Representations of orders, lattices, algebras over commutative rings, Fibrations, degenerations in algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Infinite-dimensional and general division rings The flat locus of Brauer-Severi fibrations of smooth orders. | 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 totally real number field of degree \(n\) and \(K\hookrightarrow \mathbb R^n\) the usual embedding. Let \(M\) be a lattice of rank \(n\) in \(K\) and \(V\) a subgroup of maximal rank \((n-1)\) of the group of those totally positive units which stabilize \(M\). Pairs \((M,V)\) describe cusps of the Hilbert modular variety associated to \(K\). In a beautiful series of papers, Hirzebruch has shown how to use the number theory of \(K\) to resolve these cusps when \(n=2\).
This papers gives completely explicit resolutions of these cusps for two infinite families of cubics. More generally, for any cubic it gives a fundamental domain, \(D\), for the action of \(V\) on \(\mathbb R^n_+\) such that \(D\) is a finite union of open simplicial cones with vertices in \(M\). Following \textit{M. Shintani} [J. Fac. Sci., Univ. Tokyo, Sect. I A 23, 393--417 (1976; Zbl 0349.12007)], this can be used to calculate, as a finite sum, the values at nonnegative integers of a certain zeta function attached to \(M\). The methods used are geometric with a mild use of algebraic topology. resolution of cusp singularities; Shintani decomposition; totally real cubic number fields; Hilbert modular variety; family of cubics; evaluation of zeta-function DOI: 10.1007/BF01359864 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Singularities of surfaces or higher-dimensional varieties, Special surfaces, \(3\)-folds, Totally real fields, Cubic and quartic extensions, Global ground fields in algebraic geometry On the resolution of cusp singularities and the Shintani decomposition in totally real cubic number fields | 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 \(n\)-dimensional Lie algebra \(\mathfrak g\) over a field \(k\supset\mathbb Q\), together with its vector space basis \(X_1^0, X_2^0,\dots, X_n^0\), we give a formula, depending only on the structure constants, representing the infinitesimal generators, \(X_i=X_i^0t\) in \(\mathfrak g\otimes_kk[[t]]\), where \(t\) is a formal variable, as a formal power series in \(t\) with coefficients in the Weyl algebra \(A_n\). Actually, the theorem is proved for Lie algebras over arbitrary rings \(k\supset\mathbb Q\).
We provide three different proofs, each of which is expected to be useful for generalizations. The first proof is obtained by direct calculations with tensors. This involves a number of interesting combinatorial formulas in structure constants. The final step in calculation is a new formula involving Bernoulli numbers and arbitrary derivatives of \(\coth(x/2)\). The dimensions of certain spaces of tensors are also calculated. The second method of proof is geometric and reduces to a calculation of formal right-invariant vector fields in specific coordinates, in a (new) variant of formal group scheme theory. The third proof uses coderivations and Hopf algebras. deformations of algebras; Lie algebras; Weyl algebra; Bernoulli numbers; representations; formal schemes N. Durov, S. Meljanac, A. Samsarov and Z. Skoda, \textit{A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, J. Algebra}\textbf{309} (2007) 318 [math/0604096] [INSPIRE]. Universal enveloping (super)algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Formal groups, \(p\)-divisible groups, Group schemes, Deformations of associative rings, Bernoulli and Euler numbers and polynomials A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl 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 It is conjectured that the Hilbert scheme of non-general type \(r\)-dimensional manifolds embedded in \(\mathbb{P}^N\) for \(N \leq 2r\) has only a finite number of components, or equivalently, that the degree of such manifolds is bounded. This was proved by Ellingsrud and Peskine for \(r = 2\), \(N = 4\), and subsequently for \(r = 3\), \(N = 5\). The proofs seem to depend on the fact that in these cases the codimension equals two. In Int. J. Math. 3, No. 3, 397-399 (1992; Zbl 0762.14022), \textit{M. Schneider} proves the conjecture for \(N\leq 2r-2\) by using the positivity of the Schur polynomials for the normal bundle and the fact that in this situation the canonical bundle is induced.
In this paper, we are using Schneider's method in order to prove boundedness for some special classes of embedded manifolds, e.g., Fano manifolds if \(N = 2r\) (proposition 1) and Fano fibrations over curves if \(N = 2r - 1\) (proposition 2). A different argument shows boundedness for scrolls over a surface, in case \(N = 2r - 1\) (proposition 4). We also give some existence results and a partial classification for the two special classes mentioned above when \(N = 2r - 1\) (propositions 3, 4 and 5). embedded manifolds; finite number of components of Hilbert schemes; Fano manifolds; Fano fibrations; scrolls Ionescu, P. and Toma, M.: Boundedness for some special families of embedded manifolds. (Contemp. Math.162, 215--225) American Mathematical Society 1994 Parametrization (Chow and Hilbert schemes), Embeddings in algebraic geometry, \(n\)-folds (\(n>4\)), Fano varieties Boundedness for some special families of embedded 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 ``We assume that there is given a finite family of projective subspaces in certain projective space. Our aim is to prove that the simple homotopy type of the union of all the subspaces in question is completely determined by the complex of the nerves resulting from the family, equipped with the filtration obtained by assigning to each simplex the Krull dimension of the corresponding intersection. This fact enables us to compute the homology groups of the union and its complement in principle.'' family of projective subspaces in projective space; nerve complex; simple homotopy type; Krull dimension; homology groups Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Homology and cohomology theories in algebraic topology, (Co)homology theory in algebraic geometry The topology of the configuration of projective subspaces in a projective space. 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 Let \(G\) be a finite group. A finite group \(X\) is called a Frattini cover of \(G\) if \(X/F=G\), where \(F\) is contained in the Frattini subgroup \(\Phi(X)\) of \(X\). Frattini covers have been studied considerably with respect to coverings of curves. Also, Frattini covers come up in studying minimal counterexamples.
The first main result of this paper is a lifting theorem for Frattini covers \(X\) of \(G=X/F\) with \(F\) a primary group of odd order. Using this result, \textit{J. G. Thompson}'s classification of the finite simple groups in which every proper subgroup is solvable [Bull. Am. Math. Soc. 74, 383-437 (1968; Zbl 0159.30804)] and a result of \textit{I. M. Isaacs} [Am. J. Math. 95, 594-635 (1973; Zbl 0277.20008)], the authors obtain the following characterization of finite solvable groups (\textit{M. J. J. Barry} [in ``On conditions related to nonsolvability'', \url{arXiv:1109.4913} (2011)] asked whether this was true): a finite group \(G\) is solvable if and only if \(x_1x_2x_3\neq 1\) for all nontrivial \(p_i\)-elements \(x_i\) of \(G\) for distinct primes \(p_i\), \(i=1,2,3\). Thompson [loc. cit.] proved this result if one considers all triples of nontrivial elements of coprime order.
In addition, the authors obtain some other characterizations of finite solvable groups, give a short proof of a theorem of \textit{W. Feit} and \textit{J. Tits} [Can. J. Math. 30, 1092-1102 (1978; Zbl 0358.20014)] about the minimal dimension of a representation of a group which has a section isomorphic to a given simple group, and discuss some connections between lifting theorems and coverings of curves over \(\mathbf C\). finite groups; Frattini covers; characterizations of finite solvable groups; representations of simple groups; coverings of curves Guralnick, RM; Tiep, PH, Lifting in Frattini covers and a characterization of finite solvable groups, J. Reine Angew. Math., 708, 49-72, (2015) Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, Extensions, wreath products, and other compositions of groups, Special subgroups (Frattini, Fitting, etc.), Ordinary representations and characters, Coverings of curves, fundamental group Lifting in Frattini covers and a characterization of finite solvable 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 \((A,\sigma)\) be a central simple algebra of even degree with an involution of orthogonal type, over a field \(F\) of characteristic not 2. Using techniques of generic algebras, the author gives a nice necessary condition for the existence of a decomposition \((A,\sigma)=(A_1,\sigma_1)\otimes(A_2,\sigma_2)\). If \((A,\sigma)\) decomposes it is known that the Clifford algebra \(C(A,\sigma)\) of \((A,\sigma)\) decomposes as a product \(C(A,\sigma)=C_+(A,\sigma)\times C_- (A,\sigma)\) of central simple algebras. Using the notations of the author, let \(\text{sp}(A,\sigma)\in F^\times/F^{\times 2}\) be the signed discriminant of the involution \(\sigma\) on \(A\) and let \(Q\) be the quaternion algebra \((\text{sp}(A_1,\sigma_1),\text{sp}(A_2,\sigma_2))_F\). Let further \(n_i\) be the index of \(A_i\) and let \(\sim\) denote Brauer equivalence. If \(4|n_i\) for \(i=1\) or 2, then \(C_+(A,\sigma)\sim A\otimes Q\) and \(C_+(A,\sigma) \sim Q\) (or vice versa) and, if \(n_i \equiv 2 \bmod 4\) for \(i=1, 2\), then \(C_+(A,\sigma) \sim A_1 \otimes Q\) and \(C_- (A,\sigma) \sim A_2 \otimes Q\) (or vice versa). Using his criterion, the author gives an example of a split central simple algebra of degree 8 with an involution which is indecomposable. central simple algebras; involutions of orthogonal type; generic algebras; Clifford algebras; signed discriminants; quaternion algebras; Brauer equivalence Tao D.: The generalized even Clifford algebra. J. Algebra 172, 184--204 (1995) Rings with involution; Lie, Jordan and other nonassociative structures, Finite-dimensional division rings, Clifford algebras, spinors, Brauer groups of schemes, Homogeneous spaces and generalizations, Computations of higher \(K\)-theory of rings, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) The generalized even Clifford 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 this paper the author gives ``approximations'' for Zariski-dense subgroups of semi-simple algebraic groups. To make it precise, let \(G\) be an almost simple, connected and simply-connected algebraic group over an algebraically closed field of characteristic different from 2 and 3. Let \(\Gamma\) be a finitely generated Zariski-dense subgroup of \(G(k)\). Denoting the adjoint action of \(G\) on \(\mathrm{Lie}\, G\), the Lie algebra of \(G\), by Ad, let \(A\) be the subring of \(k\) generated by the traces \(\text{tr Ad}\,\gamma\), \(\gamma\in\Gamma\). The author shows that \(G\) has a structure \(G_A\) of a group scheme over \(A\) and for a suitable choice of \(b\) in \(A\), the author shows that \(A_b\) is an affine \(k\)-algebra and regular, and \(G_{A_b}(A_b)\cap \Gamma\) is of finite index in \(\Gamma\). The author exhibits a certain normal subgroup \(\Gamma'\) of \(\Gamma\) contained in \(G_{A_b}(A_ b)\cap \Gamma\) such that the reduction of \(\Gamma'\) modulo most maximal ideals \(M\) of \(A_b\) is \(G_{A_b}(A_b/M)\). (In the process of proving this, the author uses classification of finite simple groups.)
Using this, it is shown that the reduction of \(\Gamma'\) modulo any cofinite ideal \(I\) of \(A_b\) (i.e., \(| A_b/I| <\infty)\) is \(G_{A_b}(A_b/I)\). Thus the author obtains the main result that \(\Gamma'\) is dense in \(G_{A_b}(\hat A_b)\) where \(\hat A_b\) denotes the profinite completion \(\varprojlim_{| A_b/I| <\infty}(A_b/I)\). Zariski-dense subgroups of semi-simple algebraic groups; adjoint action; Lie algebra B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. (2) 120 (1984), no. 2, 271-315. Classical groups (algebro-geometric aspects), Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Strong approximation for Zariski-dense subgroups of semi-simple 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 Let \(A\) be a central simple algebra of degree \(n\) over a field \(F\) with an orthogonal involution \(*\). In analogy with the classical construction of the Brauer-Severi variety \(BS(A)\) of \(A\), the author defines a Brauer- Severi variety \(IV(A,*)\), which he calls the involution variety, for the pair \((A,*)\) and the main topic of this paper is to discuss properties of \(IV(A,*)\). The function field \(K\) of \(IV(A,*)\) is a generic isotropic splitting field for \(IV(A,*)\), i.e. it splits \(A\) and \(*\) is over \(K\) the adjoint involution of an isotropic quadratic form. The kernel of the map of Brauer groups \(\text{Br}(F)\to\text{Br}(K)\) is the subgroup generated by the class of \(A\), except when \((A,*)\) is the tensor product of two quaternion algebras \(Q_ 1\) and \(Q_ 2\) with canonical involutions, in which case the kernel is generated by the classes of \(Q_ 1\) and \(Q_ 2\). This case is quite special (``\(D_ 2= A_ 1+ A_ 1\)'') and has to be treated separately. Further the author gives some criteria to determine when \(K\), which has transcendence degree \(n-2\), can be embedded into \(K_ B\), the function field \(K_ B\) of \(BS(A)\), which has transcendence degree \(n-1\). Finally the author computes the Quillen \(K\)- theory of \(IV(A,*)\) and applies the result to obtain an index reduction formula for the function field \(K\) of \(IV(A,*)\). tensor product of quaternion algebras; central simple algebras; orthogonal involution; Brauer-Severi variety; involution variety; function fields; generic isotropic splitting field; Brauer groups; Quillen \(K\)-theory D. Tao, ''A variety associated to an algebra with involution'',J. Algebra,168, 479--520 (1994). Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Rings with involution; Lie, Jordan and other nonassociative structures, Brauer groups of schemes, Computations of higher \(K\)-theory of rings, Homogeneous spaces and generalizations A variety associated to an algebra with 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 0702.00033.]
This article consists of three parts. The first of them contains general information about abelian varieties. In the second part, Lie algebras are discussed, which correspond to the Mumford-Tate groups and Galois groups acting in the Tate modules of abelian varieties. These sections are written as a short and incomplete survey for nonspecialists. -- The third part is devoted to abelian varieties over number fields. It is proved a finiteness theorem for the dimension of an abelian variety with fixed dimension of the endomorphism algebra and of rank of the semisimple component of the \(\ell\)-adic Lie algebra of the Galois group acting in the Tate module. Lie algebras; Mumford-Tate groups; Tate modules; dimension of an abelian variety Yu. G. Zarhin, Abelian varieties of Lie algebras , Mathematics and Modelling, Research Computing Center of the USSR Academy of Sciences, Pushchino, 1990, English translation will appear in Selecta Math. Soviet, pp. 57-99. Arithmetic ground fields for abelian varieties, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Abelian varieties and 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 purpose of this paper is the classification of 2-simple prehomogeneous vector spaces of type I. A prehomogeneous vector space (G,\(\rho\),V) is called a 2-simple prehomogeneous vector space when (1) \(G=GL(1)^{\ell}\times G_ 1\times G_ 2\) with simple algebraic groups \(G_ 1\) and \(G_ 2\), (2) \(\rho\) is the composition of a rational representation \(\rho '\) of \(G_ 1\times G_ 2\) of the form
\[
\rho '=\rho_ 1\otimes \rho_ 1'+...+\rho_ k\otimes \rho_ k'+(\sigma_ 1+...+\sigma_ s)\otimes 1+1\otimes (\tau_ 1+...+\tau_{\ell})
\]
with \(k+s+t=\ell\), where \(\rho_ i\), \(\sigma_ i\) (resp. \(\rho_ j'\), \(\tau_ j)\) are nontrivial irreducible representations of \(G_ 1\) (resp. \(G_ 2)\), and the scalar multiplications \(GL(1)^{\ell}\) on each irreducible component \(V_ i\) for \(i=1,...,\ell\) where \(V=V_ 1\oplus...\oplus V_{\ell}\). We say that a 2-simple prehomogeneous vector space (G,\(\rho\),V) is of type I if \(k\geq 1\) and at least one of \((GL(1)\times G_ 1\times G_ 2,\rho_ i\otimes \rho_ i')\) \((i=1,...,k)\) is a non-trivial prehomogeneous vector space. If \(k\geq 1\) and all \((GL(1)\times G_ 1\times G_ 2,\rho_ i\otimes \rho_ i')\) \((i=1,...,k)\) are trivial prehomogeneous vector spaces, it is called of type II. In a previous paper [\textit{T. Kimura}, \textit{S.-i. Kasai}, \textit{M. Taguchi} and \textit{M. Inuzuka}, Trans. Am. Math. Soc. 308, No.2, 433-494 (1988)], all 2-simple prehomogeneous vector spaces of type II have been classified. Together with the previous paper, this paper completes a classification of all 2-simple prehomogeneous vector spaces. 2-simple prehomogeneous vector spaces of type I; simple algebraic groups; rational representation; irreducible representations Tatsuo Kimura, Shin-ichi Kasai, Masaaki Inuzuka, and Osami Yasukura, A classification of 2-simple prehomogeneous vector spaces of type \?, J. Algebra 114 (1988), no. 2, 369 -- 400. Linear algebraic groups over the reals, the complexes, the quaternions, Representation theory for linear algebraic groups, Homogeneous spaces and generalizations A classification of 2-simple prehomogeneous vector spaces of type 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 The survey covers recent developments in toric geometry since 1995 (for the state of research that time see \textit{D. A. Cox} [American Mathematical Society. Proc. Symp. Pure Math. 62 (pt.2), 389--436 (1997; Zbl 0899.14025)]. One of its goals is to introduce the reader to the Lectures of the Summer School on the Geometry of toric varieties held at the Fourier Institute (Grenoble, 2000) and published in \textit{L. Bonavero} and \textit{M. Brion} (eds.), Séminaires et Congrés. 6 (2002; Zbl 1005.00028). The main topics under consideration are: the minimal model program for toric varieties [\textit{J. Wisniewski}, Sémin. Congr. 6, 249--272 (2002; Zbl 1053.14002)], toric singularities and their resolution [\textit{D. I. Dais}, Sémin. Congr. 6, 187--192 (2002; Zbl 1047.14039); ibid. 155--186 (2002; Zbl 1047.14038)], the McKay correspondence and \(G\)-Hilbert schemes of \({\mathbb C}^n\) [\textit{Y. Ito}, Sémin. Congr. 6, 213--225 (2002; Zbl 1054.14065); \textit{A. Craw} and \textit{M. Reid}, Sémin. Congr. 6, 129--154 (2002;Zbl 1080.14502)], polytopes and polytopal algebras in toric geometry [\textit{W. Bruns} and \textit{J. Gubeladze}, Sémin. Congr. 6, 43--127 (2002; Zbl 1083.14057)], toric quotients and embeddings into toric varieties [\textit{J. Hausen}, Sémin. Congr. 6, 193--212 (2002; Zbl 1050.14045)], toric varieties over number fields [\textit{Y. Tschinkel}, Sémin. Congr. 6, 227--247 (2002; Zbl 1064.14018)].
Among important results discussed in the survey it is worth to mention the following. The toric flip theorem is the main tool for the minimal model program. Classification results on toric Fano varieties were obtained by Batyrev, Bonavero, A.~Borisov et al. Various types of toric singularities are presented in terms of cones of a fan and the problem of constructing a canonical crepant resolution is considered. For instance, if \(X={\mathbb C}^3/G\) is an Abelian quotient singularity, then its canonical crepant toric resolution \(\widetilde{X}\) is given by the \(G\)-Hilbert scheme of \({\mathbb C}^3\). Given a torus action \(H:X\), under certain technical conditions there is an equivariant embedding of \(X\) into a smooth toric variety \(Z\) acted on by a bigger torus such that all maximal open subsets of \(X\) having a good quotient are obtained by intersecting \(X\) with some toric open subvariety of \(Z\) having a good quotient (and these toric open subsets can be described in combinatorial terms). Important results of Tschinkel, Batyrev, Peyre et al provide the asymptotics for the number of rational points with bounded height on toric varieties over number fields.
Among other subjects discussed more briefly there are toric ideals [\textit{B. Sturmfels}, American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 437-449 (1997; Zbl 0914.14022)], quotient presentations and homogeneous coordinates on toric varieties, cohomology theories on toric varieties [\textit{D. Eisenbud, M. Mustata} and \textit{M. Stillman}, J. Symb. Comput. 29, No.4-5, 583-600 (2000; Zbl 1044.14028), \textit{V. V. Batyrev} and \textit{D. A. Cox}, Duke Math. J. 75, No.2, 293-338 (1994; Zbl 0851.14021), \textit{G. Barthel; J.-P. Brasselet, K.-H. Fieseler} and \textit{L. Kaup}, Contemp. Math. 241, 45-68 (1999; Zbl 0970.14028), \textit{M. Brion}, Banach Cent. Publ. 36, 25-44 (1996; Zbl 0878.14035)], toric fibrations et al. toric variety; fan; polytope; resolution of singularities; quotient space; cohomology; minimal model program; McKay correspondence; \(G\)-Hilbert scheme Cox, D.: Update on toric geometry. Sémin. congr. 6, 1-41 (2002) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Update on toric 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 [For the entire collection see Zbl 0711.00011.]
Es werden endlichdimensionale \(\lambda\)-adische Darstellungen \(\rho\) von Galoisgruppen über Zahlkörpern studiert. Unter geeigneten Irreduzibilitätsvoraussetzungen läßt sich die von Im \(\rho\) erzeugte Liealgebra in ihr Zentrum und eine halbeinfache Liealgebra vom Rang r aufspalten. Der Verf. beweist unter Annahme einer Weil- Riemannschen Vermutung, daß der Darstellungsgrad beschränkt ist durch eine Konstante, die nur abhängt von r und dem in der Weil- Riemannschen-Vermutung vorkommenden Exponenten. Auf dem Weg über die Darstellung auf dem Tate-Modul einer abelschen Varietät erhält der Verf. eine interessante Abschätzung ihrer Dimension durch den Rang ihrer Endomorphismenalgebra. Tate module; \(\ell \)-adic representations; Galois groups; Weil-Riemann conjecture; \(\ell \)-adic Lie algebras; dimension of Abelian variety Langlands-Weil conjectures, nonabelian class field theory, Galois theory, Arithmetic ground fields for abelian varieties Finiteness theorems for dimensions of irreducible \(\lambda\)-adic 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 Associated to a generalized Cartan matrix A, one can construct Lie algebras (in general infinite dimensional) and groups which are a generalization of semi-simple Lie algebras and Lie groups and called Kac- Moody Lie algebras and Kac-Moody groups associated with A. Many of the theory of semi-simple Lie groups have been generalized to these groups.
In this note, to each commutative ring k and the matrix A, the author associates in a functorial way a group ind-scheme (which is a generalization of the Chevalley scheme). Using the existence of this group and some of his methods given in [``Formule de Weyl et Demazure, et Théorème de Borel-Weil-Bott pour les algèbres de Kac-Moody générales'', Astérisque 159/160 (1988)], the author proves a technical theorem from which he can deduce the projective normality of Schubert varieties and the existence of some factors in the tensor product of two G-modules. generalized Cartan matrix; Kac-Moody Lie algebras; Kac-Moody groups; projective normality of Schubert varieties; tensor product of two G- modules O. Mathieu : Construction du groupe associé aux algèbres de Kac-Moody . Comptes Rendus 306 série I (1988) 227-230. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Group schemes, Infinite-dimensional Lie groups and their Lie algebras: general properties Construction du groupe de Kac-Moody et applications. (Construction of the Kac-Moody group with applications) | 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 start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group \(G\), the derived and the stable categories of representations of a subgroup \(H\) can be constructed out of the corresponding category for \(G\) by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry.
In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup \(H\) can be extended to \(G\). We show that the presheaves of plain, derived and stable representations all form stacks on the category of finite \(G\)-sets (or the orbit category of \(G\)), with respect to a suitable Grothendieck topology that we call the \textit{sipp topology}.
When \(H\) contains a Sylow subgroup of \(G\), we use sipp Čech cohomology to describe the kernel and the image of the homomorphism \(T(G)\to T(H)\), where \(T(-)\) denotes the group of endotrivial representations. restrictions of representations; extensions of representations; stacks; modular representations; finite groups; ring objects; descent; endotrivial representations; categories of finite \(G\)-sets; derived categories; stable categories P. Balmer, Stacks of group representations. J. Eur. Math. Soc. (JEMS) 17 (2015), no. 1, 189--228.MR 3312406 Zbl 06419400 Modular representations and characters, Generalizations (algebraic spaces, stacks), Grothendieck topologies and Grothendieck topoi, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Derived categories, triangulated categories, Étale and other Grothendieck topologies and (co)homologies Stacks of group 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 Locally all holomorphic Poisson structures on a complex manifold are well understood. In local coordinates \((z_1, \ldots z_n)\), we have
\[\{ z_a, z_b \} = g_{ab},\]
and in a neighbourhood of any generic point one can always find coordinates such that \(g_{ab}\) are constants. However, studying or classifying global holomorphic Poisson structures is a very hard problem on a given compact complex manifold.
In the paper under review, the author studies Poisson structure of a specific form, namely,
\[\{z_a(x), z_b(y) \} = g_{ab}\, \delta'(x-y) + \sum_{1\leq c \leq n} h^c_{ab}z'(x) \, \delta(x-y),\]
where the coordinates \(z_a\) are promoted to fields \(z_a(x)\), and both \(g_{ab}\) and \(h^c_{ab}\) are holomorphic in these fields. The author studies global holomorphic Poisson structures of this specific form on projective spaces \(\mathbb{C}P^n\). In particular, a family of Poisson structures of hydrodynamic type on the loop space of \(\mathbb{C}P^{n-1}\) are constructed. This family is parametrised by the moduli space of elliptic curves. In order to lift these Poisson structures to the loop space of \(\mathbb{C}^n\), the modular parameter \(\tau\) needs to be promoted to a field \(\tau(x)\) that satisfies
\[ \{\tau(x), \tau(y) \} =0,\]
and
\[\{ \tau(x), z_a(y)\} = 2 \pi \, z_a(y)\, \delta'(x-y),\]
where \(z_a\) are coordinates on \(\mathbb{C}^n\). Poisson structures; projective spaces; Poisson structures of hydrodynamic type; elliptic \(r\)-matrix; quadratic Poisson structures; elliptic functions; modular forms Poisson manifolds; Poisson groupoids and algebroids, Elliptic curves Poisson structures on loop spaces of \(\mathbb{C} P^n\) and an \(r\)-matrix associated with the universal elliptic curve | 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 deals with the Fourier-Mukai tecnique in the equivariant setting.
For any smooth complex projective variety \(Y\) provided with the action of a finite group \(G\), denote by \(\text{D}^{G}(Y)\) the bounded derived category of \(G\)-linearised coherent sheaves on \(Y\) or equivalently the triangulated category consisting of \(G\)-linearised objects of the bounded derived category \(\text{D}^{b}(Y)\) of coherent sheaves on \(Y\). Lemma 5 states that any object \((P,\rho)\) of \(\text{D}^{b}(Y)\) naturally induces an exact functor \(FM^{G}_{(P,\rho)}:\text{D}^{G}(X) \rightarrow \text{D}^{G}(X')\). Moreover if \(P\) is the kernel of an equivalence between \(\text{D}^{b}(X)\) and \(\text{D}^{b}(X')\), then \(FM^{G}_{(P,\rho)}\) is an also an equivalence.
As a consequence, derived equivalent smooth complex projective surfaces have punctual Hilbert schemes derived equivalent (Proposition 8). More precisely, let \(K\) be the Kernel of en equivalence between \(\text{D}^{b}(Z)\) and \(\text{D}^{b}(W)\), where \(Z\) and \(W\) are amooth complex projective surfaces. The exterior product \(p^{*}_{1}(K)\otimes\cdots\otimes p^{*}_{n}(K)\in D^{b}(Z^{n}\times W^{n})\) is the kernel of an equivalence between \(\text{D}^{b}(Z^{n})\) and \(\text{D}^{b}(W^{n})\). Moreover it has an obvious \(S_{n\:\Delta}\)-linearisation (\(S_{n}\) is the symmetric group of order \(n\)). It follows by Lemma 5 that \(\text{D}^{S_{n}}(Z^{n})\) and \(\text{D}^{S_{n}}(W^{n})\) are equivalent. A theorem of \textit{T. Bridgeland, A. King} and \textit{M. Reid} [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] implies that, for a smooth complex surface \(T\) the triangulated category \(\text{D}^{S_{n}}(T^{n})\) is equivalent to the bounded derived category of the Hilbert scheme \(\text{Hilb}^{n}(T)\) parametrizing 0-dimensional subschemes of length \(n\) on \(T\). Therefore \(\text{Hilb}^{n}(Z)\) and \(\text{Hilb}^{n}(Z')\) are derived equivalent.
Proposition 8 implies that birational punctual Hilbert schemes of \(K3\) sufaces are derived equivalent (Proposition 10). If \(S\) and \(S'\) are projective \(K3\) such that \(\text{Hilb}^{n}(S)\) and \(\text{Hilb}^{n}(S')\) are birational, then \(S\) and \(S'\) have the same transcendental lattice, hence by a theorem of \textit{D. O. Orlov} [J. Math. Sci., New York 84, No. 5, 1361--1381 (1997; Zbl 0938.14019)], they are derived equivalent. By Proposition 8 the same holds for \(\text{Hilb}^{n}(S)\) and \(\text{Hilb}^{n}(S')\).
In the general case where the finite group \(G\) acts on a smooth projective variety \(X\), denote by \(\text{Aut}(\text{D}^{b}(X))^{G}\) the group of autoequivalence of \(\text{D}^{b}(X)\) which are invariant under the action of \(G\) and set \(\text{Aut}^{G_{\Delta}}(\text{D}^{b}(X)):=\{(P,\rho)\in \text{D}^{G_{\Delta}}(X\times X): P\in \text{Aut}(\text{D}^{b}(X))\}\). The main theorem of the paper compares these groups with the automorphism group \(\text{Aut}(\text{D}^{G}(X))\).
Theorem 6. Suppose that the finite group \(G\) acts faithfully on the smooth projective variety \(X\).
(1) The construction of inflation gives a group homomorphism \text{inf} which fits in the following exact sequence, where \(Z(G)\subset G\) is the centre of \(G\):
\[
0\longrightarrow Z(G)\longrightarrow\text{Aut}^{G_{\Delta}}(\text{D}^{b}(X)) \overset\inf{\longrightarrow}\text{Aut}(\text{D}^{G}(X)).
\]
(2) Forgetting the \(G_{\Delta}\)-linearisation gives a group homomorphism which sits in the following exact sequence; here \(G_{\text{ab}}:= G/[G,G]\) is the abelianisation, which is non-canonically isomorphic to \(\text{Hom}(G,\mathbb{C}^{*})=H^{1}(G,\mathbb{C}^{*})\):
\[
0\longrightarrow G_{\text{ab}}\longrightarrow\text{Aut}^{G_{\Delta}}(\text{D}^{b}(X)) \overset{\text{for}}{\longrightarrow} \text{Aut}(\text{D}^{b}(X))^{G}\longrightarrow H^{2}(G,\mathbb{C}^{*}).
\]
Fourier-Mukai transforms; derived categories; Hilbert schemes; groups of autoequivalences; equivariant sheaves Ploog D.: Equivariant autoequivalences for finite group actions. Adv. Math. 216(1), 62--74 (2007) Group actions on varieties or schemes (quotients), Derived categories, triangulated categories Equivariant autoequivalences for finite group actions | 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 connected reductive group over \(\mathbb{C}\) and \(\mathfrak g\) its Lie algebra. Let \(\mathcal B\) be the variety of all Borel subalgebras of \(\mathfrak g\). For a nilpotent element \(N\) in \(\mathfrak g\), set \({\mathcal B}_N=\{{\mathfrak b}\in{\mathcal B}\mid N\in{\mathfrak b}\}\). We sometimes call the varieties \({\mathcal B}_N\) Springer fibers since they are the fibers of the Springer resolution \({\mathcal Y}\to{\mathcal N}\), \((N,{\mathfrak b})\to N\), where \(\mathcal N\) is the set of all nilpotent elements in \(\mathfrak g\) and \({\mathcal Y}=\{(N,{\mathfrak b})\mid N\in{\mathcal N},\;N\in{\mathfrak b}\}\).
In [\textit{D. Kazhdan} and \textit{G. Lusztig}, Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] we see that the equivariant \(K\)-groups \(K^{F_N}({\mathcal B}_N)\) and \(K^{F_N}({\mathcal B}_N\times{\mathcal B}_N)\) are interesting in representations of affine Hecke algebras. In this paper we give a partition of \({\mathcal B}_N\) for type \(A_{n-1}\). Applying it, we determine the equivariant \(K\)-groups \(K^{F_N}({\mathcal B}_N)\) and \(K^{F_N}({\mathcal B}_N\times{\mathcal B}_N)\). For type \(G_2\) we also determine the equivariant \(K\)-groups \(K^{F_N}({\mathcal B}_N)\) and \(K^{F_N}({\mathcal B}_N\times{\mathcal B}_N)\). For completeness we also give the \(K\)-groups for type \(B_2\). connected reductive groups; Lie algebras; Borel subalgebras; Springer fibers; equivariant \(K\)-groups; representations of affine Hecke algebras Nanhua Xi, A partition of the Springer fibers \Cal B_{\?} for type \?_{\?-1},\Cal B\(_{2}\),\?\(_{2}\) and some applications, Indag. Math. (N.S.) 10 (1999), no. 2, 307 -- 320. Linear algebraic groups over the reals, the complexes, the quaternions, \(K\)-theory of schemes, Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations A partition of the Springer fibers \({\mathcal B}_N\) for type \(A_{n-1}\), \(B_2\), \(G_2\) and some applications | 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 superb history of 20th century mathematical analysis. One is taken on a tour of all the major areas with a lot of clear detail and many quotes from those who took part in the developments.
In the introduction the scope and the definition of analysis is discussed with many quotes, including one by Temple (1981) who says that the only fault of the French analysts is that they did not provide an effective definition of their subject, and one by Penrose (1978) who claims that the picture of physical space led to the basic ideas of continuity and smoothness in analysis.
In each topic considered the chapters are divided into ``Evolution 1900-1950'' and ``Flashes 1950-2000''.
The first chapter is General Topology which is fairly thorough but unfortunately, only contains a very small section at the end on Robinson's non-standard analysis that seeks to justify the infinitesimal. We proceed through chapters on Integration and Measure to Functional Analysis. In the latter I was particular interested in the historical development of distributions, although I would have expected some reference to Lighthill's work. The work of Dirac and Schwartz are well covered and there is an interesting comment from the latter to the effect that if he had not found the theory of distributions it would soon have been discovered during this time because of the recent work of other mathematicians. ( This was from a book in 1997).
The next chapter is on Harmonic Analysis linking to physics through von Neumann's work but also mentioning the classical work of Hardy and Littlewood. Throughout these chapters there are so many topics that one is forced to concentrate on those for which one has a firm interest and I was particularly interested in the next chapter on Lie Groups. Many aspects are covered and inevitably minor errors will occur. I note that on page 181 the last equation should be an expression for \(C\) not for \(\exp(A)\) \(\exp(B)\) which is equal to \(\exp(C)\).
The next few chapters are written in the same style and cover: The Theory of Functions; Ordinary and Partial Differential Equations; Differential Topology; Probability and Algebraic Geometry. Two topics that I found very interesting were the definition of random variables as defined by Kolmogorov and the clear description by Halmos; that a random variable is a function attached to an experiment and once the experiment has been performed the value of the function is known. The discussion of the Riemann hypothesis is also fascinating.
In conclusion a book from which most mathematicians would find a lot to interest them. analysis; history; 20th Century; general topology; integration and measure; functional analysis; distributions; harmonic analysis; Lie groups; theory of functions; ordinary differential equations; partial differential equations; differential topology; probability; algebraic geometry; Riemann hypothesis Pier J P, Mathematical analysis during the 20th century (2001) Development of contemporary mathematics, Research exposition (monographs, survey articles) pertaining to history and biography, History of real functions, History of measure and integration, History of abstract harmonic analysis, History of topological groups, History of functions of a complex variable, History of ordinary differential equations, History of partial differential equations, History of algebraic topology, History of manifolds and cell complexes, History of probability theory, History of algebraic geometry, History of \(K\)-theory, History of number theory Mathematical analysis during the 20th century | 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 Square-tiled surfaces are covers of the square torus, ramified over at most one point. Affinely deforming the squares into parallelograms yields a curve in the moduli space of curves, called arithmetic Teichmüller curve. For these Teichmüller curves, the classification problem is only solved for genus 2 surfaces with a single ramification point. For genus two square-tiled covers with two ramification points we have three invariants: the spin, the torsion order of the branch points and the degree of a minimal intermediate covering. It is conjectured that these are the only invariants, i.e., that the set of genus two degree \(d\) covers of the torus with given torsion order and spin is irreducible.
The authors of this paper propose to attack this conjecture by first computing the class of these covers in the Picard group of a pseudo-Hilbert modular surface and in the second step to argue that this class is not too divisible and that potential summands cannot be Teichmüller curves. In the paper, the authors perform the first step for any odd degree \(d\).
Moreover, as a corollary of the computation of the class of arithmetic genus two Teichmüller curves in the Picard group of pseudo-Hilbert modular surfaces, they give a closed formula for the number of genus 2 square-tiled surfaces with fixed torsion order and spin.
The strategy for the proof of previous results is the following: instead of locating a Teichmüller curve inside the compactified pseudo-HIlbert modular surface \(X_{d^{2}}\), whose open part \(X_{d^{2}}^{0}\) parametrizes abelian surfaces with multiplication by a pseudo-quadratic order, they locate the branch points of the covering map from the flat surface to the torus inside the universal family of abelian surfaces over the open subset \(X_{d^{2}}^{0}\).
The image of the intersection of three divisorial conditions in \(X_{d^{2}}\) is the Teichmüller curve. They move the intersection calculations to a reasonable compactification \(A_{d^{2}}\) of the previous universal family of abelian surfaces. The family \(A_{d^{2}}\) comes with some obvious divisors given by Jacobi forms. Carefully computations have to be performed in the boundaries of these compactifications. square-tiled surface; Teichmüller curve; pseudo-Hilbert modular surfaces; compactification of moduli spaces; Jacobi forms; Theta functions; Intersection products Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Theta series; Weil representation; theta correspondences, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Theta functions and abelian varieties Cutting out arithmetic Teichmüller curves in genus two via theta 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 We prove lifting theorems for complex representations \(V\) of finite groups \(G\). Let \(\sigma=(\sigma_1,\ldots,\sigma_n)\) be a minimal system of homogeneous basic invariants and let \(d\) be their maximal degree. We prove that any continuous map \(\overline{f} \colon{\mathbb R}^m \to V\) such that \(f = \sigma \circ \overline{f}\) is of class \(C^{d-1,1}\) is locally of Sobolev class \(W^{1,p}\) for all \(1 \le p < d/(d-1)\). In the case \(m=1\) there always exists a continuous choice \(\overline{f}\) for given \(f\colon{\mathbb R} \to \sigma(V) \subseteq{\mathbb C}^n\). We give uniform bounds for the \(W^{1,p} \)-norm of \(\overline{f}\) in terms of the \(C^{d-1,1} \)-norm of \(f\). The result is optimal: in general a lifting \(\overline{f}\) cannot have a higher Sobolev regularity and it even might not have bounded variation if \(f\) is in a larger Hölder class. Sobolev lifting over invariants; complex representations of finite groups; \(Q\)-valued Sobolev functions Representations of Lie and linear algebraic groups over real fields: analytic methods, Lipschitz (Hölder) classes, Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems, Geometric invariant theory Sobolev lifting over invariants | 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 construct a reasonable singular homology theory on the category of schemes of finite type over an arbitrary field \(k\). Let \(X\) be a CW-complex. The theorem of \textit{A. Dold} and \textit{R. Thom} [Ann. Math., II. Ser. 67, 239-281 (1958; Zbl 0091.37102)] shows that \(H_i(X,{\mathbb{Z}})\) coincide with \(\pi_i\) of the simplicial Abelian group \(\Hom_{\text{top}} (\Delta_{\text{top}}^{\circ}, \coprod_{d=0}^\infty S^d(X))^{+},\) where \(S^d(X)\) is the \(d\)-th symmetric power of \(X\), \(\Delta_{\text{top}}^i\) is the usual \(i\)-dimensional topological simplex and for any Abelian monoid \(M\) denote by \(M^+\) the associated Abelian group.
Conjecture. If \(X\) is a variety over \({\mathbb{C}}\) then the evident homomorphism
\[
\Hom(\Delta^\circ, \coprod_{d=0}^\infty S^d(X))^+\rightarrow \Hom_{\text{top}} (\Delta_{\text{top}}^{\circ}, \coprod_{d=0}^\infty S^d(X))^+
\]
induces isomorphisms \(H_i^{\text{sing}}(X,{\mathbb{Z}}| n)\cong H_i(X({\mathbb{C}}),{\mathbb{Z}}| n)\).
The authors prove that the conjecture is true. Also, they prove a rather general version of the rigidity theorem of \textit{A. Suslin} [Invent. Math. 73, 241-245 (1983; Zbl 0514.18008); Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 222-244 (1987; Zbl 0675.12005)], \textit{O. Gabber} (unpublished), \textit{H. A. Gillet} and \textit{R. W. Thomason} [J. Pure Appl. Algebra 34, 241-254 (1984; Zbl 0577.13009)]. One of the main results of the paper is that if \(F\) is any \(qfh\)-sheaf on the category of schemes of finite type over an algebraically closed field \(k\) of characteristic zero, then \(H_{\text{sing}}^*(F,{\mathbb{Z}|}n)= \text{Ext}_{qfh}^*(F,{\mathbb{Z}|}n)\). singular homology; abstract algebraic varieties; category of schemes of finite type Suslin, A.; Voevodsky, V., \textit{singular homology of abstract algebraic varieties}, Invent. Math., 123, 61-94, (1996) Singular homology and cohomology theory, Varieties and morphisms Singular homology of abstract algebraic 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 Algebraic groups are treated here from a group theoretical point of view and the obtained results are compared with the analogous issues in the theory of Lie groups. The main body of the text is devoted to a classification of algebraic groups and Lie groups having only few subgroups or few factor groups of different type. In particular, the diversity of the nature of algebraic groups over fields of positive characteristic and over fields of characteristic zero is emphasized. This is revealed by the plethora of three-dimensional unipotent algebraic groups over a perfect field of positive characteristic, as well as, by many concrete examples which cover an area systematically. In the final section, algebraic groups and Lie groups having many closed normal subgroups are determined. group varieties; nilpotent Lie groups; solvable Lie groups; solvable Lie algebras; nilpotent Lie algebras; chains of subgroups; lattices of subgroups; closed normal subgroups Linear algebraic groups over arbitrary fields, General properties and structure of complex Lie groups, General properties and structure of real Lie groups, Solvable, nilpotent (super)algebras, Subgroup theorems; subgroup growth, Chains and lattices of subgroups, subnormal subgroups, Group varieties, Nilpotent and solvable Lie groups Algebraic groups and Lie groups with few factors. | 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 0533.00008.]
This work studies certain prehomogeneous vector spaces that arise in a natural way in the structure of complex semi-simple Lie algebras. The associated zeta functions are important objects in this study. Let \({\mathfrak g}\) be a semi-simple Lie algebra and let \({\mathfrak g}_ i\) (i in \({\mathbb{Z}})\) be a \({\mathbb{Z}}\)-gradation of \({\mathfrak g}\), i.e., a sequence of vector subspaces such that [\({\mathfrak g}_ i,{\mathfrak g}_ j]\subset {\mathfrak g}_{i+j}\). Let \(G_ 0\) be the subgroup (assumed to be parabolic) of the adjoint group G of \({\mathfrak g}\) corresponding to \({\mathfrak g}_ 0\). The natural action of \(G_ 0\) on \({\mathfrak g}_ 1\) is prehomogeneous, i.e., there exists a \(G_ 0\)-orbit in \({\mathfrak g}_ 1\) which is open in the Zariski sense.
The author's first result (obtained also by Vinberg as follows) is: \({\mathfrak g}_ 1\) decomposes into a finite number of \(G_ 0\)-orbits. The author goes on to demonstrate the equivalence (in the irreducible case) of the notion of regularity and the existence of certain \(s\ell_ 2\)- triplets. Thus he finds the classification of regular, prehomogeneous spaces of this type. Also it is shown that when [\({\mathfrak g}_ 1,{\mathfrak g}_ 1]=0\) the local zeta function associated with the prehomogeneous space is interpreted as an intertwining integral of a degenerate principal series of representations of G. Certain orbits are also shown to be symmetric spaces. prehomogeneous vector spaces; complex semi-simple Lie algebras; zeta functions; degenerate principal series Rubenthaler, H.: Espaces préhomogènes de type parabolique. Lect. math. Kyoto univ. 14, 189-221 (1982) Simple, semisimple, reductive (super)algebras, Homogeneous spaces and generalizations Espaces préhomogènes de type parabolique | 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 continues the study, begun in [J. Reine Angew. Math. 483, 75-161 (1997; Zbl 0859.11032)], of the \(L\)-functions associated to cusp forms of cohomological type on unitary groups. Arithmetic properties of the theta correspondence for automorphic forms on pairs of unitary groups of Hermitian vector spaces over imaginary quadratic fields are studied. We specifically consider the correspondence taking automorphic forms of general discrete series type on groups of signature \((n-1,1)\) to holomorphic automorphic forms on groups of general signature. It is shown that the correspondence preserves arithmeticity up to multiplication by an explicit scalar, which can be expressed in terms of periods of elliptic curves with complex multiplication-CM periods. The scalar depends on the normalization of the theta correspondence and on the central character of the initial representation. Arithmeticity of holomorphic automorphic forms is defined by a version of Shimura's criterion; arithmeticity of general discrete series automorphic forms is defined in terms of coherent cohomology.
The technique is recursive and ultimately comes down to a calculation of theta lifts from \(U(1)\) to \(U(1)\), whose arithmetic properties are expressed in terms of special values of Hecke \(L\)-functions. A final section explains the relevance of the present results to \textit{P. Deligne}'s conjecture on special values of motivic \(L\)-functions [see Proc. Symp. Pure Math. 33, No. 2, 313-346 (1979; Zbl 0449.10022)]. automorphic representations; unitary groups; periods; \(L\)-functions; theta functions; cohomology of arithmetic groups; theta correspondence; automorphic forms; pairs of unitary groups; Hermitian vector spaces Harris, M.: Cohomological automorphic forms on unitary groups, I: Rationality of the theta correspondence, Proc. symp. Pure math 66.2, 103-200 (1999) Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Theta series; Weil representation; theta correspondences, Arithmetic aspects of modular and Shimura varieties, Other groups and their modular and automorphic forms (several variables), Modular and Shimura varieties, Cohomology of arithmetic groups Cohomological automorphic forms on unitary groups. I: Rationality of the theta correspondence | 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 version of Bézout's theorem for non-commutative analogues of the projective spaces \(\mathbb{P}^n\) is proved. graded modules; Hilbert series; rank functions; global dimension; cohomological dimensions; Auslander-Gorenstein algebras; regular algebras; Picard groups; Grothendieck groups I. MORI AND S. P. SMITH, Bézout's theorem for noncommutative projective spaces, J. Pure Appl. Algebra 157 (2001), 279-299. Graded rings and modules (associative rings and algebras), Homological dimension in associative algebras, Noncommutative algebraic geometry, Grothendieck groups, \(K\)-theory, etc., Rings arising from noncommutative algebraic geometry Bézout's theorem for non-commutative projective spaces | 0 |
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