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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 [For the entire collection see Zbl 0534.00007.]
If F is a field and n is an integer, prime to char(F), then \(H^ 1(F,\mu_ n)=F^{\times}/F^{\times n},\) and we have a map \(R_{n,F}:K_ 2(F)/nK_ 2(F)\to H^ 2(F,\mu_ n\otimes \mu_ n),\) which is called the norm residue homomorphism. Here \(\mu_ n\) denotes the n-th roots of unity. Theorem 1 claims that \(R_{n,F}\) is an isomorphism. The proof [see \textit{A. S. Merkur'ev} and the author, Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)] is based on the computation of certain K-cohomology groups of Severi-Brauer varieties, using a careful study of \(K_ 2\) of division algebras. The case \(n=2\) has been done in April 1981 by Merkur'ev. Among other results announced in the note is Hilbert's 90 for \(K_ 2\) and a canonical isomorphism between the \(\ell\)-primary component of the second Chow group of any smooth projective variety X over an algebraically closed field and the subgroup of \(H^ 3(X_{et},{\mathbb{Q}}_{\ell}/{\mathbb{Z}}_{\ell}(2))\) consisting of cohomology classes with zero restriction at the generic point. Hilbert theorem 90 for K2; norm residue homomorphism; roots of unity; K- cohomology groups of Severi-Brauer varieties; \(K_ 2\) of division algebras; second Chow group; cohomology classes with zero restriction A. S. Merkur'ev and A. A. Suslin, ''The norm residue homomorphism,'' Preprint LOMI, P-6-82, Leningrad (1982). Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of global fields, Étale and other Grothendieck topologies and (co)homologies, Grothendieck groups, \(K\)-theory, etc., (Equivariant) Chow groups and rings; motives Norm residue homomorphism | 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 and \(G\) a finite group. The essential dimension \(\text{ed}_F(G)\) of \(G\) over \(F\) is defined to be the smallest integer \(d\geq 0\) such that one requires \(d\) algebraically independent parameters to define a Galois \(G\)-algebra over any finite extension of \(F\). This notion was introduced by \textit{J. P. Buhler} and \textit{Z. Reichstein} in their paper ``On essential dimension of a finite group'', [Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003)]. The computation of \(\text{ed}_F(G)\) is a hard problem for even specific finite groups. \textit{M. Florence}, in a recent paper [Invent. Math. 171, No. 1, 175--189 (2008; Zbl 1136.14035)], calculated the exact value of \(\text{ed}_F(C)\) for a cyclic \(p\) group \(C\), assuming that \(F\) contains a primitive \(p\)th root of unity and subsequently \textit{P. Brosnan, Z. Reichstein} and \textit{A. Vistoli} computed \(\text{ed}_F(G)\) for a class of nonabelian \(p\)-groups [''Essential dimension and algebraic stacks'', preprint, \url{arXiv:math/0701903}].
In the paper under review the authors prove a generalization of these results.
Theorem 4.1: Let \(G\) be a finite \(p\)-group and \(F\) be a field containing a primitive \(p\)th root of unity. Then \(\text{ed}_F(G)\) is the smallest integer \(n\) such that \(G\) has a faithful representation \(G\rightarrow \text{GL}_n(F)\).
The authors prove as a corollary (Theorem 5.2) that
\[
\text{ed}_F(G_1\times G_2)=\text{ed}_F(G_1)+\text{ed}_F(G_2)
\]
for any two \(p\)-groups \(G_1\) and \(G_2\).
Some results of independent interest on central simple algebras and Brauer-Severi varieties are also proven in the paper. essential dimension; finite \(p\) groups; central simple algebras; Brauer-Severi varieties Karpenko, NA; Merkurjev, AS, Essential dimension of finite \(p\)-groups, Invent. Math., 172, 491-508, (2008) Separable extensions, Galois theory, Linear algebraic groups over arbitrary fields, Finite-dimensional division rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Essential dimension of finite \(p\)-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 We study the distribution of the trace of the seven-dimensional representation of the exceptional compact simple Lie group \(G\) of type \(G_2\). The interest for this distribution comes from its relevance to the equidistribution of several families of exponential sums involving a seven degree binomial phase, established by N. Katz. Firstly, we give a construction of the algebraic group of type \(G_2\) and of its Lie algebra, whose \(G\) is the compact form. We then define the Steinberg map of \(G\), defined from the traces of the fundamental representations, inducing a homeomorphism from the alcove of \(G\) (the simplex parametrizing conjugacy classes) to a compact set in the affine space. By combining Weyl's integration formula and the Steinberg map, we obtain an explicit expression for the probability density function of the distribution of the trace function on \(G\) in terms of Gauss hypergeometric function, and some other special functions. This answers a question raised by J.-P. Serre and N. Katz. compact Lie groups of type \(G_2\); distribution of the trace of matrices; random matrices; Weyl's integration formula; Steinberg map; equidistribution Estimates on exponential sums, Structure of families (Picard-Lefschetz, monodromy, etc.), Exceptional groups, Representations of Lie and linear algebraic groups over real fields: analytic methods, Probability measures on groups or semigroups, Fourier transforms, factorization, Random matrices (probabilistic aspects), Well-distributed sequences and other variations, Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics The distribution of the trace in the compact group of type \(\mathbf{G}_2\) | 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 This article contains the details of the following two announcements: C. R. Acad. Sci., Soc. R. Can. 3, 273-278 (1981; Zbl 0495.14015) by these authors and Can. J. Math. 34, 169-180 (1982; Zbl 0477.14019) by the second author. The authors construct the coordinate ring A of a curve with one singular point P via Cartesian squares \(A\to^{f}\bar A\to^{\pi}\prod^{s}_{i=1}k[t_ i]/t^ n_ i;\quad A\to D\to^{g}\prod^{s}_{i=1}k[t_ i]| t^ n_ i\) (f and g are inclusions and \(\pi\) is onto), where \(\bar A\) is the normalization of A and k is a field. They compute several invariants of the local ring \(A_ P\), which only depend on g. In particular, there is shown an algorithm for computing the Hilbert function H of P. Thus they decided in several cases when H can have a temporary decrease. In particular, there is a generic homogeneous ordinary singular point such that H can decrease and \(H(1)=4\). Furthermore, the authors discuss the Cohen-Macaulay type of \(A_ P\) and its relationship to the Cohen-Macaulay type of its reduced tangent cone. In the case of generic homogeneous ordinary singularities both are the same, but in other cases, they may differ. This answers some questions posed by \textit{A. V. Geramita} and \textit{F. Orecchia} in J. Algebr 78, 36-57 (1982; Zbl 0502.14001) and by \textit{A. V. Geramita} and \textit{P. Maroscia} in C. R. Math. Acad. Sci., Soc. R. Can. 4, 179-184 (1982; Zbl 0493.14001). Furthermore the authors show that the K-theory of A, \(K_ i(A)\), \(i\leq 1\), depends on the number of irreducible components of Spec A (and on the inclusion). ordinary singularities of curves; coordinate ring of a curve; Hilbert function; Cohen-Macaulay type; K-theory Gupta, S. K.; Roberts, L. G., Cartesian squares and ordinary singularities of curves, \textit{Commun. Algebra}, 11, 2, 127-182, (1983) Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Grothendieck groups, \(K\)-theory and commutative rings, Applications of methods of algebraic \(K\)-theory in algebraic geometry Cartesian squares and ordinary singularities of 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 Let \(k\) be an algebraically closed field. Let \(B\) be the Borel subgroup of \(\text{GL}_n(k)\) consisting of nonsingular upper triangular matrices. Let \(\mathfrak b=\text{Lie\,}B\) be the Lie algebra of upper triangular \(n\times n\) matrices and \(\mathfrak u\) the Lie subalgebra of \(\mathfrak b\) consisting of strictly upper triangular matrices. We classify all Lie ideals \(\mathfrak n\) of \(\mathfrak b\), satisfying \(\mathfrak u'\subseteq\mathfrak n\subseteq\mathfrak u\), such that \(B\) acts (by conjugation) on \(\mathfrak n\) with a dense orbit. Further, in case \(B\) does not act with a dense orbit, we give the minimal codimension of a \(B\)-orbit in \(\mathfrak n\). This can be viewed as a first step towards the difficult open problem of classifying of all ideals \(\mathfrak n\subseteq\mathfrak u\) such that \(B\) acts on \(\mathfrak n\) with a dense orbit.
The proofs of our main results require a translation into the representation theory of a certain quasi-hereditary algebra \(\mathcal A_{t,1}\). In this setting we find the minimal dimension of \(\text{Ext}^1_{\mathcal A_{t,1}}(M,M)\) for a \(\Delta \)-good \(\mathcal A_{t,1}\)-module of certain fixed \(\Delta\)-dimension vectors. general linear groups; Borel subgroups; Lie algebras of matrices; Lie ideals; dense orbits; quasi-hereditary algebras; dimension vectors Goodwin, S.M., Hille, L.: Prehomogeneous spaces for Borel subgroups of general linear groups. Transform. Groups (to appear) Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients) Prehomogeneous spaces for Borel subgroups of general 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 See the review of the English original (1986) in Zbl 0618.22011. 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 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, Homogeneous complex manifolds, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), Harmonic analysis on homogeneous spaces, Grassmannians, Schubert varieties, flag manifolds Loop groups. (Gruppy petel'). With an appendix by Segal and G. Wilson. Transl. from the 2nd English ed. and with a preface by A. V. Zelevinskij and A. O. Radul | 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 will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves \(\mathcal{C}_{\mathcal{H}_{g, n / k}} \rightarrow \mathcal{H}_{g, n / k}\), when \(g \geq 3\) and the image of the \(\ell\)-adic cyclotomic character \(G_k \rightarrow \mathbb{Z}^\times\) is infinite. Furthermore, we will study the nonabelian phenomena associated to the universal hyperelliptic curves. For example, we will show that the section conjecture holds for \(\mathcal{C}_{\mathcal{H}_{g / k}} \rightarrow \mathcal{H}_{g / k}\) and the unipotent analogue of the conjecture holds for the pointed cases. This work is an extension of Hain's original work on the rational points of universal curves to the hyperelliptic case. algebraic geometry; arithmetic geometry; fundamental groups; moduli space of hyperelliptic curves; hyperelliptic curves; hyperelliptic mapping class groups; Lie algebras; unipotent completions Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Elliptic curves over global fields, Arithmetic ground fields for curves, Solvable, nilpotent (super)algebras, Coverings of curves, fundamental group On the sections of universal hyperelliptic 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 See the preview in Zbl 0666.14002. variation of complex structures; variation of Lie algebras; simple- elliptic singularities; 1-parameter families Seeley, C., Yau, S.S.-T.: Variation of complex structures and variation of Lie algebras. Invent. Math. 99, 545--565 (1990) Deformations of singularities, Deformations of complex singularities; vanishing cycles, Algebraic moduli problems, moduli of vector bundles, Simple, semisimple, reductive (super)algebras Variation of complex structures and variation of Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author constructs a minimal complex projective nonsingular algebraic surface S of general type with \(\pi_ 1(S)=\{1\},\) \(p_ g(S)=0,\) \(K^ 2_ S=1.\) This answers negatively the question on the rationality of surfaces with \(\pi_ 1(S)=\{1\},\) \(p_ S=0,\) \(K^ 2>0,\) arising from a conjecture of Severi. Moreover the above example fills a gap in the classification of surfaces with \(p_ g=0\), \(K^ 2=1\). Actually, for such surfaces, it has to be \(| Tors| \leq 5\) and surfaces with \(Tors={\mathbb{Z}}_ m,\) \(2\leq m\leq 5\) occur, while surfaces with \(Tors={\mathbb{Z}}_ 2\oplus {\mathbb{Z}}_ 2\) do not. The above surface is the first example with trivial Tors. To construct the surface let Y be a simply connected surface with \(\grave p_ g=4,\) \(K^ 2=10\) described by \textit{F. Catanese} [Invent. Math. 63, 433-465 (1981; Zbl 0472.14024)] (and previously considered by G. Van der Geer and D. Zagier) and \(D_{10}\) the dihedral group \(\{\alpha,\beta | \quad a^ 2=\beta^ 5=1,\quad \alpha \beta \alpha =\beta^ 4\}.\) The author observes that \(D_{10}\) acts on \(Y: \alpha\) has a finite fixed locus, while \(\beta\) acts freely on Y, so that the quotient \(X=Y/D_{10}\) has 4 nodes. The surface S is the resolution of X. Hilbert modular surfaces; minimal complex projective nonsingular algebraic surface S of general type; classification of surfaces Barlow, R, A simply connected surface of general type with \(p_g=0\), Invent. Math., 79, 293-301, (1985) Families, moduli, classification: algebraic theory, Special surfaces, Global theory and resolution of singularities (algebro-geometric aspects) A simply connected surface of general type with \(p_ g=0\) | 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 Zariski closures of orbits of representations of quivers of type \(A\), \(D\) or \(E\). With the help of Lusztig's canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth. quantum groups; representations of quivers; singularities; canonical basis Caldero, P., Schiffler, R.: Rational smoothness of varieties of representations for quivers of Dynkin type. Ann. Inst. Fourier 54(2), 295--315 (2004) Quantum groups (quantized enveloping algebras) and related deformations, Representations of quivers and partially ordered sets, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Rational smoothness of varieties of representations for quivers of Dynkin type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper is divided in three parts.
In the first one, we develop the theory of \(\mathfrak{D}\)-modules on ind-schemes of pro-finite type. This allows to define \(\mathfrak{D}\)-modules on (algebraic) loop groups and, consequently, the notion of strong loop group action on a DG category.
In the second part, we construct the functors of Whittaker invariants and Whittaker coinvariants, which take as input a DG category acted on by \(G((t))\), the loop group of a reductive group \(G\). Roughly speaking, the Whittaker invariant category of \(\mathcal{C}\) is the full subcategory \(\mathcal{C}^{N((t)),\chi}\subseteq \mathcal{C}\) consisting of objects that are \(N((t))\)-invariant against a fixed non-degenerate character \(\chi:N((t))\) a of conductor zero. (Here \(N\) is the maximal unipotent subgroup of \(G\).) The Whittaker coinvariant category \(C_{N((t)),{\chi}}\) is defined by a dual construction. In the third part, we construct a functor \(\Theta :\mathcal{C}_{N((t)),{\chi}}\to \mathcal{C}^{N((t)),{\chi}}\), which depends on a choice of dimension theory for \(G((t))\). We conjecture this functor to be an equivalence. After developing the Fourier-Deligne transform for Tate vector spaces, we prove this conjecture for \(G=GL_n\). We show that both Whittaker categories can be obtained by taking invariants of \(\mathcal{C}\) with respect to a very explicit pro-unipotent group subscheme (not indscheme!) of \(G((t))\). group actions on categories; loop groups; indschemes of pro-finite type; Whittaker invariants; geometric Langlands; Fourier-Deligne transform D. Beraldo, Loop group actions on categories and Whittaker invariants, arXiv:1310.5127. Geometric Langlands program: representation-theoretic aspects, Loop groups and related constructions, group-theoretic treatment, Geometric Langlands program (algebro-geometric aspects) Loop group actions on categories and Whittaker 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 In this article we give an algorithm for determining the generators and relations for the rings of semi-invariant functions on irreducible components of \(\text{Rep}_A(\beta)\) when \(A\) is a (acyclic) gentle algebra and \(\beta\) is a dimension vector. These rings of semi-invariants turn out to be semigroup rings to which we can associate a so-called matching graph. Under this association, generators for the semigroup can be seen by certain walks on this graph, and relations are given by certain configurations in the graph. This allows us to determine degree bounds for the generators and relations of these rings. algorithms; generators and relations; rings of semi-invariant functions; gentle algebras; semigroup rings; matching graphs Carroll, AT; Weyman, J, Semi-invariants for gentle algebras, Contemp. Math., 592, 111-136, (2013) Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Vector and tensor algebra, theory of invariants, Graphs and abstract algebra (groups, rings, fields, etc.), Semigroup rings, multiplicative semigroups of rings Semi-invariants for gentle 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 paper under review continues the study of the authors about singular quiver Grassmannians, providing desingularizations of irreducible components of arbitrary quiver Grassmannians over Dynkin quivers.
Given a Dynkin quiver \(\mathcal Q\), being a directed graph whose unoriented underlying graph is a Dynkin diagram, a representation \(M\) of \(\mathcal Q\), and a dimension vector \(e\), the quiver Grassmannian \(Gr_e(M)\) is the variety of subrepresentations of \(M\) of dimension vector \(e\). When \(M\) has good homological properties, \(Gr_e(M)\) is smooth. However, the general analysis needs a desingularization of the quiver Grassmannian.
The construction of a desingularization lies on the definition of an algebra \(B_{\mathcal Q}\) for the quiver \(\mathcal Q\) which has global dimension at most two, such that the original module category \(\mathrm{mod\,}k\mathcal Q\) embeds into the subcategory \(\mathrm{mod\,}B_{\mathcal Q}\) of objects of projective and injective dimensions at most one, where all non-trivial extensions in \(\mathrm{mod\,}k\mathcal Q\) vanish after the embedding. This is to avoid the natural embedding \(M\mapsto\Hom(-,M)\) of \(\mathrm{mod\,}k\mathcal Q\) into \((\mathrm{mod\,}k\mathcal Q)^{op}\) which give projective functors of dimension two, in general.
A quiver \(\widehat{\mathcal Q}\) is constructed from \(B_{\mathcal Q}\) and, for every representation \(M\) over \(\mathcal Q\), a representation \(\widehat M\) over \(\widehat{\mathcal Q}\) arises. This, together with a fully faithful functor \(\Lambda\) from the category of representations of \(\mathcal Q\) to the one in \(B_{\mathcal Q}\) with good homological properties, completes the ingredients of the construction.
Then, it is constructed the desingularization map \(\pi_{[N]}\colon Gr_{\dim\widehat N}(\widehat M)\to Gr_e(M)\), where \(\widehat N\) is a representation over \(\widehat{\mathcal Q}\) coming from an isomorphism class of representations \([N]\) over \(\mathcal Q\) of dimension vector \(e\). The variety \(Gr_{\dim\widehat N}(\widehat M)\) is smooth with irreducible equidimensional connected components, and the fibers of the map can be described in terms of a quiver Grassmannian over \(\widehat{\mathcal Q}\) itself, once we know the irreducible components of the singular \(Gr_e(M)\). The article finishes by showing some particular examples of the construction. quiver Grassmannians; desingularizations; Dynkin quivers; Auslander algebras; flag varieties; Auslander-Reiten theory; quiver representations; categories of representations; irreducible components Cerulli Irelli, G., Feigin, E., Reineke, M.: Desingularization of quiver Grassmannians for Dynkin quivers. Adv. Math. \textbf{245}, 182-207 (2013) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Singularities of surfaces or higher-dimensional varieties Desingularization of quiver Grassmannians for Dynkin quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We consider the problem to classify function germs \((\mathbb{C}^2,0)\to(\mathbb{C},0)\) that are equivariant simple with respect to nontrivial actions of the group \(\mathbb{Z}_3\) on \(\mathbb{C}^2\) and on \(\mathbb{C}\) up to equivariant automorphism germs \((\mathbb{C}^2,0)\to(\mathbb{C}^2,0)\). The complete classification of such germs is obtained in the case of nonscalar action of \(\mathbb{Z}_3\) on \(\mathbb{C}^2\) that is nontrivial in both coordinates. Namely, a germ is equivariant simple with respect to such a pair of actions if and only if it is equivalent to one of the following germs:
\[
\begin{aligned}(x,y)&\mapsto x^{3k+1}+y^2,\;k\geqslant1;\\ (x,y)&\mapsto x^2y+y^{3k-1},\;k\geqslant2;\\ (x,y)&\mapsto x^4+xy^3;\\ (x,y)&\mapsto x^4+y^5.\end{aligned}
\]
classification of singularities; simple singularities; group action; equivariant functions Local complex singularities, Singularities of surfaces or higher-dimensional varieties On the classification of function germs of two variables that are equivariant simple with respect to an action of the cyclic group of order three | 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 Using \(K\)-theory, we construct a map \(\pi:T_Y\mathrm{Hilb}^p(X)\rightarrow H_y^p(\Omega_{X/\mathbb Q}^{p-1})\) from the tangent space to the Hilbert scheme at a point \(Y\) to the local cohomology group. We use this map \(\pi\) to answer (after slight modification) a question by [\text it{M. Green} and \text it{P. Griffiths}, On the tangent space to the space of algebraic cycles on a smooth algebraic variety. Princeton, NJ: Princeton University Press (2005; Zbl 1076.14016)] on constructing a map from the tangent space \(T_Y\mathrm{Hilb}^p(X)\) to the Hilbert scheme at a point \(Y\) to the tangent space to the cycle group \(TZ^p(X)\). deformation of cycles; tangent spaces to cycle groups; \(K\)-theory; Chern character; tangent spaces to Hilbert schemes; Koszul complex; Newton class; absolute differentials Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles, Parametrization (Chow and Hilbert schemes), \(K\)-theory in geometry \(K\)-theory, local cohomology and tangent spaces to Hilbert 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 A concept of central simple \(G\)-algebras is introduced, as well as an equivalence relation among central simple \(G\)-algebras. From it, a set of equivalence classes is obtained which generalizes the Brauer group of a field, as well as the Brauer-Wall group. It is shown that the elements of this set are the natural objects that control the Clifford theory with Schur indices for a finite group \(H\) with a normal subgroup \(N\) with \(G\simeq H/N\). Schur index; central simple algebras; Brauer groups; Brauer-Wall groups; Clifford theory; Schur indices; finite groups Turull, A.: Clifford theory with Schur indices. J. Algebra 170, 661--677 (1994) Ordinary representations and characters, Finite-dimensional division rings, Brauer groups of schemes, Automorphisms and endomorphisms, Group rings of finite groups and their modules (group-theoretic aspects), Finite rings and finite-dimensional associative algebras Clifford theory with Schur indices | 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 \(Q=(Q_0, Q_1)\) be a quiver (i.e. a finite oriented graph), where \(Q_0\) is the set of vertices of \(Q\) and \(Q_1\) is the set of arrows of \(Q\). We denote by \(KQ\) the path \(K\)-algebra of \(Q\). If \(X\) is a finitely generated right \(KQ\)-module and \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector we denote by \(\text{Gr}_{KQ}(X,\beta)\) the algebraic Grassmann variety of all submodules \(Y\) of \(X\) of codimension \(\beta\), that is, the dimension vector of \(X/Y\) is \(\beta\).
Let \(b_Q(-,-):\mathbb{Z}^{Q_0}\times\mathbb{Z}^{Q_0}\to\mathbb{Z}\) be the bilinear form associated with \(Q\). Given a dimension vector \(\beta\in\mathbb{N}^{Q_0}\) we call the affine variety
\[
\text{Rep}_{KQ}(\beta)=\prod_{\gamma:i\to j}\text{Hom}_K(K^{\beta(i)},K^{\beta(j)})
\]
the configuration space of representations of \(Q\) of dimension vector \(\beta\). Given \(y\in\text{Rep}_{KQ}(\beta)\) we denote by \(K_y\) the right \(KQ\)-module corresponding to \(y\). By a rank of a \(KQ\)-homomorphism \(X\to K_y\) we mean the dimension vector of its image. If \(X\) is a finitely generated right \(KQ\)-module, \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector and \(y\in\text{Rep}_{KQ}(\beta)\) then \(\text{Hom}_{KQ}(X,K_y)\) is a finite dimensional vector space and the function \(y\mapsto\dim\text{Hom}_{KQ}(X,K_y)\) is an upper semicontinuous function on \(\text{Rep}_{KQ}(\beta)\). It follows that the minimum value of this function, denoted by \(\text{hom}(X,\beta)\), is also its general value. For any \(\alpha\in\mathbb{N}^{Q_0}\) the set of all homomorphisms \(X\to K_y\) of rank at most \(\alpha\) is a closed subset of \(\text{Hom}_{KQ}(X,K_y)\). It follows that there is a unique maximal rank \(\gamma_{X,y}\) of homomorphisms \(X\to K_y\), and that the set of homomorphisms of rank \(\gamma_{X,y}\) is an open subset of \(\text{Hom}_{KQ}(X,K_y)\). The function \(y\mapsto\gamma_{X,y}\) is constant on a non-empty open subset of \(\text{Rep}_{KQ}(\beta)\), and its general value is denoted by \(\gamma_{X,\beta}\).
One of the main results of the paper asserts that if \(X\) is a finite dimensional right \(KQ\)-module and \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector then \(\text{hom}(X,\beta)=b_Q(\gamma_{X,\beta},\beta)-\dim U\) for some open subset \(U\) of the Grassmann variety \(\text{Gr}_{KQ}(X,\gamma_{X,\beta})\).
As a consequence the author derives the following two corollaries concerning the asymptotic behaviour of \(\text{hom}(X,\beta)\) as \(\beta\) increases: (a) If \(X\) is a finitely presented right \(KQ\)-module and \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector then
\[
\lim_{r\to\infty}\text{hom}(X,r\beta)=\max\{\widetilde\beta(X/Y)\mid Y\subseteq X\text{ a finitely presented submodule\}}
\]
where \(\widetilde\beta:K_0(KQ)\to\mathbb{Z}\) is the natural homomorphism induced by \(\beta\). (b) If \(\beta\in\mathbb{N}^{Q_0}\) is a dimension vector and \(X\) is a finitely presented right \(KQ\)-module which is \(\beta\)-semistable then \(\text{hom}(X,r\beta)=0\). These results are related to a work of A. Schofield on homological epimorphisms from the path algebra \(KQ\) to a simple artinian ring. path algebras; maximal rank of homomorphisms; quivers; finitely generated right modules; dimension vectors; Grassmann varieties; bilinear forms; affine varieties; configuration spaces of representations; finitely presented right modules; simple Artinian rings Crawley-Boevey, W, On homomorphisms from a fixed representation to a general representation of a quiver, Trans. Am. Math. Soc., 348, 1909-1919, (1996) Representations of quivers and partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Algebraic theory of abelian varieties, Vector and tensor algebra, theory of invariants On homomorphisms from a fixed representation to a general representation of a quiver | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The present paper studies Noetherianity of the inverse limits of the sequence dual to a sequence of Lie algebras. The main results concern diagonal embeddings \(G_i\hookrightarrow G_{i+1}\), where \(G_i\) are four types of subgroups of the general linear group.
Let \(G\) be its direct limit and consider the associated sequence \(\mathfrak g_1\hookrightarrow\mathfrak g_2\hookrightarrow\dots\), where \(\mathfrak g_i\) is the Lie algebra of \(G_i\). Now, we let \(V\) be the inverse limit of the sequence obtained by dualizing the previous sequence. Then \(V\) has a natural action of \(G\). In the main result is proved that for these four types of groups the space \(V\) is \(G\)-Noetherian, i.e. for every descending sequence \(V\supseteq X_1\supseteq X_2\supseteq X_3\supseteq\dots\) of \(G\)-stable closed subsets of \(V\) there is an \(i\in\mathbb N\) such that \(X_i = X_j\) for all \(j \geq i\). Noetherianity; locally finite Lie algebras; classical groups Commutative Noetherian rings and modules, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients) Noetherianity up to conjugation of locally diagonal inverse limits | 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 known since the work of \textit{C. Peskine} and \textit{L. Szpiro} [Inv. Math. 26, 271-302 (1974; Zbl 0298.14022)] that the intersection of two \(c\)-codimensional arithmetically Cohen-Macaulay linked schemes is a \((c+1)\)-codimensional arithmetically Gorenstein (aG) scheme. In this paper we study under which conditions a 3-dimensional aG scheme comes as such an intersection and we investigate relationships between the graded Betti numbers and Hilbert functions for these schemes. In particular, we show that this property holds in \(\mathbb{P}^3\) for aG schemes on smooth quadric surfaces and, for some Hilbert functions, for aG schemes lying on a smooth surface of minimal degree. arithmetically Gorenstein scheme; intersection of schemes; graded Betti numbers; Hilbert functions Ragusa, A.; Zappalà, G., Properties of 3-codimensional Gorenstein schemes, Comm. Algebra, 29, 1, 303-318, (2001) Low codimension problems in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Properties of 3-codimensional Gorenstein 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 \(k\) be algebraically closed field, and let \(Q=(Q_0,Q_1,s,e) \) be a Dynkin quiver of type \(\mathbb A_n\) for \( n\geq 1\). Let \({\mathbf d}\in\mathbb N^{Q_0}\) be a dimension vector. It was previously known [\textit{S. Abeasis, A. Del Fra, H. Kraft}, Math. Ann. 256, 401--418 (1981; Zbl 0477.14027)] that, in the case where \(Q\) is an equioriented Dynkin quiver and \(k\) is of characteristic zero, the orbit closures in \(\text{rep}_Q({\mathbf d})\) are normal and Cohen-Macauley varieties and have rational singularities. Here the same result is proved when \(Q\) has arbitrary orientation.
To accomplish this, the authors prove the following. Let \(Q'\) and \(Q\) be Dynkin quivers of type \(\mathbb A\), and let \(A=kQ'\) (resp. \(B=kQ)\) be the path algebras of quivers \(Q'\) (resp. \(Q\)). If we assume there is a full embedding of translation quivers \(F: \Gamma_B\to\Gamma_A\), then there exists a functor \({\mathcal F}: \text{mod }B\to\text{mod }A\) which is hom-controlled. This short paper is dedicated to proving this proposition. The desired result above follows from a paper by \textit{G. Zwara} [Proc. Lond. Math. Soc. (3) 84, No. 3, 539--558 (2002; Zbl 1054.16009)]. Dynkin quivers; orbit closures; Cohen-Macaulay varieties; rational singularities; path algebras; translation quivers Bobiński, Grzegorz; Zwara, Grzegorz, Normality of orbit closures for Dynkin quivers of type \(\mathbb{A}_n\), Manuscripta Math., 105, 1, 103-109, (2001) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers Normality of orbit closures for Dynkin quivers of type \(\mathbb A_n\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper grew out of the author's work [J. Reine Angew. Math. 375/376, 47-66 (1987; Zbl 0628.14023)] on the interpretation of deformations of simple elliptic singularities by singular del Pezzo surfaces. Since the cases arising are classified by subsystems of root systems, it seemed natural to seek direct relations between the geometry of the cases arising and properties of the root systems involved. In pursuing this, it is necessary to have at hand a complete list of the subsystems in question, so a direct proof of the classification of these subsystems was sought.
The paper starts by introducing basic notions, including those of \(\mathbb{Z}\)- and \(\mathbb{Q}\)-closed subsystems, then develops an algorithm in terms of the Coxeter diagram for listing all subsystems of a given root system.
In \S2 we classify subsystems up to equivalence by the Weyl group. This is easy to do directly for the classical cases and \(G_ 2\), so most time is spent on \(F_ 4\) and the \(E_ n\). For \(\mathbb{Q}\)-closed subsystems this is done by classifying subsets of the vertex set of the diagram by ``moves''. Analyzing the effect of these moves throws up a notion ``strict on the right'' which reappears in \S3.
We then present various situations (simple and simple-elliptic singularities; intersections of two quadrics) where deformations of singularities can be described in terms of root systems, and we seek to relate the geometry to the combinatorics. In particular, where we have a curve \(\Gamma\), we can characterize the cases when \(\Gamma\) is reducible.
It is well known that the root systems \(B_ l\), \(C_ l\), and \(F_ 4\) can be obtained by ``folding'' \(A_{2l-1}\), \(D_{l+1}\), and \(E_ 6\) respectively. In \S4 we give an axiomatic description of this process which allows us to investigate its effects on subsystems. We then proceed to geometrical applications analogous to those above. algorithm for subsystems of a root system; simple elliptic singularities; singular del Pezzo surfaces; Coxeter diagram; intersections of two quadrics; deformations of singularities C.\ T.\ C. Wall, Root systems, subsystems and singularities, J. Alg. Geom. 1 (1992), 597-638. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Computational aspects of algebraic surfaces, Deformations of complex singularities; vanishing cycles Root systems, subsystems and 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 Le présent article peut être considéré comme un chapitre dans la théorie classique des algèbres simples centrales.
Soit K un corps. Une algèbre simple centrale A/K de dimension finie sur son centre K s'écrit comme une algèbre de matrices \(M_ r(D)\) sur un corps gauche D de centre K. On définit l'indice \(i(A)=i(D)=\sqrt{[D:K]}\), puis \(ms(A)=r\) (ici \(ms=matrix\) size), enfin le degré \(\deg (A)=ri(A)=\sqrt{[A:K]}.\)
Soit L/K une extension finie de corps, et A/L une algèbre simple centrale. Les auteurs s'intéressent aux K-algèbres simples centrales B dans lesquelles on peut plonger l'algèbre A (vue comme K-algèbre).
Ils introduisent les invariants \(d_ K(A)=\min_ B\deg_ K(B)\); \(ms_ K(A)=\min_ Bms_ K(B)\). On a toujours l'inégalité \(\deg_ L(A).[L:K]\leq d_ K(A)\). Pour obtenir des informations plus précises, les auteurs introduisent d'autres invariants plus agréables. Le premier, \(r_ K(A)\), est le minimum des i(C) pour C/L simple centrale et de même classe que A dans le quotient de groupes de Brauer Br(L)/Res(Br(K)). Le second, \(e_ K(A)\), est l'exposant de la classe de A dans Br(L)/Res(Br(K)). Il y a aussi un troisième invariant, \(k_ K(A)\), utile dans l'étude de \(ms_ K(A)\). Les auteurs établissent la formule \(d_ K(A)=[L:K].\deg_ L(A).r_ k(A).\)
Comme l'invariant \(r_ K(A)\) (à la différence de \(d_ K(A))\) se calcule composante p-primaire par composante p-primaire, cela permet aux auteurs de montrer que \(d_ K(A)\) divise \(\deg_ K(B)\) pour toute K- algèbre simple centrale B dans laquelle A se plonge. Ce résultat de divisibilité vaut aussi pour \(ms_ K(A)\), grâce à une formule analogue.
Sur des corps raisonnables comme les corps de nombres, exposant et indice d'une algèbre simple centrale coïncident. Les auteurs font une analyse systématique de leurs invariants sur de tels corps. Ils montrent en particulier qu'alors \(d_ K(A)=e_ K(A)\) ou \(2e_ K(A)\), et montrent que l'égalité \(d_ K(A)=2e_ K(A)\), bien qu'exceptionnelle, peut être réalisée. Il y a des résultats analogues pour \(ms_ K(A)\) sur de tels corps. embeddings of central simple algebras; exponent; index; degree; invariants; Brauer groups; number fields DOI: 10.1016/0021-8693(90)90277-U Finite-dimensional division rings, Brauer groups of schemes Minimal embeddings of central simple 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 paper is the first part of the author's doctoral dissertation (with the same title) defended at the University of Bucharest in 1981. Let \(f: X\to Y\) be a morphism of finite type from a noetherian, universally japanese scheme X to an arbitrary scheme Y. One of the main results of this part of the paper is the following: if \(f_*(F)\) is an \({\mathcal O}_ Y\)-module of finite type for every \({\mathcal O}_ X\)-module F, then f is a proper morphism. One of the main obstruction in dealing with such kind of questions is a systematic need to study the schemes with 1-codimensional closed points.
These schemes will also occur in \(parts\quad II\quad and\quad III\) of this paper [ibid. 38, 438-451 and 477-510 (1986)] in connection with the study of the finite generatedness of subalgebras of an algebra of finite type over a field. As an application of the main result, the author is able to strengthen some criteria for affineness (due to Serre and Goodman and Landman). morphism of finite type; universally japanese scheme; proper morphism; schemes with 1-codimensional closed points Adrian Constantinescu. Proper morphisms and finite generation of subalgebras. I. Proper morphisms of schemes. Stud. Cerc. Mat., 38(4):321--341, 1986. Birational geometry, Schemes and morphisms Proper morphisms and finite generation of subalgebras. I: Proper morphisms of 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 See the preview in Zbl 0725.13004. Euclidean domains; algebras of finite type over a field; diophantine geometry; integral points on curves; Euclidean algorithm; generalized Jacobian varieties Commutative Artinian rings and modules, finite-dimensional algebras, Euclidean rings and generalizations, Rational points Euclidean rings of affine 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 article develops results concerning the projective McKay correspondence for the action of a finite group \(\widetilde{G}\subset \text{PSL}(2,{\mathbb C})\) on \({\mathbb P}^1,\) analogous to those obtained by \textit{A. Kirrilov} [Mosc. Math. J. 6, No. 3, 505--529 (2006; Zbl 1171.14302)]. More precisely, let \(G\subset\text{SL}(2,{\mathbb C})\) be a group such that \(\widetilde{G}/{\pm I}=G,\) where \(I\) is the identity matrix. Let \(\Gamma\) be the affine Dynkin graph associated to \(G\) and let \(\Pi_\Gamma\) denote the preprojective algebra. There is an equivalence between the bounded derived category of \(\widetilde{G}-\)equivariant coherent sheaves on the cotangent bundle \(T^*{\mathbb P}^1\) and the bounded derived category of finitely generated \(\Pi_\Gamma-\)modules. This provides a link with the \(2-\)dimensional affine McKay correspondence (see \textit{T. Bridgeland, A. King, M. Reid} [J. Am. Math. Soc. 14, No.3, 535--554 (2001; Zbl 0966.14028)]). That is, there exists a sequence of equivalences
\[
D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\simeq D^b(\Pi_\Gamma)\simeq D^b_G({\mathbb C}^2)\simeq D^b(Y),
\]
where \(Y\) is the minimal resolution of \({\mathbb C}^2/G.\)
The equivalence \(R\Phi_h:D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\longrightarrow D^b(\Pi_\Gamma)\) constructed in the article, depends on a height function \(h.\) Such function assigns an integral number to each vertex of graph \(\Gamma\) and every two height functions are connected by a series of Bernstein-Gelfand-Ponomarev reflections, [cf. \textit{I. N. Bernstein, I. M. Gelfand} and \textit{V. A. Ponomarev}, Russ. Math. Surv. 28, No.2, 17-32 (1973; Zbl 0279.08001)].
Let \({\mathcal D}\) denote the subcategory of \(D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\) consisting of objects supported along the zero section. Define \({\mathcal B}_h\) to be the heart associated to the t-structure obtained by restricting the pull-back of the standard t-structure on \(D^b(\Pi_\Gamma)\) to the category \({\mathcal D}.\) Theorem 5.8 shows that hearts \({\mathcal B}_h, {\mathcal B}_{\sigma_i^{\pm} h},\) where \({\sigma_i^{\pm} h}\) denotes the height function obtained by reflection of \(h\) at the vertex \(i\) of \(\Gamma\), are related by a spherical twist. Moreover, the spherical twist between the hearts can be realized as tilting (Theorem 5.8 and Proposition 5.9).
The proof of above results involves the theory of Koszul algebras. In particular, it is shown that the preprojective algebra \(\Pi_\Gamma\) is the Koszul dual of the \(\text{Ext}-\)algebra of a \(\Gamma-\)collection of spherical objects in \(D^b_{\widetilde{G}}(T^*{\mathbb P}^1)\).
Section 3 of the article contains a detailed review of the results by A. Kirillov. McKay correspondence; Dynkin quiver; quiver representations; cotangent bundle of projective line; Koszul duality; reflection functor; spherical twist; derived equivalence McKay correspondence, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Representations of quivers and partially ordered sets, Quadratic and Koszul algebras The projective McKay 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 Cet ouvrage est consacré à l'arithmétique des ``corps cyclotomiques'', dans le prolongement logique des travaux de Kummer sur les corps \(\mathbb Q(\mu_{p^n})\) (des racines \(p^n\)-ième de l'unité) en rapport avec le théorème de Fermat (en réalité, comme nous allons le préciser, l'ouvrage déborde largement ce cadre, sans toutefois englober une théorie des corps abéliens). Une des motivations de l'auteur semble, en effet, avoir été ce qu'il appelle lui-même ``la conjecture de Vandiver'' \((p\) ne divise pas le nombre de classes réelles de \(\mathbb Q(\mu_p))\): ``\dots The terminology ``Vandiver's conjecture'' seemed appropriate to me. In any case I believe it. (\dots) Proving the Vandiver conjecture (\dots) is therefore one of the major problems of algebraic number theory today''. Il en résulte que, lorsqu'il aborde la question des \(p\)-classes d'idéaux, l'auteur privilégie les résultats de structure découlant de l'hypothèse de Vandiver (essentiellement des résultats de Galois-monogénéité de ces \(p\)-groupes de classes); cet aspect de l'ouvrage montre que l'auteur n'a pas en vue l'arithmétique abélienne générale pour laquelle les problèmes majeurs actuels semblent plutôt constitués par la ``non cyclicité'' des groupes de classes.
Ceci étant dit, on retiendra que l'auteur traite de façon remarquable (et générale), et avec les apports techniques les plus récents: (i) la théorie des fonctions \(L\) \(p\)-adiques abéliennes, (ii) la théorie d'Iwasawa et ses applications à une ``cyclotomic theory'', (iii) enfin les lois explicites de réciprocité. De façon précise:
Chapitre 1: Sommes des Gauß et de Jacobi, théorème de Stickelberger, et théorèmes d 'annulation, par les nombres de Bernoulli, des \(\chi\)-composantes des \(p\)-groupes de classes imaginaires.
Chapitre 2: Exposé des propriétés des mesures de Bernoulli, étude des idéaux de Stickelberger, étude plus générale de la notion de distribution, assortie de nombreux exemples. Des calculs d'indices d'idéaux de Stickelberger généralisent un résultat d'Iwasawa.
Chapitre 3: Formule analytique du nombre de classes \(h\) (non redémontrée), décomposition de \(h\) sous la forme \(h^+h^-\) et leurs expressions classiques [suit à peu près l'exposé de \textit{H. Hasse}, ``Über die Klassenzahl Abelscher Zahlkörper.'' Berlin: Akademie-Verlag (1952; Zbl 0046.26003) et celui de \textit{Z. I. Borevich} et \textit{I. R. Shafarevich}, ``Number theory.'' Moscow: Nauka (1964; Zbl 0121.04202); German translation Basel 1966]; s'achève par une majoration de \(h^-\) pour les corps \(\mathbb Q(\mu_p)\).
Chapitre 4: Exposé général de la théorie des fonctions \(L\) \(p\)-adiques abéliennes, vues d'après Mazur comme transformées de Mellin d'une mesure convenable. L'auteur introduit l'isomorphisme d'Iwasawa (entre mesures et séries formelles de l'algèbre d'Iwasawa); il explicite la formule analytique \(p\)-adique du nombre de classes de Leopoldt ainsi que le calcul de \(L_p(1,\chi)\).
Chapitre 5: Excellent exposé de la théorie d'Iwasawa, qui reprend des points de vue développés par Serre; les bases algébriques sont données en détail, (structure des modules de type fini sur l'algèbre d'Iwasawa \(\Lambda = \mathbb Z_p[[X]] )\). Étude des \(p\)-classes d'idéaux dans une \(\mathbb Z_p\)-extension. L'auteur introduit ensuite la théorie de Galois et du corps de classes infinies pour l'étude du groupe de Galois de la \(p\)-extension \(p\)-ramifiée maximale d'un corps de nombres \((\Lambda\)-structure, existence des \(\mathbb Z_p\)-extensions).
Chapitre 6 est relatif au cas particulier de la \(\mathbb Z_p\)-extension cyclotomique d'un corps. Étude du \(p\)-groupe des classes de \(\cup \mathbb Q(\mu_{p^n})\): \(C = \varprojlim C_n\), \(C_n\) \(p\)-groupe des classes de \(\mathbb Q(\mu_{p^n})\). La technique consiste en un passage à la limite projective des méthodes du ``Spiegelungssatz'' de \textit{H.-W. Leopoldt} [J. Reine Angew. Math. 199, 165--174 (1958; Zbl 0082.25402)], qui donne, sous l'hypothèse de Vandiver, la \(\Lambda\)-cyclicité de \(C\).
Chapitre 7: Étude (pour l'extension cyclotomique fondamentale locale) de la limite projective \(U\) des groupes des unités locales; l'exposé suit la récente généralisation du travail de Kummer, par Coates et Wiles; soit \(V\) l'image dans \(U\) des unités cyclotomiques, l'auteur donne la structure de \(U/V\) (sous la forme \(U(\chi)/V(\chi)\cong \Lambda/g_\chi \Lambda\), où \(g_\chi\) est la série d'Iwasawa attachée à \(L_p(s,\chi)\), \(\chi\) caractère de \(\mathbb Q(\mu_p))\).
Chapitres 8 et 9: Théorie des groupes formels de Lubin-Tate, et détermination des lois explicites de réciprocité (d'après un travail de Coates et Wiles). Ici aussi la théorie exposée est très complète et offre une synthèse appréciable d'un grand nombre d'articles publiés sur le sujet. La bibliographie, sans oublier les articles fondamentaux, met en évidence les articles significatifs les plus récents dont l'auteur a pu avoir connaissance (une référence sur six est ``à paraître'').
Ce livre bénéficie de l'importante culture de son auteur; de ce fait il renouvelle beaucoup le style de cet aspect de l'arithmétique (grace, entre autre, à un choix judicieux des définitions et du vocabulaire). Par son originalité, il devrait donc s'avérer très utile à beaucoup de théoriciens des nombres. Vandiver conjecture; p-adic abelian L-functions; research survey; explicit reciprocity laws; cyclotomic fields; Jacobi sums; Iwasawa invariant; Iwasawa theory; Bernoulli numbers; Gauss sums; Stickelberger ideals; distribution; Bernoulli measure; analytic class number formula; p-adic analytic class number formula of Leopoldt; p-adic class group; Spiegelungssatz; Lubin-Tate formal groups; projective limit of local unit groups; cyclotomic units S.~Lang, \emph{Cyclotomic fields}, Springer-Verlag, New York-Heidelberg, 1978, Graduate Texts in Mathematics, Vol. 59. zbl 0395.12005; MR0485768 Cyclotomic extensions, Research exposition (monographs, survey articles) pertaining to number theory, Iwasawa theory, Units and factorization, Class numbers, class groups, discriminants, Class field theory; \(p\)-adic formal groups, Formal groups, \(p\)-divisible groups, Zeta functions and \(L\)-functions, Trigonometric and exponential sums (general theory) Cyclotomic 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 This monograph is essentially about the representation theory of a sheaf of reductive Lie algebras on a generalization \(J(X;L,d)\) of the classical Jacobian to smooth projective surfaces \(X\), whose closed points are pairs \((\mathcal{E}, [e])\), \(\mathcal{E}\) torsion-free sheaves of rank 2 on \(X\), \(c_1(\mathcal{E})=L\), \(L\) a fixed divisor on \(X\), \(c_2(\mathcal{E})=d \geq 0\), \(e\) a global section of \(\mathcal{E}\) with homothety class \([e]\), \(Z_e = (e=0)\), and the sheaf of Lie algebras in question is obtained from reductive Lie subalgebra \(\tilde{\mathcal{G}}(\mathcal{E}, [e])\) of \(\mathfrak{gl}(H^0(\mathcal{O}_{Z_e}))\). Among the many applications developed in this work (analog of a Lie algebraic aspect of the classical Jacobian, analog of a variation of Hodge structure à la Griffiths, analog of an infinitesimal Torelli theorem, toric geometry, action of affine Lie algebras on the direct sum of cohomology rings of Hilbert schemes, \dots) chief among those is probably the connection with Langlands Duality, and it is very much in this spirit that this work was initiated. It is also having Langlands duality in mind that we recommend reading this work lest the reader be quickly sidetracked by peripheral results that are certainly bold and tantalizing connections to various subfields of Algebraic Geometry and Representation Theory, but which unfortunately are not fully exploited and somewhat obscure the author's original goal, that of providing new insights into the Langlands program.
Two connected results of the author that are worthy of attention are for one thing that \(J(X;L,d)\) yields a finite collection \(\mathcal{V}\) of quasi-projective subvarieties of \(X^{[d]}\), every element \(\Gamma\) of which determines a finite collection of nilpotent orbits in \(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C})\), \(d[\Gamma] \leq d\) intrinsically associated to \(\Gamma\), and second that the same \(\Gamma\)'s determine a finite collection \( ^L R(\Gamma)\) of irreducible representations of the Langlands dual group \( ^L \mathfrak{sl}_{d[\Gamma]}(\mathbb{C}) = \mathrm{PGL}_{d[\Gamma]}(\mathbb{C})\). On a certain subset \(\breve{J}\) of \(J(X;L,d)\), \(H^0(\mathcal{O}_{Z_e})\) has a certain direct sum decomposition, and with the ring structure on \(H^0(\mathcal{O}_{Z_e})\), we get a reductive Lie subalgebra \(\tilde{\mathcal{G}}(\mathcal{E}, [e])\) of \(\mathfrak{gl}(H^0(\mathcal{O}_{Z_e}))\), the semisimple part of which is denoted by \(\mathcal{G}(\mathcal{E}, [e])\). We also have a morphism of schemes \(J(X;L,d) \rightarrow{\pi} X^{[d]}\) that sends a point \((\mathcal{E}, [e])\) to \([Z_e]\). One can attach a nilpotent element \(D^+(\nu)\) of \(\mathcal{G}(\mathcal{E}, [e])\) to every tangent vector \(\nu\) of \(\breve{J}\) along the fibers of \(\pi\), at a point \((\mathcal{E}, [e])\). If one denotes by \(T_{\pi}(\mathcal{E}, [e])\) the space of all such vectors of \(\breve{J}\) at \((\mathcal{E}, [e])\), one obtains a linear map
\[
D^+_{(\mathcal{E}, [e])}: T_{\pi}(\mathcal{E}, [e]) \rightarrow \mathcal{N}(\mathcal{G}(\mathcal{E}, [e])).
\]
The nilpotent cone of \(\mathcal{G}(\mathcal{E}, [e])\) being partitioned into a finite set one gets the first result.
The loop version of this map has values in the infinite Grassmannian \(\mathrm{Gr}(\mathcal{G}(\mathcal{E}, [e]))\) of \(\mathcal{G}(\mathcal{E}, [e])\) and one obtains a loop version of the first result where now \(\Gamma\)'s determine a finite collection of orbits in \(\mathrm{Gr}(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C}))\), and taking the intersection cohomology complexes of those orbits one passes to the category of perverse sheaves on \(\mathrm{Gr}(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C}))\), from which the second result follows after making use of the geometric version of the Satake isomorphism of [\textit{V. Ginzburg}, ``Perverse Sheaves on a Loop Group and Langlands Duality'', \url{arXiv:alg-geom/9511007}] and \textit{I. Mirkovic} and \textit{K. Vilonen} [Ann. Math. (2) 166, No. 1, 95--143 (2007; Zbl 1138.22013)].
One other result that is worthy of attention is the following. \(J(X;L,d)\) determines a finite collection \(\mathcal{P}(X;L,d)\) of perverse sheaves on \(X^{[d]}\), intersection cohomology complexes associated to local systems \(\mathcal{L}_{\lambda}\) on \(\Gamma\), \(\Gamma \in \mathcal{V}\), \(\mathcal{V}\) as in the first result above, \(\mathcal{L}_{\lambda}\) corresponding to a representation \(\pi_1(\Gamma, [Z]) \rightarrow \mathrm{Aut}(H^{\bullet}(B_{\lambda}, \mathbb{C}))\), \(B_{\lambda}\) a Springer fiber over the nilpotent orbit \(O_{\lambda}\) of \(\mathfrak{sl}_{d[\Gamma]}(\mathbb{C})\), \(\lambda\) a partition of \(d[\Gamma]\). \(\mathcal{P}(X;L,d)\) gives rise to a distinguished collection \(C(X;L,d)\) of irreducible perverse sheaves on \(X^{[d]}\) and one denotes by \(\mathcal{A}(X;L,d)\) the abelian category of finite direct sums of \(C[n]\), \(n \in \mathbb{Z}\), \(C \in C(X;L,d)\). \(J(X;L,d)\) also comes equipped with a Cartier divisor \(\Theta(X;L,d)\) parametrizing pairs \((\mathcal{E}, [e])\), \(\mathcal{E}\) not locally free, and letting \(J^0(X;L,d) = J(X;L,d)\setminus \Theta(X;L,d)\), \(\mathcal{T}^*_{J^0(X;L,d) / X^{[d]}}\) the sheaf of relative differentials of \(J^0\) over \(X^{[d]}\), then the author proves there is a natural map \(H^0(\mathcal{T}^*_{J^0(X;L,d) / X^{[d]}}) \rightarrow \mathcal{A}(X;L,d)\), which one can see as a deformation theoretic result instrumental in reformulating the classical Langlands correspondence into the geometric Langlands correspondence. Jacobian; Hilbert scheme of points; period map; Torelli problem; Springer resolution; Langlands duality; perverse sheaves; Griffiths period domain; affine Lie algebras; variation of Hodge structure Parametrization (Chow and Hilbert schemes), Variation of Hodge structures (algebro-geometric aspects), Torelli problem, Families, moduli, classification: algebraic theory, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) Nonabelian Jacobian of projective surfaces. Geometry and representation 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 The main object of the paper under review is an algebraic torus \(T\) defined over a \(p\)-adic field \(k\). The goal is to construct an integer model of \(T\) (i.e. an \(O\)-scheme \(\mathcal T\) with generic fibre \(T\) where \(O\) is the ring of integers of \(k\)) which would possess certain finiteness and minimality properties. Note that classic Néron-Raynaud models of tori (cf. \textit{S. Bosch}, \textit{W. Lütkebohmert}, and \textit{M. Raynaud} [Néron models. Berlin: Springer (1990; Zbl 0705.14001)]) may fail to be \(O\)-schemes of finite type. The authors construct such models and compute the reduction (i.e. the special fibre of \(\mathcal T\)) in a number of cases thus generalizing some results of \textit{E. Nart} and \textit{X. Xarles} [Arch. Math. 57, No. 5, 460--466 (1991; Zbl 0782.14042)]. They also present a global analogue of this construction and give some applications to computing local volumes and class numbers.
There are several remarks. First, the definition of \(A_0\) (formula (2) on p. 883) should be corrected as follows:
\[
A_0=\{f\in k[T]\mid f(u)\in O_L\;\forall u\in U_L\;\forall L/k,\;[L:k]<\infty\}
\]
(see the author [Algebraic groups and their birational invariants. Translations of Mathematical Monographs. 179. Providence, RI: AMS (1998; Zbl 0974.14034)]). Second, we refer to the reviewer and \textit{J.-J. Sansuc} [C. R. Acad. Sci., Paris, Sér. I 324, No. 3, 307--312 (1997; Zbl 0920.14024)]\ for some further development.
Finally, one has to remark that non Russian speaking readers may encounter problems with understanding the translation. To make their life easier, here is a sample of English-Russian-English translation: ``strongly planar'' (p. 882) should mean ``faithfully flat''. algebraic tori over \(p\)-adic fields; integer models; reductions; schemes of finite type; local volumes; class numbers Voskresenskii V. E., Izvestiya: Mathematics 59 (5) pp 881-- Linear algebraic groups over local fields and their integers, Linear algebraic groups over global fields and their integers, Algebraic field extensions, Minimal model program (Mori theory, extremal rays), Group schemes, Local ground fields in algebraic geometry Integral structures in algebraic tori | 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 is the following Theorem 1.1. Let \(n,m\geq 1\) be relatively prime positive integers with \(m>1\). Let \(\mathbb{Z}'\) be the localization of \(\mathbb{Z}\) at a multiplicative set of nonzero integers whose elements are relatively prime to \(m\). Let \(\mu_{\text{tor}}(R)\) denote the group of roots of unity in a commutative ring \(R\). For \(m>2\), the natural multiplication map
\[
\mu_{\text{tor}}(\mathbb{Z}[\zeta_n])\times\mu_m\to\mu_{\text{tor}} (\mathbb{Z}'[\zeta_n][x])/(x^{m-1}+\cdots+x+1)
\]
is an isomorphism, where \(\mu_m\) denotes the cyclic group of order \(m\) generated by \(x\). When \(m=2\), this map is surjective with order-2 kernel that has \((-1,-1)=(-1,x)\) as its unique nontrivial element. finite étale Galois coverings; connected schemes; Abelian Galois groups; automorphisms of finite order; roots of unity Linear algebraic groups over adèles and other rings and schemes, Affine algebraic groups, hyperalgebra constructions Finite-order automorphisms of a certain torus. | 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 consider elliptic curves with the equation
\[
E^u: y^2= x^3+ ux^2- 16x
\]
as well as their quadratic twists \(E^u_n\) by a squarefree integer \(n\), with
\[
u^2+ 65= p_1p_2\cdots p_l,
\]
where \(p_1,p_2,\dots, p_l\) are prime numbers.
For the case when \(l<3\) and \(n\) is of the form \(n= 4k+1\), \(k\in\mathbb{Z}\), as well as all prime divisors of \(n\) are of the form \(4\lambda+ 3\), \(\lambda\in\mathbb{Z}\), the authors have produced a complete study of sizes of Selmer groups of \(E^u_n\) by using the number of even partitions of certain graphs. Further related results are also obtained. rank of elliptic curves; twists of elliptic curves; Selmer groups; graph; Neumann-Setzer type elliptic curves Elliptic curves over global fields, Elliptic curves, Zeta functions and \(L\)-functions of number fields, Applications of graph theory to circuits and networks Partitions of graphs and Selmer groups of elliptic curves of Neumann-Setzer 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 \(Q\) be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid \(\mathcal{CM}(Q)\), as introduced by M. Reineke, and the generic composition algebra \(\mathcal C_q(Q)\), as introduced by C. M. Ringel, specialised at \(q=0\). In this thesis we continue their work. We show that if \(Q\) is a Dynkin quiver or an oriented cycle, then \(\mathcal C_0(Q)\) is isomorphic to the monoid algebra of \(\mathbb Q\mathcal{CM}(Q)\). Moreover, if \(Q\) is an acyclic, extended Dynkin quiver, we show that there exists a surjective homomorphism \(\Phi\colon\mathcal C_0(Q)\to\mathbb Q\mathcal{CM}(Q)\), and we describe its non-trivial kernel.
Our main tool is a geometric version of BGP reflection functors on quiver Grassmannians and quiver flags, that is varieties consisting of filtrations of a fixed representation by subrepresentations of fixed dimension vectors. These functors enable us to calculate various structure constants of the composition algebra.
Moreover, we investigate geometric properties of quiver flags and quiver Grassmannians, and show that under certain conditions, quiver flags are irreducible and smooth. If, in addition, we have a counting polynomial, these properties imply the positivity of the Euler characteristic of the quiver flag. representations of quivers; Hall polynomials; Hall algebras; Schur roots; composition monoids; extended Dynkin quivers; quiver flag varieties; composition algebras; reflection functors; quiver Grassmannians Wolf, S.: The Hall Algebra and the Composition Monoid. arXiv:0907.1106 Representations of quivers and partially ordered sets, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds The Hall algebra and the composition monoid. | 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 the second in a series of four papers on fixed point ratios for actions of classical groups [see \textit{T. C. Burness}, part I, ibid. 309, No. 1, 69-79 (2007; see the preceding review Zbl 1128.20003), for more details on the set-up].
The main theorem of these papers is the following: if \(G\) is a finite almost simple classical group and \(\Omega\) is a faithful transitive non-subspace \(G\)-set, then either \(\text{fpr}(x)\lesssim|x^G|^{-\frac 12}\) for all \(x\in G\) of prime order, or the pair \((G,\Omega)\) is one of a small number of known exceptions.
The proof of this result first reduces to primitive actions and then considers the different possibilities for the stabiliser \(G_\omega\) of a point \(\omega\in\Omega\); in the paper under review the result is proved when \(G_\omega\) lies in a maximal subgroup from one of the Aschbacher families \(\mathcal C_i\), where \(4\leq i\leq 8\) [see \textit{M. Aschbacher}, Invent. Math. 76, 469-514 (1984; Zbl 0537.20023)]. Prior to the main proof, the author includes a section containing many preliminary results which may be of interest independent of their applications in these papers. finite classical groups; fixed point ratios; primitive permutation groups; monodromy groups; permutation representations; finite almost simple groups; maximal subgroups of classical groups T. C. Burness, Fixed point ratios in actions of finite classical groups, II, Journal of Algebra 309 (2007), 80--138. Primitive groups, Linear algebraic groups over finite fields, Representation theory for linear algebraic groups, Riemann surfaces; Weierstrass points; gap sequences, Group actions on varieties or schemes (quotients) Fixed point ratios in actions of finite classical groups. 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 See the preview in Zbl 0739.14003. finite fundamental groups; invariants of surfaces of general type B. Moishezon and M. Teicher, Finite fundamental groups, free over \(\mathbb{Z}\)/c\(\mathbb{Z}\), Galois covers of \(\mathbb{C}\)\(\mathbb{P}\) 2, Mathematische Annalen 293 (1992), 749--766. Coverings in algebraic geometry Finite fundamental groups, free over \(\mathbb{Z}/c\mathbb{Z}\), for Galois covers of \(\mathbb{C}\mathbb{P}^2\). | 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 Let \(M_0\) be a smooth complex threefold with trivial canonical bundle \(\omega_{M_0}\) acted on by a finite group \(G\) of automorphisms acting trivially on \(\omega_{M_0}\). A conjecture of Dixon-Harvey-Vafa-Witten says that there always exists a desingularization \(M_0/G\) with trivial canonical bundle, and predicts its Euler number. This conjecture has a local form from which it follows: If \(G\) is a finite subgroup of \(SL(3,{\mathbb{C}})\), there exists a crepant (i.e. with trivial canonical bundle) desingularization of \({\mathbb{C}}^3/G\), and its Euler number is the number of conjugacy classes of \(G\). The conjecture is now completely solved: There is a list of all finite subgroups of \(SL(3,{\mathbb{C}})\) due to Miller, Blichfeldt and Dickson, and a case-by-case analysis, of which this paper is part, shows that there always exists a crepant resolution, constructed by an explicit sequence of blow-ups. The formula for the Euler number follows from these explicit constructions; it also has an independent general proof by the generalization of the MacKay correspondence to dimension \(3\) by \textit{Y. Ito} and \textit{M. Reid}, in: Higher-dimensional complex varieties, Proc. Int. Conf., Trento 1994, 221-240 (1996; Zbl 0894.14024). The construction of a crepant resolution of \({\mathbb{C}}^3/H_{168}\) is pretty straightforward: One makes \(4\) successive blow-ups of singular curves until all the singular points disappear. The fact that the resolution obtained in this way is crepant follows from a remark of Reid (1979). Calabi-Yau threefolds; McKay correspondence; crepant resolutions; complex threefold; finite group of automorphisms; Euler number G. Markushevich, ''Resolution of \(\mathbb{C}\)3/H 168,''Math. Ann.,308, 279--289 (1997). \(3\)-folds, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Birational automorphisms, Cremona group and generalizations Resolution of \(\mathbb{C}^3/H_{168}\) | 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 Soient \(k\) un corps de caractéristique 0 et \(X\) une \(k[[t]]\)-variété (éventuellement singulière) plate, purement de dimension relative \(d\). Nous prouvons la rationalité des séries de Poincaré motiviques et de fonctions Zêta d'Igusa motiviques, associées à \(X\), à l'aide de l'intégration motivique, du théorème de désingularisation plongée d'Hironaka, de la théorie des modèles de Néron faibles pour les schémas formels et d'un théorème d'élimination des quantificateurs en théorie des modèles. motivic zeta functions; resolution of singularities; formal schemes Sebag J. , Rationalité des séries de Poincaré et des fonctions zêta motiviques , Manuscripta Math. 115 ( 2 ) ( 2004 ) 125 - 162 , (in French). MR 2098466 | Zbl 1073.14524 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta functions and \(L\)-functions, Motivic cohomology; motivic homotopy theory Rationality of Poincaré series and motivic zeta 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 Trying to resolve singularities of algebraic varieties in positive characteristic, the second author introduced Rees algebras several years ago. A Rees algebra over a smooth variety \(V\) over a field \(k\) is a graded subsheaf \(\mathcal G = \sum _{i=1}^{\infty} I_i W^i\) of \({{\mathcal O}_V} [W]\), locally finitely generated, where \(W\) is an indeterminate. The singular locus \(\mathrm{sg}(\mathcal G)\) is the set of points \(x \in V\) such that \({\nu} _x (I _i)\), the order of the ideal \(I_i{\mathcal O}_{V,x}\) is \(\geq i\). If one is able to solve Rees algebras by means of a finite sequence of suitable monoidal transformations (i.e., reach a situation where the singular locus is empty) it is possible to resolve singularities of algebraic varieties. This is usually done, following Hironaka, with the aid of Hilbert-Samuel functions. Under appropriate assumptions (which are not really restrictive) one may find, given a Rees algebra \({\mathcal G}\) over a smooth \(d\)-dimensional variety \(V\), a \textit{transversal projection} onto a smooth \(d'\)-dimensional variety \(V'\), \(d'<d\), that is a smooth surjective morphism \(\beta : V \to V'\) with some nice properties, and a useful Rees algebra \({\mathcal R}={\mathcal R}_{\mathcal G, \beta}\) over \(V'\), called the \textit{elimination algebra} of \(\mathcal G\), relative to \(\beta\). In characteristic zero one shows that a resolution of \(\mathcal G'\) induces one of \(\mathcal G\), so we may use induction on the dimension to resolve \(\mathcal G\). This method is an alternative to the use of \textit{subvarieties of \(V\) of maximal contact}, a more ``classical'' technique, available in characteristic zero only. If the characteristic is positive, the method of projections still might work, but the situation is more complicated, there are problems that are not fully solved yet. Villamayor and some of his former students (A. Bravo, A. Benito, S. Encinas, etc.) have introduced and studied certain invariants of Rees algebras, trying to overcome those problems.
The most basic one, due to Hironaka, in the context of Rees algebras and using the notation above, is \(\mathrm{ord}(\mathcal G)(x)=\)min\(\{\nu_x(I_n) / n : n \in {\mathbb N}\}\) (for \(x \in \mathrm{sg}(\mathcal G))\). Let \(\tau (x)\) be the Hironaka invariant (for Rees algebras, its definition uses the \textit{tangent cone} of \(\mathcal G\) at \(x\)). If \(\tau(x) \geq e\) for all \(x \in \mathrm{sg}(\mathcal G)\) one gets a transversal projection \(\beta\) as above with \(V'\) of dimension \(d-e\). Then define \(\mathrm{Ord}^{(d-e)}(\mathcal G)(x)=\mathrm{ord}(\mathcal R _{\mathcal G, \beta}(x))\).
In previous work the authors showed useful applications of Ord, but trying to surmount the mentioned difficulties, further refinements seem necessary. In this paper the authors discuss a refinement, the function \(\text{H-ord}^{(d-e)}(\mathcal G):\mathrm{sg}(\mathcal G) \to {\mathbb Q}\). Its definition involves a ``presentation'' of \(\mathcal G\) in terms of the elimination algebra \(\mathcal R _{\mathcal G, \beta}\) and certain auxiliary monic polynomials with coefficients in the completion of the local ring of \(\mathcal O _{V',\beta(x)}\). To show the existence of such presentations the authors prove a variant, or generalization, of Weierstrass' Preparation Theorem. These functions are not necessarily upper semicontinuos (as often invariants are), though they satisfy though the inequality \( {\text{H-ord}} ^{(d-e)}(\mathcal G)(x) \leq {\text{Ord}}^{(d-e)}(\mathcal G)(x) \).
After discussing the necessary background to properly define these functions and proving some of its properties (which is pretty technical work) the authors prove their main result. This is an improvement upon a theorem in [\textit{A. Bravo} and \textit{O. Villamayor U.}, Adv. Math. 224, No. 4, 1349--1418 (2010; Zbl 1193.14019)], that essentially says that, under some extra hypotheses which are not really restrictive, with the technique of transversal projections and elimination algebras, by induction of the dimension, given a Rees algebra \(\mathcal G\) one can find a sequence of permissible transformations
\[
\mathcal G = {\mathcal G}_0 \leftarrow \cdots \leftarrow {\mathcal G}_r
\]
where either \(\mathrm{sg}({\mathcal G}_r)=\emptyset\), or the \(\tau\)-invariant increases, or \({\mathcal R} _{{\mathcal G}_r,\beta_r}\) (the elimination algebra, relative to a suitably induced projection \(\beta_r\)) is \textit{monomial} (a particularly simple type of algebra). In characteristic zero this is enough to obtain, with some extra work, a resolution of \(\mathcal G\). In characteristic \(p >0\) the third (monomial) case is more complicated. In this paper the authors obtain new results. Namely, (assuming the base field is perfect, if the H-ord function of \(\mathcal G_r\) satisfies an equality involving ord of a certain attached monomial algebra, then also in this case we are led to a resolution of \(\mathcal G\) or an increase in the value of \(\tau\). A similar result when \(\tau \geq e=1\) had been obtained in [\textit{A. Benito} and \textit{O. E. Villamayor U.}, Compos. Math. 149, No. 8, 1267--1311 (2013; Zbl 1278.14019)].
With these techniques they prove that it is possible to prove an embedded desingularization theorem for an algebraic surface \(X\) embedded in a smooth variety \(V\), over a perfect field. In previous work the authors had showed a similar result in case \(\dim V =3\). resolution of singularities; Hilbert-Samuel function; Rees algebra; invariant; transversal projection Benito, A., Villamayor U., O.E.: On elimination of variables in the study of singularities in positive characteristic. arxiv.1103.3462 Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Multiplicity theory and related topics, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics On elimination of variables in the study of singularities in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\{G_ i\}\) be a set of finite groups, \(\{\rho_{ij}:G_ i\to GL(V_ j)\}\) a set of linear representations (over \(\mathbb{C})\) with no trivial components (i.e., \(V_ j^{\rho_{ij}(G_ i)}=0)\), \(\{R_{ij}=S(V_ j)^{\rho_{ij}(G_ i)}\}\) the set of corresponding algebras of invariants. This paper considers the cases in which codim\( R_{ij}\) and \(d(R_{ij})\to\infty\) as \(\dim_ \mathbb{C} V_ j\to\infty\) (here codim\( R_{ij}\) and \(d(R_{ij})\) are the codimension and defect of the algebra \(R_{ij})\). In particular it is proved that such a case occurs when \(\{G_ i\}\) is the set of all groups of the form \(SL_ n(q)\) (for all \(n\) and \(q)\) and \(\{\rho_{ij}\}\) is the set of all representations with nontrivial components. finite groups; linear representations; algebras of invariants; codimension; defect Ordinary representations and characters, Vector and tensor algebra, theory of invariants, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Representations of finite symmetric groups On algebras of invariants of finite groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is an excellent survey of geometrical properties of the class of flag manifolds and of the Duistermaat-Heckman integration formula as it applies to this class. The appendix collects relevant results on Lie algebras and representation theory. representation of Lie algebras; loop groups; survey; flag manifolds; Duistermaat-Heckman integration formula R. F. Picken, J. Math. Phys., 31, 616--638 (1990). Integration on manifolds; measures on manifolds, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Loop groups and related constructions, group-theoretic treatment The Duistermaat-Heckman integration formula on flag manifolds | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The thesis under review may be viewed as a step forward in understanding the semi-universal deformation of complex analytic isolated complete intersection singularities. One main question is: given such a singularity, which singularities does it deform to ? The answer depends on a detailed knowledge of the geometry of the deformation, and the author does an exhaustive study for simple singularities on space curves. These were classified by \textit{M. Giusti} [Singularities, Summer. Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part I, 457-494 (1983; Zbl 0525.32006)].
The main result is that for the \(S_{\mu}, T_ 7, T_ 8\) and \(T_ 9\) singularities (Giusti's notation), the base space of the semi-universal deformation is isomorphic to a quotient X/W of a torus embedding X by a Weyl group W, such that the discriminant of the group action maps to the discriminant of the deformation. This parallels other known descriptions of the discriminant, for example as done by \textit{E. Brieskorn}, in Actes Congr. intern. Math. (1970), part 2, 279-284 (1981; Zbl 0223.22012) for simple hypersurface singularities [see also \textit{P. Slodowy}, ''Simple singularities and simple algebraic groups,'' Lect. Notes Math. 815 (1980; Zbl 0441.14002)].
The novelty is the introduction of torus embeddings which are constructed from extended Dynkin diagrams corresponding to generalized root systems. This construction, based on recent work by \textit{E. Looijenga} [Invent. Math. 61, 1-32 (1980; Zbl 0436.17005)], is done from a general point of view and should be applicable to other situations. The explicit study of the deformations uses classical algebraic geometry, e.g. results on families of hyperelliptic curves and del Pezzo surfaces. semi-universal deformation of complex analytic isolated complete; intersection singularities; simple singularities on space curves; torus embeddings; Dynkin diagrams; root systems; semi-universal deformation of complex analytic isolated complete intersection singularities Deformations of singularities, Singularities of curves, local rings, Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Complete intersections Torus embeddings and deformations of simple singularities and space 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 In this paper, we develop the theory of residually finite rationally \(p\) (RFR \(p)\) groups, where \(p\) is a prime. We first prove a series of results about the structure of finitely generated RFR \(p\) groups (either for a single prime \(p\), or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR \(p\) groups, which we use to study the boundary manifolds of algebraic curves \(\mathbb{CP}^2\) and in \(\mathbb{C}^2\). We show that boundary manifolds of a large class of curves in \(\mathbb{C}^2\) (which includes all line arrangements) have RFR \(p\) fundamental groups, whereas boundary manifolds of curves in \(\mathbb{CP}^2\) may fail to do so. residually finite rationally \(p\) group; graph of groups; \(3\)-manifold; plane algebraic curve; boundary manifold; Alexander varieties; BNS invariant Residual properties and generalizations; residually finite groups, Plane and space curves, Geometric group theory, Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Homology with local coefficients, equivariant cohomology, Covering spaces and low-dimensional topology, General topology of 3-manifolds Residually finite rationally \(p\) 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 This paper is a continuation of our paper ``Simply connected algebraic surfaces of positive index'' [Invent. Math. 89, 601-643 (1987; Zbl 0627.14019)]. There we constructed a series of surfaces of positive and zero indices denoted \(\hbox{Gal}(X_{ab})\) for \(a\), \(b\in \mathbb{Z}\), \(a\geq 5\), \(b\geq 6\). For \(a,b\) relatively prime we got \(\pi_ 1(\hbox{Gal}(X_{ab}))=0\). This gave a counterexample to the Bogomolov- Watershed conjecture that surfaces of general type and positive index are not simply connected. In this paper we compute \(\pi_ 1(\hbox{Gal}(X_{ab}))\) for arbitrary \(a\), \(b\) and prove that \(\pi_ 1(\hbox{Gal}(X_{ab}))\) is a finite commutative group free over \(\mathbb{Z}/c\mathbb{Z}\) for \(c=\hbox{g.c.d.}(ab)\) with \(2ab\) generators.
This study is related to the problem of finding new invariants of surfaces of general type distinguishing different components of moduli spaces.
We use all the notations, results and subresults of our paper cited above. finite fundamental groups; invariants of surfaces of general type Boris Moishezon and Mina Teicher, Finite fundamental groups, free over \?/\?\?, for Galois covers of \?\?², Math. Ann. 293 (1992), no. 4, 749 -- 766. Coverings in algebraic geometry Finite fundamental groups, free over \(\mathbb Z/c\mathbb Z\), for Galois covers of \(\mathbb C\mathbb P^2\) | 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 main goal of this paper is to discuss a certain set of ordered bases of the Milnor lattice of a simple singularity which I call distinguished [up to the ordering they correspond to the ''weakly distinguished'' bases of \textit{S. M. Husein-Zade} ; cf. Russ. Math. Surv. 32, No.2, 23-69 (1977); translation from Usp. Mat. Nauk 32, No.2(194), 23-65 (1977; Zbl 0363.32010)], by means of the corresponding root system \((=set\) of vanishing cycles). It is shown that every ordered basis of the Milnor lattice, which consists of vanishing cycles (i.e. of roots), is distinguished. At the same time I prove a conjecture of Husein-Zade (in the case of a simple singularity) concerning the orbits of the braid- group on these bases. The orbits are explicitly determined. Dynkin diagram; ordered bases of the Milnor lattice of a simple singularity; root system; vanishing cycles; braid-group Voigt E., Ausgezeichnete Basen von Milnorgittern einfacher Singularitäten, Bonner Math. Schriften 160 (1985). Singularities in algebraic geometry, Local complex singularities, Cycles and subschemes Ausgezeichnete Basen von Milnorgittern einfacher Singularitäten | 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 surveys some of the most important results concerning the description of the automorphism groups of classes of algebras. The author's attention is concentrated on the polynomial, free, and, especially, on generic matrix algebras. The principal achievements in the study of the automorphisms of these algebras are discussed; in addition to that they are supplemented by rather interesting heuristic comments. The bibliography is quite complete.
The paper undoubtedly is very interesting for the specialists in the area. It is of great interest, too, for researchers in PI theory as well as in Algebraic combinatorics. automorphism groups of polynomial algebras; free Lie algebras; algebras with polynomial identities; generic matrix algebras; free associative algebras V. Drensky, Automorphisms of polynomial, free and generic matrix algebras, inTrends in Ring Theory, Proc. Conf. Miskolc, 1996 (V. Dlab and L. Márki, eds.), CMS Conference Proceedings22, AM. Math. Soc. Providence, 1998, pp. 13--26. Automorphisms and endomorphisms, Trace rings and invariant theory (associative rings and algebras), Polynomials over commutative rings, Automorphisms of curves, Identities, free Lie (super)algebras, Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Automorphisms, derivations, other operators for Lie algebras and super algebras Automorphisms of polynomial, free and generic matrix algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A=k\langle x_1,\dots,x_m\rangle\) be the free associative \(k\)-algebra, \(k\) algebraically closed. A representation of \(A\) on a finite dimensional vector space \(W\) of dimension \(d\) consists of a tuple \(\varphi_\ast\in\text{End}_k(W)^m.\) The orbits of \(\text{GL}(W)\) in \(\text{End}(W)^m\) correspond bijectively to the isomorphism classes of \(d\)-dimensional representations of \(A\).
Fix another \(k\)-vector space \(V\) of dimension \(n\), together with a basis \(v_1,\dots,v_n.\) Then \(X=\text{Hom}(V,W)\oplus\text{End}(W)^m\) parametrizes \(d\)-dimensional representations of \(A\), together with a fixed linear map from \(V\) to \(W\), and again the group \(\text{GL}(W)\) acts on \(X\) via base change in \(W\).
Define a tuple \((f,\varphi_\ast)\) to be stable if the image of \(f\) generates \(W\) as a representation of \(A\), and let \(X^s\) denote the subset of \(X\) consisting of stable tuples. Then \(H_{d,n}^{(m)}=X^s/\text{GL}(W)\) is the quotient variety of \(X^s\) by \(\text{GL}(W).\) It is then known that \(H_{d,n}^{(m)}\) is smooth and irreducible, of dimension \(N=nd+(m-1)d^2,\) and that the set \(X^s\) of stable tuples is a principal \(\text{GL}(W)\) bundle over \(H_{d,n}^{(m)}.\)
The variety \(H_{d,n}^{(m)}\) has several interpretations, that is, the \(k\)-points parameterize each of the following sets:
(1) Equivalence classes of \(d\)-dimensional representations \(W\) of \(A\) together with an \(n\)-tuple of vectors generating \(W\) as a representation of \(A.\)
(2) Equivalence classes of \(d\)-dimensional representations \(W\) of \(A,\) together with a surjective \(A\)-homomorphism from the free representation \(A^n\) to \(W\).
(3) \(A\)-subrepresentations of codimension \(d\) of the free representation \(A^n.\)
(4) Isomorphism classes of stable representations of the quiver \(Q_n^{(m)}.\)
In particular, the variety \(H_{d,1}^{m}\) parametrizes left ideals of codimension \(d\) in \(A\). Thus it can be viewed as a noncommutative Hilbert scheme for the free algebra in \(m\) generators, in the same way as the Hilbert scheme \(\text{Hilb}^d(\mathbb{A}^m)\) parametrizes ideals of codimension \(d\) in the polynomial ring \(k[x_1,\dots,x_m].\) Denote the quotient variety \(\text{End}(W)^m//\text{GL}(W)\) by \(V_d^{(m)}.\) Then \(V_d^{(m)}=\text{Spec}(k[\text{End}(W)^m]^{\text{GL}(W)}),\) such that its \(k\)-points are in bijection with the semisimple representations of \(A\).
By a result of C. Procesi, the ring \(R=k[\text{End}(W)^m]^{\text{GL}(W)}\) is generated by the functions \((\varphi_1,\dots,\varphi_m)\mapsto\text{tr}(\varphi_{i_1},\dots,\varphi_{i_s})\) for sequences \((i_1,\dots,i_s)\in \{1,\dots,m\}.\) In fact, \(R\) is already generated by such functions for \(s\leq d^2+1.\)
One of the problems in dealing with the varieties \(V_d^{(m)}\) is that they are highly singular, except in the cases \(m=1,\) and \(d=2=m\). Although no explicit desingularizations of the varieties \(V_d^{(m)}\) are known (except in case \(d=2\)), the noncommutative Hilbert schemes \(H_{d,n}^{(m)}\) form a class of closely related smooth varieties: The obvious map \(X^s\rightarrow\text{End}(W)^m,\) is \(\text{GL}(W)\)-equivariant. Thus it descends to a projective morphism \(p:H_{d,n}^{(m)}\rightarrow V_d^{(m)}\) on the level of quotients by \(\text{GL}(W).\) The fibres of \(p\) are difficult to determine in general, but they are at least tractable using the Luna stratification of \(V_d^{(m)}\) and the theory of nullcones of quiver representations. The morphism \(p\) extends the canonical map from the Hilbert scheme \(\text{Hilb}^d(\mathbb{A}^m)\) to the \(d\)th symmetric power \((\mathbb{A}^m)^d/S_d.\)
The main aim of this paper is to prove the following results on the geometry of the varieties \(H_{d,n}^{(m)}\)
The variety \(H_{d,n}^{(m)}\) has a cell decomposition, whose cells are parametrized by \(m\)-ary forests with \(n\) roots and \(d\) nodes.
As a consequence, the Betti numbers in the cohomology of \(H_{d,n}^{(m)}(\mathbb{C})\) can be described in a compact form by assembling the Poincaré polynomials into a generating function. This leads to the result that the cohomological Euler characteristic of \(H_{d,n}^{(m)}\) is given by
\[
\chi(H_{d,n}^{(m)})=\frac{n}{(m-1)d+n}\left(\begin{matrix} md+n-1\\d\end{matrix}\right).
\]
The last main result of the article considers the asymptotic behavior of both the Euler characteristic and the Poincaré polynomials:
After a suitable normalization, the distribution of the Betti numbers of \(H_{d,1}^{(m)}\) for large \(d\) has the Airy distribution as a limit law.
The author thoroughly defines words and forests and uses this to construct a cell decomposition. This is very well written, and the applications that follows are then easy to understand. That is, the normal forms for representations and submodules, and in some sense the intersection theory of \(H_{d,n}^{(m)}\). Also, the generating functions are well treated. The article ends with the final result about the asymptotics of the Euler characteristic of \(H_{d,n}^{(m)}\). representations of free algebras; moduli of representations; Hilbert schemes; Betti numbers Reineke, M.: Cohomology of non-commutative Hilbert schemes. Algebras Represent. Theory \textbf{8}, 541-561 (2005) Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets, Noncommutative algebraic geometry Cohomology of noncommutative Hilbert 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 A boundary singularity is a singularity of a function on a manifold with boundary. The relation between the characteristic polynomial of the mono\-dromy and the Poincaré series of the ambient hypersurface singularity for such a singularity is investigated. The simple boundary singularities arise from simple hypersuface singularities, and there is a generalization of the McKay correspondence for these cases. For seven of the 12 exceptional uni\-nodal boundary singularities a direct relation between the Poincaré series of the ambient singularity and the characteristic polynomial of the monodromy is given. boundary singularities; unimodal singularities; simple singularities; monodromy; McKay correspondence W. Ebeling and S. M. Gusein-Zade, On indices of 1-forms on determinantal singularities, Proceedings of the Steklov Institute of Mathematics 267 (2009), 113--124. Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Local complex singularities Poincaré series and monodromy of the simple and unimodal boundary 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 This is a generalization of the classic work of Beilinson, Lusztig and MacPherson [\textit{A. A. Beilinson} et al., Duke Math. J. 61, No. 2, 655--677 (1990; Zbl 0713.17012)]. In this paper (and an Appendix) we show that the quantum algebras obtained via a BLM-type stabilization procedure in the setting of partial flag varieties of type \texttt{B}/\texttt{C} are two (modified) coideal subalgebras of the quantum general linear Lie algebra, \( \dot{\mathbf{U}}^j \) and \( \dot{\mathbf{U}}^i \). We provide a geometric realization of the Schur-type duality of Bao-Wang [\textit{H. Bao} and \textit{W. Wang}, A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs. Astérisque 402. Paris: Société Mathématique de France (SMF) (2018; Zbl 1411.17001)] between such a coideal algebra and Iwahori-Hecke algebra of type \texttt{B}. The monomial bases and canonical bases of the Schur algebras and the modified coideal algebra \( \dot{\mathbf{U}}^j \) are constructed.
In an Appendix by three authors (Bao, Li and Wang), a more subtle 2-step stabilization procedure leading to \( \dot{\mathbf{U}}^i \) is developed, and then monomial and canonical bases of \( \dot{\mathbf{U}}^i \) are constructed. It is shown that \( \dot{\mathbf{U}}^i \) is a subquotient of \( \dot{\mathbf{U}}^j \) with compatible canonical bases. Moreover, a compatibility between canonical bases for modified coideal algebras and Schur algebras is established.
For Part II, see \textit{Z. Fang} and \textit{Y. Li}, Trans. Am. Math. Soc., Ser. B 2, 51--92 (2015; Zbl 1339.17012). quantum algebras; coideal subalgebras; quantum general linear Lie algebra; geometric realization of the Schur-type duality; Iwahori-Hecke algebra of type B Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Geometric Schur duality of classical 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 Algebraic stacks are objects which generalize schemes from the viewpoint of fibered categories. As it has turned out, over the past decades, stacks are especially well adapted to the study of classification problems in algebraic geometry via geometric invariant theory.
In the present survey article, the author summarizes some general structure results about the particular class of Deligne-Mumford (DM) stacks of finite type over a field of characteristic 0, together with their applications to certain moduli spaces. After a brief introduction to stacks in general, a concrete description of smooth Deligne-Mumford stacks in dimension 1 is given, with special emphasis placed on orbifold curves over \(\mathbb{C}\) and their role in the classical moduli theory of vector bundles on compact Riemann surfaces. This is followed by a more detailed discussion of smooth Deligne-Mumford stacks, orbifolds, and quotient stacks of arbitrary dimension over a base field \(k\). Then the study turns to possibly singular Deligne-Mumford stacks, culminating in a new characterization of those stacks that are isomorphic to the stack quotient of a quasi-projective scheme by a reductive algebraic group acting linearly. This leads to the suggestion that a Deligne-Mumford stack over a field of characteristic 0 should be called ``(quasi-)projective'' if it admits a (locally) closed embedding into a smooth proper Deligne-Mumford stack with projective coarse moduli space. In this context, the class of (quasi-)projective Deligne-Mumford stacks is completely characterized in different ways. More precisely, it is outlined that a separated Deligne-Mumford stack with quasi-projective coarse moduli space over a field of characteristic 0 is quasi-projective (in the above sense) if and only if it enjoys several equivalent, well-studied properties, namely: being a quotient stack, satisfying the so-called resolution hypothesis, admitting a finite flat covering by a scheme, or possessing a generating sheaf. At the end of the paper, these general structure results are illustrated by various examples of concrete moduli stacks known from the recent literature.
Throughout the entire article, a basic reference is the fundamental work ``Brauer Groups and Quotient Stacks'' by \textit{D. Edidin, B. Hassett, A. Kresch} and \textit{A. Vistoli} [Am. J. Math. 123, No. 4, 761--777 (2001; Zbl 1036.14001)], together with numerous other, mostly very recent original research papers by various authors. algebraic stacks; Deligne-Mumford stacks; quotient stacks; orbifolds; moduli spaces; Brauer groups of schemes; geometric invariant theory Kresch, A., \textit{on the geometry of Deligne-Mumford stacks}, Algebraic geometry (Seattle 2005), part 1, 259-271, (2009), American Mathematical Society, Providence, RI Generalizations (algebraic spaces, stacks), Algebraic moduli problems, moduli of vector bundles, Brauer groups of schemes, Geometric invariant theory, Vector bundles on curves and their moduli, Homotopy theory and fundamental groups in algebraic geometry On the geometry of Deligne-Mumford stacks | 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 authors study Lie algebras of formal groups. In the special case considered by Ritt, the Lie algebras correspond biunivoquely to formal groups. The basic object is the algebra K[P] of differential operators where K is a field and P is a Lie ring acting as derivations of K. It turns out that K[P] has a twisted Hopf algebra structure - it is a Hopf K/k-algebra, where k is a subfield of K on which P acts trivially. The authors introduce first in general the Hopf K/k-algebra B, B-algebra, B-coalgebra and B-bialgebra. Thus one can define a formal Ritt B-group to be a complete topological B-bialgebra A which both has a unique maximal ideal and is finitely generated as a topological B-algebra. Under some additional finiteness conditions on B and A, the Lie algebra of the formal group is linearly compact. This allows the application of the methods and results of \textit{V. Guillemin} [J. Differ. Geom. 2, 313-345 (1968; Zbl 0183.261)] and \textit{R. J. Blattner} [Trans. Am. Math. Soc. 144(1969), 457-474 (1970; Zbl 0295.17002)]. The authors use these techniques to classify these K-algebras which admit a simple linearly compact K[P]-algebra structure. Then, they study the set of non- isomorphic K[P]-structures on each such K-algebra as follows. Fix one easily constructed structure as a reference point and each additional structure is described by a kind of 1-cocycle - a flat connection with values in the Lie algebra. Finally, the authors look at the question: Can formal Ritt groups be considered to be formalizations (completions of the local ring at identity) of some algebraic structures? The answer is ''yes'' if their Lie algebras are finite dimensional and ''no'' if their Lie algebras are simple of Cartan type. Ritt formal group; Hopf algebra; formalization of algebraic structure; algebra of differential operators; Lie algebras of formal groups W. Nichols and B. Weisfeiler, ''Differential formal groups of J. F. Ritt,''Am. J. Math.,104, 943--1005 (1982). Formal groups, \(p\)-divisible groups, Modules of differentials, Lie algebras of linear algebraic groups, Linear algebraic groups over arbitrary fields Differential formal groups of J. F. Ritt | 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 preliminary version in Rep., Akad. Wiss. DDR, Inst. Math. R-MATH- 04/83 (1983) has been reviewed in Zbl 0513.10029. Eisenstein series; spectral resolution; Selberg trace formula; index of signature; Dolbeault operator; Hilbert modular varieties; signature defects of cusp singularities; values of L-series; Hirzebruch conjecture; dimension formula Müller, W, Signature defects of cusps of Hilbert modular varieties and values of \(L\)-series at \(s=1\), J. Differ. Geom., 20, 55-119, (1984) 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, Index theory and related fixed-point theorems on manifolds, Representation-theoretic methods; automorphic representations over local and global fields Signature defects of cusps of Hilbert modular varieties and values of L- series at \(s=1\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, the classification of real simple singularities of surfaces is investigated with respect to the actions of complex conjugations on exceptional loci. It is given a procedure for determining the action of complex conjugation on a non-singular fiber of a semi- universal deformation of a real simple singularity of surfaces. Dynkin diagram; complex conjugations on exceptional loci; semi-universal deformation of a real simple singularity of surfaces Singularities of surfaces or higher-dimensional varieties, Deformations of singularities, Real algebraic and real-analytic geometry, Real-analytic manifolds, real-analytic spaces, Deformations of complex singularities; vanishing cycles, Singularities in algebraic geometry, Complex singularities Real simple singularities of 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 Let \(p\) be a prime and \(D\) a nonabelian finite simple group such that \(p\) does not divide \(| \text{Aut}(D)|\). The main theorem of this paper says that the wreath product \(D \wr \mathbb{Z}_ p\) can be realized as the Galois group of a finite unramified covering of the affine line of characteristic \(p \neq 0\). To prove it, the concept of group enlargements is introduced. Namely, a group \(G\) together with an exact sequence \(1 \to H \to G \to J \to 1\) is called an enlargement of a group \(\Theta\) by \(J\) if there exists \(\Delta \triangleleft H\) with \(\bigcap_{g \in G}g \Delta g^{-1} = \{1\}\) and \(H/\Delta \cong \Theta\). Then it is shown that, under the same assumptions on \(p\) and \(D\) as above, the only enlargements of \(D\) by \(Z_ p\) are \(D \times Z_ p\) and \(D \wr \mathbb{Z}_ p\). Grothendieck's description of tame fundamental groups of curves combined with techniques from a series of papers by the author conclude the proof of the above result. Galois theory; algebraic fundamental groups; finite simple group; wreath product; Galois group; unramified covering of the affine line; group enlargements; enlargements; tame fundamental groups of curves Abhyankar, S. S.: Group enlargements. CR acad. Sci. Paris 312, 763-768 (1991) Extensions, wreath products, and other compositions of groups, Plane and space curves, Separable extensions, Galois theory Group enlargements | 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 \(\mathbb{S}:=\text{Res}_{\mathbb{C}/ \mathbb{R}}\mathbb{G}_{m \mathbb{C}}\) be the algebraic group over \(\mathbb{R}\) obtained by restriction of scalars from \(\mathbb{C}\) to \(\mathbb{R}\) of the multiplicative group. A Shimura datum is a pair \((G,X)\), with \(G\) a connected reductive affine algebraic group over \(\mathbb{Q} \), and \(X\) a \(G(\mathbb{R})\)-conjugacy class in the set of morphisms of algebraic groups \(\text{Hom} (\mathbb{S},G_\mathbb{R})\), satisfying three special conditions [see \textit{B. Moonen} in: Galois representations in arithmetic algebraic geometry, Proc. Symp. Durham 1996, Lond. Math. Soc. Lect. Note Ser. 254, 267-350 (1998; Zbl 0962.14017); definition 1.4].
Definition 1.1. Let \((G,X)\) be a Shimura datum, \(K\) an open compact subgroup of \(G(\mathbb{A}_f)\), and \(Z\) an irreducible, closed subvariety of \(\text{Sh}_K (G,X)_\mathbb{C}\). Then \(Z\) is a subvariety of Hodge type if there is a Shimura datum \((G',X')\), a morphism of Shimura data \(f:(G',X')\to(G,X)\), and an element \(g\) of \(G(\mathbb{A}_f)\) such that \(Z\) is an irreducible component of the image of the map:
\[
\text{Sh} (G', X')_\mathbb{C} @>\text{Sh}(f)>> \text{Sh}(G,X)_\mathbb{C} @>\cdot g>> \text{Sh} (G, X)_\mathbb{C} \to\text{Sh}_K(G,X)_\mathbb{C}.
\]
Definition 1.2. Let \((G,X)\) be a Shimura datum. For \(h\) in \(X\) we let \(\text{MT}(h)\) be the Mumford-Tate group of \(h\), i.e., the smallest algebraic subgroup \(H\) of \(G\) such that \(h\) factors through \(H_\mathbb{R}\). A point \(h\) in \(X\) is called special if \(\text{MT}(h)\) is commutative (in which case it is a torus). For \(K\) a compact open subgroup of \(G(\mathbb{A}_f)\), a point in \(\text{Sh}_K(G,X)_\mathbb{C}\) is special if its preimages in \(\text{Sh} (G,X)_\mathbb{C}\) are of the form \((h,g)\) with \(h\) in \(X\) special. Equivalently, the special points in \(\text{Sh}_K (G,X)_\mathbb{C}\) are the zero dimensional subvarieties of Hodge type.
Conjecture 1.3 (André-Oort). Let \((G,X)\) be a Shimura datum. Let \(K\) be a compact open subgroup of \(G(\mathbb{A}_f)\) and let \(S\) be a set of special points in \(\text{Sh}_K(G,X) (\mathbb{C})\). Then every irreducible component of the Zariski closure of \(S\) in \(\text{Sh}_K (G,X)_\mathbb{C}\) is a subvariety of Hodge type.
Some remarks are in order at this point. André stated this conjecture as a problem for curves containing infinitely many special points in general Shimura varieties [\textit{Y. André}, ``\(G\)-functions and geometry'' (1989; Zbl 0688.10032); X.4]. Independently, \textit{F. Oort} [in: Arithmetic geometry, Proc. Symp., Cortona 1994, Symp. Math. 37, 228-234 (1997; Zbl 0911.14018)] raised the question for general subvarieties of the moduli spaces of principally polarized abelian varieties.
\textit{B. Edixhoven} [Compos. Math. 114, 315-328 (1998; Zbl 0928.14019)] proved the conjecture for the moduli space of pairs of elliptic curves, assuming the generalized Riemann hypothesis (GRH) for imaginary quadratic fields.
In this article, we prove conjecture 1.3, assuming GRH, for Hilbert modular surfaces. The method of proof is basically the same as the author's earlier one (loc. cit.), but now we do use more advanced techniques. The two main results of the article are described in section 2:
Theorem 2.1. Assume GRH. Let \(C\subset S_\mathbb{C}\) be an irreducible closed curve containing infinitely many CM points. Then \(C\) is of Hodge type.
Theorem 2.2. Let \(C\subset S_\mathbb{C}\) be an irreducible closed curve containing infinitely many CM points corresponding to abelian varieties that lie in one isogeny class (the isogenies are not required to be compatible with the multiplications by \(O_K)\). Then \(C\) is of Hodge type.
The reason for which we state and prove theorem 2.2 is that it has an interesting application to transcendence of special values of certain hypergeometric functions via work of Wolfart, Cohen and Wüstholz, without having to assume GRH.
In section 4 we recall an important result of André, relating the generic Mumford-Tate group of a variation of Hodge structure to its algebraic monodromy group (i.e., the Zariski closure of the image of monodromy). We use it to prove that for a curve in a Hilbert modular surface that is not of Hodge type and that does contain a special point, the connected algebraic monodromy group is maximal, i.e., \(\text{SL}_{2,K}\).
Section 5 introduces the Hecke correspondence \(T_p\) associated to a prime number \(p\). We use a very powerful result of Nori in order to prove that for \(C\) a curve with maximal algebraic monodromy group, \(T_pC\) is irreducible if \(C\) is large enough.
The main result of section 6 says that the size of the Galois orbit of a special point \(x\) grows at least as a positive power of the discriminant \(\text{discr} (R_x)\) of the ring of endomorphisms (commuting with the real multiplications) of the corresponding abelian variety.
Section 7 gives an upper bound for the number of points in intersections of the form \(Z_1 \cap T_gZ_2\), with \(Z_1\) and \(Z_2\) fixed subvarieties of a general Shimura variety, and with \(T_g\) a varying Hecke correspondence. Finally, section 8 combines all these preliminary results. Hilbert modular surface; Shimura datum; generic Mumford-Tate group; variation of Hodge structure; Hecke correspondence B. Edixhoven, On the André-Oort conjecture for Hilbert modular surfaces, Moduli of abelian varieties (Texel Island 1999), Birkhäuser, Basel (2001), 133-155. Modular and Shimura varieties, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic aspects of modular and Shimura varieties On the André-Oort conjecture for Hilbert modular 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 Let \(T\) be a ring, \(P\) a finitely generated \(T\)-module, and \(I\) the trace ideal of \(P\) in \(T\). The main general result of this interesting article asserts that \(K_0(\text{End }P_T)\) is isomorphic to a subgroup of \(K_0(T,I)\). This fact is then applied to the situation where a finite group \(G\) acts on a ring \(S\), with \(|G|^{-1}\in S\). Namely, letting \(T=S*G\) denote the skew group ring associated with this action and taking \(P=S_T\), one has \(\text{End }P_T\cong S^G\), the ring of \(G\)-invariants in \(S\). Thus, as a corollary, the author is able to prove triviality of \(K_0(S^G)\), that is, \(K_0(S^G)=\langle[S^G]\rangle\), for various rings of invariants \(S^G\), commutative and noncommutative.
Most notably, this is established for \(G\)-actions on the symmetric algebra \(S=S(V)\) of a finite-dimensional \(G\)-representation \(V\) over an algebraically closed field of characteristic 0 under the assumption that \(V\) decomposes into fixed point free and trivial constituents only. This provides a positive answer to a special case of an open question, posed by \textit{H. Kraft} [in: CMS Conf. Proc. 10, 111-123 (1989; Zbl 0703.14009)] as to whether \(K_0(S(V)^G)\) is always trivial, for any rational representation \(V\) of a reductive algebraic group \(G\). finitely generated modules; trace ideals; finite groups; skew group rings; actions; rings of invariants; symmetric algebras; rational representations; reductive algebraic groups M. P. Holland, \(K\)-theory of endomorphism rings and of rings of invariants , J. Algebra 191 (1997), no. 2, 668-685. Grothendieck groups, \(K\)-theory, etc., Automorphisms and endomorphisms, Endomorphism rings; matrix rings, \(K_0\) of other rings, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups \(K\)-theory of endomorphism rings and of rings of 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 In this survey the author presents some basic notions and facts on intersection cohomology and perverse sheaves. The intersection cohomology was originally introduced by Goresky and MacPherson from the PL point of view. The sheaf-theoretic aspect of intersection cohomology appeared later on. The author presents the historical development of this theory, emphasizing the second point of view.
He exhibits three axiomatic characterizations of intersection homology. This leads to a proof of the topology invariance of intersection cohomology. He treats Verdier's duality, ending the survey by studying the important notion of perverse sheaf; and in particular on algebraic varieties (decomposition theorem for morphisms, Riemann-Hilbert correspondence, \(\ldots\)). In particular, the representation theory of reductive algebraic groups is stressed. intersection cohomology; perverse sheaves; Verdier's duality; algebraic varieties; Riemann-Hilbert correspondence; representation theory of reductive algebraic groups Borel, A.: Introduction to middle intersection cohomology and perverse sheaves, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991), Proc. Sympos. Pure Math., vol. 56. Am. Math. Soc., Providence, RI, 1994, pp. 25--52. MR MR1278699 (95h:55006) Intersection homology and cohomology in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Representation theory for linear algebraic groups Introduction to middle intersection cohomology and perverse sheaves | 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 show that all singularities occurring in minimal degenerations of matrix pencils are Cohen-Macaulay and regular in codimension 1. matrix pencils; orbit closures; singularities; Kronecker algebras; path algebras of quivers; module varieties; degenerations Bender, J., Bongartz, K.: Minimal singularities in orbit closures of matrix pencils. Linear Algebra Appl. 365, 13--24 (2003) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Minimal singularities in orbit closures of matrix pencils. | 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 There are many well-known results on the correspondence between an affine algebraic group and its Lie algebra. This paper manages to extend several of these results to the case where the algebraic group is pro-affine, i.e. the inverse limit of affine algebraic groups.
The extended results are as follows. Let \(K\) be an algebraically closed field. Let \(f:G\rightarrow H\) be a surjective morphism of connected pro-affine algebraic groups over \(K\). Let \(f^{o}:\mathcal{L}\left( G\right) \rightarrow \mathcal{L}\left( H\right) \) be the differential of \(f\). Then \(f\) is separable (i.e. \(K[G]\) is separable over \(K[H],\) where \(K[G]\) and \(K[H]\) are the representing algebras of \(G\) and \(H\) respectively) if and only if \( f^{o}\) is surjective. Furthermore, if \(f\) is separable, then \(\ker f^{o}= \mathcal{L}\left( \ker f\right) \) and \(f\) induces an isomorphism \(G/\ker f\rightarrow H\) of algebraic groups.
More results are obtained in the special case where char \(K=0.\) If \(K\) is of characteristic zero then for any two connected algebraic subgroups \(A\) and \(B \) of \(G\) we have that \(A\subset B\) if and only if \(\mathcal{L}\left( A\right) \subset \mathcal{L}\left( B\right) .\) Furthermore, \(A\) is normal in \(G\) if and only if \(\mathcal{L}\left( A\right) \) is an ideal of \(\mathcal{L} \left( G\right) .\) An example is given to show that these results do not hold when \(K\) is of positive characteristic. Lie algebras of pro-affine groups; separable morphisms Lie algebras of linear algebraic groups, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Group schemes, Linear algebraic groups over arbitrary fields Lie algebras and separable morphisms in pro-affine 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 \(\Gamma\) be one of the congruence subgroups \(\Gamma_ 0(N)\), \(\Gamma_ 1(N)\), or \(\Gamma\) (N), and let J be the Jacobian of the modular curve X associated to \(\Gamma\). The author studies the structures of the subgroup \(C_{\Gamma}\) of J generated by divisor classes supported only at the cusps of X. The subgroups \(C_{\Gamma}\) were first shown to be finite by Yu. Manin and \textit{V. G. Drinfel'd} [Funct. Anal. Appl. 7, 155-156 (1973; Zbl 0285.14006)]. The structure of \(C_{\Gamma}\) was, for \(\Gamma =\Gamma_ 0(N)\) (and for certain N, for \(\Gamma =\Gamma_ 1(N))\), studied by \textit{A. P. Ogg} [Proc. Symp. Pure Math. 24, 221-231 (1973; Zbl 0273.14008)]. Using a different approach, \textit{D. S. Kubert} and \textit{S. Lang} [Modular units (1981; Zbl 0492.12002)] determined the cuspidal groups attached to \(\Gamma_ 1(N)\) and \(\Gamma\) (N).
In this paper the author provides a new method for computing the groups \(C_{\Gamma}\) by relating the structure of the cuspidal groups to the arithmetic of special values of L-functions attached to weight two Eisenstein series. More precisely, the cuspidal group \(C_{\Gamma}\) is generated by subgroups \(C_{\Gamma}(E)\) that the author associates to weight two Eisenstein series E for \(\Gamma\). It is the subgroups \(C_{\Gamma}(E)\) that are computed by relating their structure to the arithmetic of the L-function of E.
It is not at all surprising that the structure of \(C_{\Gamma}(E)\) is related to the arithmetic of the L-function of E: The congruence formulae (due to Mazur for \(\Gamma_ 0(p)\), to the author and the reviewer for \(\Gamma_ 1(p)\), and to the author in the general case) for the special values of the L-functions of X are expressed in terms of the L-function of E, and the compatibility of these formulas with the Birch and Swinnerton-Dyer conjecture is shown by doing a descent on J using the ideal (of the Hecke algebra) annihilating \(C_{\Gamma}(E).\)
The method described by the author for computing \(C_{\Gamma}(E)\) is quite valuable in that it may extend to many other settings where Eisenstein series occur. cuspidal divisor class group; group of modular units; modular; curves; congruence subgroups; Jacobian; cuspidal groups; arithmetic of special values of L-functions; weight two Eisenstein series; congruence formulae Glenn Stevens, The cuspidal group and special values of \(L\)-functions, Trans. Am. Math. Soc.291 (1985), p. 519-550 Holomorphic modular forms of integral weight, Congruences for modular and \(p\)-adic modular forms, Jacobians, Prym varieties The cuspidal group and special 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 For a finite subgroup G of SL(2,k) and char k\(=0\), the McKay correspondence is a bijection between the vertices of the Dynkin diagram and the non trivial representations of G. The multiplication table by the natural representation \(G\to SL(2,k)\) is ''given by'' the Dynkin diagram. All this may be seen geometrically. The first Chern class of a non trivial indecomposable reflexive module on \(S=k^ 2/G\) (k in any characteristic) is dual to some specific exceptional curve whereas the multiplication table by \(\Omega^ 1_ S=Ext^ 1_{{\mathcal O}_ S}({\mathfrak m},{\mathcal O}_ S)\) (\({\mathfrak m}\) is the maximal ideal) is ''given by'' the Dynkin diagram. In this announcement, the authors study in characteristic p\(>0\) the multiplication by \(Ext^ 1_{{\mathcal O}_ S}({\mathfrak m},{\mathcal O}_ S)\). The Dynkin diagram still describes the multiplication in good characteristic (i.e. rank of the module is prime to the p). In the remaining cases, the authors give explicitly the correction term. rational double points; McKay correspondence; Dynkin diagram; reflexive module; characteristic p Singularities in algebraic geometry, Finite ground fields in algebraic geometry Sur la règle de McKay en caractéristique positive. (On McKay's rule 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 Let \(k\) be a base field, and let \(G\) be an algebraic group over \(k\). Let \(T\) be a \(G\)-torsor defined over \(K/k\). The `essential dimension' \(\text{ed}(T)\) is defined as the minimum of the transcendence degrees of \(K_0\) such that \(T\) descends to \(K_0\). The essential dimension of \(G\) is defined to be the supremum of all \(\text{ed}(T)\) where \(T\) runs over all \(G\)-torsors over all fields \(K/k\). For a prime \(p\), one can define the essential dimension at \(p\) of a \(G\)-torsor \(T\) as the minimum of \(\text{ed}(T_L)\) for \(L/K\) a prime-to-\(p\) extension. It is denoted \(\text{ed}(T;p)\). The essential dimension at \(p\) of \(G\) is defined to be the supremum of all \(\text{ed}(T;p)\) where \(T\) runs over all \(G\)-torsors over all fields \(K/k\).
The calculation of \(\text{ed}(\text{PGL}_n)\) is an open problem. A related problem is the calculation of \(\text{ed}(\text{PGL}_n;p)\). If \(p^s\) is the largest power of \(p\) dividing \(n\), it can be shown that \(\text{ed}(\text{PGL}_n;p)=\text{ed}(\text{PGL}_{p^s};p)\). Hence it is sufficient to compute the latter. In an earlier paper [Algebra Number Theory 3, No. 4, 467-487 (2009; Zbl 1222.11056)], the authors had shown that \(\text{ed}(\text{PGL}_{p^s};p)\leq p^{2s-1}-p^s+1\). In the present paper, using methods of lattices with finite group actions, they prove a stronger upper bound; namely that \(\text{ed}(\text{PGL}_{p^s};p)\leq 2p^{2s-2}-p^s+1\). They do this by reducing the statement to an upper bound on the essential dimension of a central simple algebra \(A\) over a field \(K\) containing a Galois extension \(F/K\). It must be noted that \textit{A. S. Merkurjev} proved [in J. Am. Math. Soc. 23, No. 3, 693-712 (2010; Zbl 1231.20048)] that this is an equality if \(s=2\). essential dimension; central simple algebras; projective linear groups; lattices; essential \(p\)-dimension; Brauer groups; Severi-Brauer varieties; \(R\)-equivalence; Chow groups; character groups of algebraic tori A. Meyer, Z, Reichstein, An upper bound on the essential dimension of a central simple algebra, J. Algebra 329 (2011), 213--221. Finite-dimensional division rings, Linear algebraic groups over finite fields, Brauer groups (algebraic aspects), Group actions on varieties or schemes (quotients), Galois cohomology of linear algebraic groups An upper bound on the essential dimension of a central simple algebra. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field. For a quiver \(Q=(Q_0 ,Q_1, s, e)\) and a dimension vector \(d\in\mathbb{Z}^{Q_0}\) we let rep\(_Q(d)\) be the vector space of representations corresponding to \(d\). Now \(\text{GL}(d)\) acts on rep\(_Q(d)\) in a natural way, hence given a representation \(V\) or \(Q\) we can consider the \(\text{GL}(d)\)-orbit in rep\(_Q(d),\) which consists of the representations isomorphic to \(V\) -- we shall write this orbit as \(\mathcal{O}_V.\) Let \(M\) and \(N\) be representations in rep\(_Q(d)\) such that \(\mathcal{O}_N\subset\mathcal{\bar{O}}_M,\) where \(\mathcal{\bar{O}}_M\) is the Zariski closure of \(\mathcal{O}_M\) in rep\(_Q(d).\) Write Sing\((M,N)\) for the set of smoothly equivalent classes of pointed varieties. The regular points form a type of singularity which we shall denote Reg. In the case where \(\mathcal{O}_N\) has codimension one in \(\mathcal{\bar{O}}_M\) it is known that Sing\((M,N) =\) Reg; furthermore of \(\mathcal{O}_N\) has codimension two and \(Q\) is a Dynkin quiver then again Sing\((M,N) =\) Reg.
Here the author investigates the codimension two case when \(Q\) is an extended Dynkin quiver. In the case where \(Q\) is the Kronecker quiver one has two types of singularities, namely \(A_r=\) Sing\((\mathcal{A}_{r+1},0)\) and \(C_r=\) Sing\((\mathcal{C}_r,0)\), where \(\mathcal{A}_r=\{ ( uv, u^r, v^r) \in k^3\mid u,v\in k\} \) and \(\mathcal{C}_r=\{(u^r, u^{r-1}v,\dots,v^r) \in k^{r+1}\mid u,v\in k\} .\) Note that \(C_1=\) Reg, \(C_2=A_1\) and the remaining types are all distinct. The main result is that for \(Q\) an extended Dynkin quiver in the codimension two case we have Sing\((M,N) \) is one of the \(A_r\) or \(C_r\)'s. If \(Q\) is a cyclic quiver then Sing\((M,N) \neq C_r\) for \(r\geq3,\) i.e. Sing\((M,N) =A_r\) or Reg. Dynkin quivers; representations of quivers; singularities of representations of quivers Zwara, G.: Codimension two singularities for representations of extended Dynkin quivers. Manuscr. Math. 123(3), 237--249 (2007) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Representations of quivers and partially ordered sets Codimension two singularities for representations of extended Dynkin quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this thesis the relations between \(k\)-algebras \(A\) and their Lie algebras of all \(k\)-derivations \(\text{Der}_ k(A)\) are studied for fields of characteristic 0. The work bases on theories introduced by J. Grabowski, H. Omori, H. Hauser and G. Müller considering equivalent problems for the global and local analytic case. A general algebraic approach to problems like that is developed, which uses the new term of a derivation module -- a pair \((A,{\mathcal L})\) of an \(k\)-algebra \(A\) and a submodule \({\mathcal L}\) of \(\text{Der}_ k(A)\) which is also a Lie subalgebra. In course of that there are given methods for constructing topological spaces as well as \(k\)-algebras from arbitrary Lie algebras, which can be interpreted here in terms of the original algebra \(A\). The main results, related to finitely generated \(k\)-algebras, are as follows:
(1) Two finitely generated, normal \(k\)-algebras are isomorphic if and only if their Lie algebras of all \(k\)-derivations are.
(2) An affine algebraic variety is smooth if and only if the Lie algebra of all \(k\)-derivations of the affine coordinate ring is simple.
(3) As application to finitely generated, local, analytic algebras: Let \((X,0)\) and \((X',0)\) be analytic germs, quasihomogeneous, isolated complete intersection singularities, which are not intersections of hypersurfaces of multiplicity 2. Let \(O_ X\) and \(O_{X'}\) be the corresponding local algebras, then \((X,0)\) and \((X',0)\) are isomorphic if and only if the Lie algebras \(\text{Der}_ \mathbb{C}(O_ X)\) and \(\text{Der}_ \mathbb{C}(O_{X'})\) are.
Further the homology and cohomology of a certain kind of graded (infinite dimensional) Lie algebras is investigated. Conditions for finiteness of the homology and cohomology for these Lie algebras are shown. Lie algebras of derivations; normal affine varieties; infinite dimensional graded Lie algebras; derivation module; local analytic algebras; analytic germs; isolated complete intersection singularities; homology; cohomology Siebert, T.: Lie-Algebren von Derivationen und affine algebraische Geometrie über Körpern der Charakteristik 0. Dissertation Berlin 1992 Lie algebras of vector fields and related (super) algebras, Graded Lie (super)algebras, Homological methods in Lie (super)algebras, Modules of differentials, Analytic algebras and generalizations, preparation theorems, Analytical algebras and rings, Local complex singularities, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Cohomology of Lie (super)algebras Lie algebra of derivations and affine algebraic geometry over fields of characteristic zero | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(k\) be an algebraically closed field of characteristic \(p>2\). Let
\[
\mathcal O_n=k[X_1,\ldots,X_n]/(X^p_1,\ldots,X^p_n),
\]
a local \(k\)-algebra of dimension \(p^n\) over \(k\). If \(p=3\) we assume that \(n>1\). Let \(\mathcal L\) be the Lie algebra of all derivations of \(\mathcal O_n\), a restricted simple Lie algebra of Cartan type \(W_n\), and denote by \(G\) the automorphism group of \(\mathcal L\). The the invariant ring \(k[\mathcal L]^G\) is freely generated by \(n\) homogeneous polynomial functions \(\psi_0,\ldots,\psi_{n-1}\) and aversion of Chevalley's Restriction Theorem holds for \(\mathcal L\). Moreover, the majority of classical results of Kostant on the adjoint action of a complex reductive group on its Lie algebra hold for the action of \(G\) on \(\mathcal L\). In particular, each fiber of the map \(\psi\colon \mathcal L\to\mathbb A^n\) sending any \(x\in\mathcal L\) to \((\psi_0(x),\ldots,\psi_{n-1}(x))\in\mathbb A^n\) is an irreducible complete intersection in \(\mathcal L\) and contains an open \(G\)-orbit. However, it is also known that the zero fiber of \(\psi\) is not a normal variety. In this paper we complete the picture by showing that Kostant's differential criterion for regularity holds in \(\mathcal L\) and we prove that a fiber of \(\psi\) is normal if and only if it consists of regular semisimple elements of \(\mathcal L\). modular Lie algebras; invariant theory Premet, A. (2014). Regular derivations of truncated polynomial rings. arXiv:1405.2426 [math.RA]. Modular Lie (super)algebras, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Coadjoint orbits; nilpotent varieties Regular derivations of truncated polynomial 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 Here we give some cohomological conditions for a morphism p:\(X\to Y\) of algebraic schemes to be of the form g\({\mathbb{O}}f\), where f:\(X\to U\) is proper, g:\(U\to Y\) is affine and \(g| B\) is finite, where \(B=\{x\in U:\dim(f^{-1}(x))>0\}.\) decomposition of morphism of algebraic schemes; finite cohomology groups; proper modification Schemes and morphisms, Birational geometry, Classical real and complex (co)homology in algebraic geometry Algebraic varieties with finite cohomology 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 article investigates lattices (i.e. finitely generated torsionfree modules) over the complete local ring of an algebraic curve singularity over an algebraically closed field. Its main interest is to classify the indecomposable lattices over tame curve singularities (i.e. over those curve singularities whose indecomposable lattices admit a classification by finitely many one-parameter families in each rank), as well as to classify tame curve singularities themselves. A solution to this problem is developed for curve singularities with large conductor (i.e. for those curve singularities whose conductor contains the radical squared of the normalization).
It turns out that, up to analytic isomorphism, there are 10 exceptionals and 1 one-parameter family of tame curve singularities with large conductor. They are characterized by the fact that a natural number which can be attached to any curve singularity with large conductor, has value 4. For each (but one) of them, the indecomposable lattices are classified via reduction to a tame matrix problem, whose solution is known from modern representation theory of finite-dimensional algebras. indecomposable lattices over tame curve singularities; large conductor; representation theory of finite-dimensional algebras Dieterich, E.: Lattices over curve singularities with large conductor. Invent. Math. 114(2), 399--433 (1993), ISSN 0020-9910 Singularities of curves, local rings, Representations of orders, lattices, algebras over commutative rings, Algebras and orders, and their zeta functions, Representation type (finite, tame, wild, etc.) of associative algebras, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) Lattices over curve singularities with large conductor | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The book under review grew out of a one-semester course that the author recently taught, in 2005 and in 2006, at the Australian National University. Geared toward fourth-year undergraduate students, this course was to provide a broad, panoramic glimpse of some of the central topics in modern algebraic geometry, without striving for a thorough introduction to the field as a whole. As the guiding subject matter for such a special, topical and wide-ranging survey course in algebraic geometry, thereby taking into account the presumed basic mathematical background knowledge of undergraduate students in their final year, the author choose the close relations between algebraic geometry and complex-analytic geometry as reflected by \textit{J.-P.Serre's} famous GAGA principle. Serre's pioneering paper ``Géométrie algébrique et géométrie analytique'' from more than fifty years ago [Ann Inst. Fourier 6, 1--42 (1955/56; Zbl 0075.30401)], now usually referred to as ``GAGA'', was not only a decisive milestone in the refoundation'of algebraic geometry by himself and A. Grothendieck, but also established crucial comparison theorems relating algebro-geometric objects (varieties, regular morphisms and algebraic sheaves) with complex-analytic objects (analytic spaces, holomorphic mappings and analytic sheaves) within the modern sheaf-theoretic framework.
In fact, a detailed explanation of Serre's cohomological GAGA theorem for coherent algebraic sheaves on projective spaces constitutes both the punchline and the ultimate goal of the present, fully elaborated and significantly extended course notes. Along the way, the author provides an introduction to those concepts, methods and techniques of modern algebraic geometry and complex analysis that are necessary to reach this goal. This approach to giving a sufficiently substantial, streamlined, accessible and inspiring introduction to some important aspects of modern algebraic geometry for beginners is not only highly original, and even fairly unique in the relevant textbook literature, but also didactically interesting and exemplary, all the more as a thorough discussion of Serre's GAGA principle is still sadly neglected in most primers of algebraic geometry, despite its fundamental importance and wide range of applications.
In view of these particular facts, the current book must be seen as out-standing, utmost valuable enhancement of the modern textbook literature in advanced undergraduate algebraic and analytic geometry, and the author is due to highest appreciation for having provided such a complementary, quite alternative introduction to the subject, very much to the benefit of interested students and instructors likewise.
Starting from scratch and throughout trying to keep the involved abstract machinery of modern algebraic geometry at a level digestible for beginners, the author has organized the material in ten chapters, each of which is subdivided into up to ten sections.
Chapter 1 introduces some of the basic objects of study and illustrates a few motivating examples. The reader gets here acquainted with algebraic and analytic subspaces of \(\mathbb{C}^n\), with the statement of Chow's theorem, and with the analytic concept of elliptic curves.
Chapter 2 is to prepare the ground for the introduction of algebraic schemes from the more familiar analytic viewpoint. This is done by discussing differentiable manifolds, first in the traditional way and then via sheaves (of functions) and ringed spaces.
Chapter 3 is devoted to affine algebraic schemes, together with their allied commutative algebra and topology, and, in the sequel, to general schemes and their morphisms. Ringed spaces over \(\mathbb{C}\) and schemes of finite type over \(\mathbb{C}\) are treated at the end of this chapter.
The comparison between the Zariski topology of a scheme locally of finite type over \(\mathbb{C}\) on the one hand, and the complex topology of its underlying analytic space (of closed points), on the other hand, is the subject of Chapter 4.
This is followed by developing the analytic theory of schemes over \(\mathbb{C}\) in the subsequent Chapter 5, where rings and sheaves of analytic functions are appropriately explained and predominantly utilized. The constructions given here are of local character, but are then globalized in the more sketchy Chapter 6 for the keen, particularly interested reader, mainly by outlining the basic facts on Fréchet spaces and a coordinate-free approach to polydiscs.
Chapter 7 turns to the crucial concept of coherent sheaves in both the algebraic and the analytic context. According to its significance for the main theme of the entire book, Serre's cohomological GAGA theorem, this chapter is worked out in greater detail, culminating in the description of the analytification of coherent algebraic sheaves and finally, in the statement of Serre's GAGA theorem, in its weak versions.
Chapter 8 and Chapter 9 provide the essential concepts and results regarding the algebraic and analytic geometry of projective spaces \(\mathbb{P}^n_{\mathbb{C}}\) and their quotients with respect to the action of affine group schemes. The sheaf-theoretic main results are explained and illustrated in Chapter 8, while their proofs, including the involved methods from invariant theory, are thoroughly carried out in Chapter 9.
The final Chapter 10, the highlight of the book, is devoted to the (almost) complete proof of Serre's GAGA theorem for projective spaces \(\mathbb{P}^n_{\mathbb{C}}\) in its following version: The analytification functor, which takes a coherent algebraic sheaf on \(\mathbb{P}^n_{\mathbb{C}}\) to its associated coherent analytic sheaf, is an equivalence of categories.
In addition to the original course notes, the present book also contains the proof of the cohomological variant of Serre's GAGA theorem, which states that a coherent algebraic sheaf on \(\mathbb{P}^n_{\mathbb{C}}\) and its analytification have canonically isomorphic \(i\)-th cohomology groups for any integer \(i\in \mathbb{Z}\).
Also, in an appendix, the predominantly technical proofs concerning some analytification results, as used in Chapters 5--7, are given as a supplement for the inquisitive reader.
All in all, the book under review is a masterpiece of expository writing in modern algebraic geometry. It is exactly what the author promised: no comprehensive text to train future algebraic geometers, but rather an attempt to convince students of the fascinating beauty, the tremendous power, and the high value of the methods of algebraic and analytic geometry.
The author has reached his declared goal in an admirable, truly brilliant manner. His utmost lucid exposition of these modern, fairly advanced topics in the field for beginners breathes his passion for the subject and for grippingly teaching it, and his style of writing bespeakes a good sympathetic understanding towards students and their needs. Numerous synoptic sections, asides for the expert reader, and additional remarks help the student grasp the vast material thoroughly, and both the careful glossary and the detailed index facilitate the active working with this highly inspiring book. Also, the author makes concrete, instructive proposals for how to use his text for different variants of self-study or teaching, which is particularly useful as its current extended contents now offer by far more than can be covered in a one-semester course.
It remains to be wished that this very special introduction to the realm of (complex) algebraic geometry find the deserved wide-spread interest within the mathematical community. algebraic schemes; complex manifolds; sheaves; transcendental methods of algebraic geometry; projective geometry; invariant theory; GAGA-type theorems; sheaf cohomology Neeman, A.: Algebraic and Analytic Geometry. Cambridge University Press, Cambridge (2007) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Schemes and morphisms, Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Analytic sheaves and cohomology groups, Actions of groups on commutative rings; invariant theory Algebraic and analytic 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 0532.00010.]
The authors discuss different algorithms for computing Hilbert functions of graded algebras. They suggest first to calculate a Gröbner basis for the corresponding ideal and then to calculate a finite resolution of the associated monomial ring over the polynomial ring to get the Hilbert function. The authors do not seem to be aware of that their resolution in lemma 2.4 is known as the Taylor resolution [\textit{D. Taylor} (unpublished thesis), cf. the reviewer in Sémin. d'Algèbre, P. Dubreil, Proc., Paris 1977/78, 31ème Année, Lect. Notes Math. 740, 272-284 (1979; Zbl 0411.13014)]. algorithms for computing Hilbert functions of graded algebras; Gröbner basis; Taylor resolution Mora, F.; Möller, H. M., The computation of the Hilbert function, (Computer Algebra, London, 1983, Lect. Notes Comput. Sci., vol. 162, (1983), Springer Berlin), 157-167 Homological methods in commutative ring theory, Software, source code, etc. for problems pertaining to commutative algebra, Multiplicity theory and related topics, Graded rings and modules (associative rings and algebras), Symbolic computation and algebraic computation, Relevant commutative algebra The computation of the Hilbert function | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper is an announcement on the recent result of the classification of generalized Igusa local zeta functions associated to irreducible matrix groups. The author gives a necessary and sufficient condition for a generalized Igusa zeta function to be simple. He introduces a decomposition diagram of a finite-dimensional representation of a simple Chevalley group \(G\) and proves the following theorem. Let \(G\) be the group of type \(A_l\) \((l\geq 1)\), \(B_l\) \((l\geq2)\), \(C_l\) \((l\geq 3)\), \(D_l\) \((l\geq 4)\), \(E_l\) \((8\geq l\geq 6)\), \(F_4\) or \(G_2\). The generalized local zeta function \(Z(s)\) associated to a representation of \(\rho\) of \(G\) is simple if and only if the decomposition diagram of \(\rho\) is connected. canonical basis; classification of generalized Igusa local zeta functions; simple; decomposition diagram Other Dirichlet series and zeta functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta functions and \(L\)-functions On simple Igusa local zeta 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 \(G\) be a split semisimple Chevalley group scheme over a base scheme \(S\), and let \(\mathfrak{g}\) be its Lie algebra. Let \(T\) be a maximal torus of \(G\), and let \(\mathfrak{t}\) be its Lie algebra. Then \(G\) acts on \(\mathfrak{g;}\) furthermore \(\mathfrak{t}\) is acted upon by its Weyl group \(W:=W_{T},\) giving rise to the a map \(\pi:\mathfrak{t}/W\rightarrow\mathfrak{g}/G\) which is referred to as the Chevalley morphism. If \(S=\text{Spec}(k)\) for \(k\) an algebraically closed field, and \(\text{char}k \nmid |W|\) then it is well-known that \(\pi\) is an isomorphism, facilitating the computation of the adjoint quotient \(\mathfrak{g}/G.\)
The focus of this work is to determine if \(\pi\) is an isomorphism for other choices of base scheme \(S\). The authors restrict their attention to simple Chevalley groups, indicating that the results can be extended easily to the semisimple case. For \(G\) a split, simple Chevalley group it is shown that \(\pi\) is a dominant morphism, and is an isomorphism provided that \(G\) is not a symplectic group\(.\) There are no restrictions on \(S\) or \(W\).
In the case where \(G=\text{Sp}_{2n},\) if we suppose \(S\) has no \(2\)-torsion, then \(\pi\) is again an isomorphism. If \(\text{Spec}(A)\) is an open subscheme of \(S\) then the ring of functions of \(\mathfrak{g}/G\) over Spec\(\left( A\right) \) is \(A[c_2,c_4,\dots,c_{2n}]\), where \(c_{2i}\) are the coefficients of the characteristic polynomial of the universal matrix over \(\mathfrak{gl}(E)\), \(E\) the natural representation of \(G\) of dimension \(2n\). Also, \(\mathfrak{g}/G\) commutes with arbitrary base change.
Similar results about the ring of functions of \(\mathfrak{g}/G\) over Spec\((A)\) are given when \(G=\text{SO}_{2n}\) and \(G=\text{SO}_{2n+1}.\) For a certain collection of functions \(\pi_{i}\) related to the Pfaffian (pf) of the universal matrix in \(\mathfrak{so}_{2n}\) we have that the ring of functions of \(\mathfrak{g}/G\) is \(A[c_{2},\dots,c_{n-2},\text{pf;} x(\pi_1) ^{\varepsilon_1}\dots(\pi_{n-1})^{\varepsilon_{n-1}}]\) in the case where \(G=\text{SO}_{2n}\); and \(A[c_{2} ,\dots,c_{n}; x(\pi_1)^{\varepsilon_1}\dots(\pi_n) ^{\varepsilon_n}]\) if \(G=\text{SO}_{2n+1},\) where \(\varepsilon_{i}=0,1\) not all zero, and \(x\) runs through the generators of the \(2\)-torsion elements in \(A\). In these cases the quotient commutes with base change \(f:S^{\prime}\to S\) if and only if \(f^{\ast}S[2] =S^{\prime}[2],\) where ``[2]'' indicates the \(2\)-torsion elements. Furthermore, if \(S\) is connected and noetherian then \(\mathfrak{g}/G\) is of finite type over \(S\) and is also flat if and only if \(S[2]\) is empty or all of \(S\). Chevalley groups; invariant theory; adjoint action; Lie algebras Chaput, P-E; Romagny, M, On the adjoint quotient of Chevalley groups over arbitrary base schemes, J. Inst. Math. Jussieu, 9, 673-704, (2010) Actions of groups on commutative rings; invariant theory, Linear algebraic groups over arbitrary fields, Group schemes On the adjoint quotient of Chevalley groups over arbitrary base 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 In projective 3-space over the complex numbers, a stationary trisecant of a non-singular curve C is a line meeting C in three points such that two of the tangents at these three points intersect. There are four classical formulas for space curves [see, for example \textit{J. G. Semple} and \textit{L. Roth}, ''Introduction to algebraic geometry'' (Oxford 1949); pp. 373- 377]. Classically, there was always the restriction of the generic case. \textit{P. Le Barz} [C. R. Acad. Sci., Paris, Sér. A 289, 755-758 (1979; Zbl 0445.14025)] proved three of the formulas without this restriction. In this article, the fourth formula is also proved. The number of stationary tangents is \(\xi =-5n^ 3+27n^ 2-34n+2h(n^ 2+4n-22-2h)\) where n is the degree and h is the number of apparent double points. The complicated computation uses similar methods to those of Le Barz (loc. cit.) involving the Chow groups of Hilbert schemes. stationary trisecant; Chow groups of Hilbert schemes Mallavibarrena , '' Validité de la formule classique des trisecantes stationnaires ''. Comptes rendus, t 303 I 16, 1986. Enumerative problems (combinatorial problems) in algebraic geometry, Curves in algebraic geometry, Questions of classical algebraic geometry, Parametrization (Chow and Hilbert schemes) Validité de la formule classique des trisécantes stationnaires. (Validity of the classical formula of stationary trisecants) | 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 set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, \textit{the differentiation functor}, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors from the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, \textit{the integration functors}. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric \textit{separability} of a given morphism of Hopf algebroids. Several examples and applications are presented. (co)commutative Hopf algebroids; affine groupoid schemes; differentiation and integration; finite dual; Kähler module; Lie algebroids; Lie groupoids; Lie-Rinehart algebras; Malgrange groupoids; Tannaka reconstruction Affine algebraic groups, hyperalgebra constructions, Coalgebras and comodules; corings, Universal enveloping algebras of Lie algebras, Groupoids (i.e. small categories in which all morphisms are isomorphisms), Topological groupoids (including differentiable and Lie groupoids), Associated Lie structures for groups, Derivations and commutative rings, General properties and structure of other Lie groups Toward differentiation and integration between Hopf algebroids and Lie algebroids | 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 \(I\equiv(I,\preceq)\) a finite partially ordered set. Assume that \(\max I=\{p_1,\dots,p_t\}\) is the set of all maximal elements of \(I\) and \(I\setminus\max I=\{1,\dots,n\}\). Given a vector \(v=(v_1,\dots,v_n,v_{p_1},\dots,v_{p_t})\in\mathbb{N}^I\), denote by \({\mathbf{Mat}}^I_v\) the irreducible affine \(K\)-variety consisting of all bipartitioned matrices of the form
\[
A=\begin{bmatrix} A_{1p_1}&A_{2p_1}&\cdots&A_{np_1}\\ \vdots &\vdots&\ddots&\vdots\\ A_{1p_t}&A_{2p_t}&\cdots&A_{np_t}\end{bmatrix}
\]
with coefficients in \(K\), where \(A_{ip_r}\) is the matrix with \(v_i\) rows, \(v_{p_r}\) columns such that \(A_{ip_r}=0\) if \(i\not\preceq p_r\). Let \({\mathbf G}^I_v\) be the parabolic algebraic group generated by the following \(I\)-elementary transformations on matrices in \({\mathbf {Mat}}^I_v\). \((E_1)\) All elementary transformations on rows inside each horizontal block of the matrix \(A\). \((E_2)\) All elementary transformations on columns from the \(i\)-th block to the \(j\)-th block, for all \(i\preceq j\in I\setminus\max I\). Finally, let \(*\colon{\mathbf G}^I_v\times{\mathbf{Mat}}^I_v\to{\mathbf{Mat}}^I_v\) be the obvious algebraic group action.
One of the main aims of the paper is to give a simple description of degenerations of \({\mathbf G}^I_v\) orbits in the variety \({\mathbf{Mat}}^I_v\). This is done in terms of homomorphisms and extensions of prinjective \(KI\)-modules of finite dimension, where \(KI\) is the incidence \(K\)-algebra of \(I\). Here, by a prinjective \(KI\)-module we mean a right \(KI\)-module \(X\) of projective dimension at most one such that there is a minimal projective presentation \(0\to P_1\to P_0\to X\to 0\), where \(P_1\), \(P_0\) are projective and \(P_1\) is a direct sum of simple modules. A complete solution is given in case the integral Tits quadratic form \(q_I\colon\mathbb{Z}^I\to\mathbb{Z}\) defined by the formula
\[
q_I(x)=\sum _{i\in I}x_i^2+\sum_{i\prec j\not\in\max I}x_ix_j-\sum_{p \in\max I}(\sum_{i\prec p}x_i)x_p
\]
is weakly positive or \(I\) is a minimal poset such that \(q_I(v)=0\), for some non-zero vector \(v\in\mathbb{N}^I\). We recall that \(q_I\) is said to be weakly positive if \(q_I(w)>0\) for any non-zero vector \(w\in\mathbb{N}^I\). partially ordered sets; finite representation type; irreducible affine varieties; bipartitioned matrices; group actions; degenerations of orbits; prinjective modules; incidence algebras; Tits quadratic forms Kosakowska, J.: Degenerations in a class of matrix varieties and prinjective modules. J. Algebra 263, 262--277 (2003) Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Ext and Tor, generalizations, Künneth formula (category-theoretic aspects), Group actions on affine varieties, Canonical forms, reductions, classification, Multilinear algebra, tensor calculus Degenerations in a class of matrix varieties and prinjective modules | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author works with semisimple Lie algebras over \(\mathbb{Z}\) and their minuscule representations connecting the root lattices for exceptional simple Lie algebras to del Pezzo surfaces. In particular, it is proven that the Lie algebra \(E_6\) has an invariant cubic polynomial of 27 variables that could be defined as follows. Let \(S\) be a cyclic 3-fold cover of \(\mathbb{P}^2\) branched over a smooth cubic curve. Introduce a variable \(Y_\ell\) for each of 27 lines of \(S\), and for each tritangent plane \(P\) let \(Y_p=\pi Y_\ell\) for \(\ell\leq P\). Then \(\Theta= \sum_x Y_{P(x)}-\sum_{P'} Y_{P'}.\)
The first sum is taken over all flex points \(x\) of the branch locus and the second over the remaining 36 tritangent planes \((P(x)\) is the Eckard plane corresponding to the flex point). Some other invariant tensors for minuscule representations of \(E_6\) and \(E_7\) are computed. minuscule representations; exceptional simple Lie algebras; del Pezzo surfaces; invariant tensors [75] Lurie J., ''On simply laced Lie algebras and their minuscule representations'', Comment. Math. Helv., 166 (2001), 515--575 Exceptional (super)algebras, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Rational and ruled surfaces On simply laced Lie algebras and their minuscule 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 Während der letzten zwanzig Jahre hat die Invariantentheorie eine Renaissance erlebt. Vor allem die geometrischen Methoden traten in den Vordergrund. Das vorliegende Buch führt in die Theorie der algebraischen Transformationsgruppen ein und bespricht alte und neue Probleme der Invariantentheorie. Dabei werden viele Ergebnisse beschrieben, die bisher nur verstreut in Zeitschriftenartikeln zu finden waren. Durch den Verzicht auf größtmögliche Allgemeinheit (alle Varietäten sind über \({\mathbb{C}}\) definiert, manche Beweise werden nur für die Gruppe GL(n,\({\mathbb{C}})\) geführt) gelingt es dem Autor, die wesentlichen Ideen zu vermitteln und den Leser rasch an aktuelle Forschungsgebiete heranzuführen.
Im Kapitel I (''Einführende Beispiele'', 44 Seiten) werden Klassifikations- und Normalformenprobleme (Quadratische Formen, Konjugationsklassen, Invarianten mehrerer Vektoren, ternäre kubische Formen) betrachtet und geometrisch interpretiert, um die folgenden Kapitel zu motivieren. - Das Kapitel II (''Gruppenoperationen, Invariantenringe und Quotienten'', 97 Seiten) bringt zunächst Grundlagen über affine algebraische Gruppen und deren Darstellungen. Im Hauptteil dieses Kapitels wird bewiesen, daß bei einer regulären Darstellung einer reduktiven Gruppe G die Algebra der invarianten Polynomfunktionen endlich erzeugt ist. Damit können die Invariantenalgebra geometrisch gedeutet und ''algebraische Quotienten'' nach reduktiven Gruppen gebildet werden. Im Abschnitt 3.4 wird ein Kriterium für Quotienten angegeben, das zusammen mit dem Normalitätskriterium aus Kapitel III, Abschnitt 3.3 eine sehr nützliche und elegante Methode zur Bestimmung von Invariantenalgebren reduktiver Gruppen ergibt. Als Anwendung wird der klassische ''1. Fundamentalsatz für GL(n,\({\mathbb{C}})''\) in geometrischer Formulierung bewiesen. - Im Kapitel III (''Darstellungstheorie und die Methode der U-Invarianten'', 82 Seiten) werden zuerst die wichtigsten Sätze der Darstellungstheorie und das Hilbert-Mumford-Kriterium für GL(n,\({\mathbb{C}})\) bewiesen und für beliebige reduktive Gruppen formuliert. Dann wird die Methode der U-Invarianten (U ist eine maximale unipotente Untergruppe einer reduktiven Gruppe) behandelt und auf Normalitätsfragen und Multiplizitätenprobleme angewendet. Den Abschluß bildet die Klassifikation aller affinen normalen SL(2,\({\mathbb{C}})\)-Einbettungen, das sind affine normale SL(2,\({\mathbb{C}})\)- Varietäten, die eine zu SL(2,\({\mathbb{C}})\) isomorphe dichte Bahn enthalten.
Im Anhang I (''Einige Grundlagen aus der algebraischen Geometrie'', 52 Seiten) sind die zum Verständnis des Buches nötigen Begriffe und Ergebnisse aus der algebraischen Geometrie zusammengestellt. - Im Anhang II (''Lineare Reduktivität der klassischen Gruppen'', 10 Seiten) wird mit dem ''Weyl'schen unitären Trick'' bewiesen, daß die Darstellungen der klassischen Gruppen vollständig reduzibel sind. algebra of invariant polynomials; algebraic quotients of reductive groups; Hilbert-Mumford group; representation of classical groups H. Kraft, \textit{Geometrische Methoden in der Invariantentheorie}, Vieweg, 1984. Geometric invariant theory, Group actions on varieties or schemes (quotients), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to group theory, Representation theory for linear algebraic groups, Vector and tensor algebra, theory of invariants, Groups acting on specific manifolds Geometrische Methoden in der Invariantentheorie | 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 finite Coxeter system \((W,S)\) we construct a certain noncommutative algebra, the so-called `bracket algebra', together with a family of commuting elements, the so-called `Dunkl elements'. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group \(W\). We prove this conjecture for classical Coxeter groups and \(I_2(m)\). We define a ``quantization'' and a multiparameter deformation of our construction and show that for Lie groups of classical type and \(G_2\), the algebra generated by Dunkl's elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in \(B_n\)-, \(D_n\)- and \(G_2\)-bracket algebras, and as an application, discover a `Pieri-type formula' in the \(B_n\)-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary \(B_n\)-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri's formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type \(A\). We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called `flat connections with constant coefficients', which describes ``a noncommutative differential geometry on a finite Coxeter group'' in the sense of S. Majid. finite Coxeter system; Dunkl elements; coinvariant algebras of Coxeter groups; multiparameter deformation; quantized bracket algebras; quantum cohomology ring; flag variety Kirillov, A., Maeno, T.: Noncommutative algebras related with Schubert calculus on Coxeter groups. Eur. J. Comb. \textbf{25}, 1301-1325 (2004). Preprint RIMS-1437, 2003 Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Reflection and Coxeter groups (group-theoretic aspects) Noncommutative algebras related with Schubert calculus on Coxeter 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 rational conformal quantum field theory, a certain finite-dimensional vector space, the so-called space of conformal blocks, is associated with a given \(n\)-pointed compact Riemann surface of genus \(g\). Physical arguments have led \textit{E. Verlinde} in 1988 [Nucl. Phys. B 300, No. 3, 360-376 (1988)] to an explicit conjectural formula for the dimension of the spaces of conformal blocks. This so-called Verlinde formula turned out to be equivalent to a dimension formula for the (complex) vector spaces of generalized theta functions on moduli spaces of semi-stable vector bundles over compact Riemann surfaces.
The first proof of the Verlinde formula, valid for \(SL_2(\mathbb{C})\)-bundles, was provided by \textit{A. Szenes} in 1993 [cf. Proc. Durham Symp. 1993, Lond. Math. Soc. Lect. Note Ser. 208, 241-253 (1995; Zbl 0823.14019)]. In the sequel, various other proofs (in varying generality and by different methodical approaches) have been presented by several authors.
The present article is a survey on the developments concerning the Verlinde formula. It describes the state-of-art as it was achieved by November 1994. Starting from the physical origin, i.e., from conformal field theories and Verlinde's initial conjecture, the author explains the approach via representations of affine Lie algebras arising in the Wess-Zumino-Witten model, and finally leads the audience to the interrelation with moduli spaces of vector bundles and their generalized theta functions. fusion rules; rational conformal quantum field theory; conformal blocks; compact Riemann surface; Verlinde formula; dimension formula; generalized theta functions; moduli spaces of semi-stable vector bundles; representations of affine Lie algebras Sorger, C., La formule de Verlinde, Séminaire Bourbaki, vol. 1994/1995, Astérisque, 237, 87-114, (1996), [Exp. No. 794, 3] Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Vector bundles on curves and their moduli, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics Verlinde's formula | 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 the translation of the second (revised) edition (1985) of the book. See the review of the first edition (1984; Zbl 0569.14003). algebra of invariant polynomials; algebraic quotients of reductive groups; Hilbert-Mumford group; representation of classical groups Kraft, H., Geometricheskie metody v teorii invariantov (Geometric Methods in Invariant Theory), Moscow, 1987. Geometric invariant theory, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector and tensor algebra, theory of invariants, Groups acting on specific manifolds Geometric methods in the theory of invariants. (Geometricheskie metody v teorii invariantov). Transl. from the 2nd German ed. by D. I. Panyushev. Transl. ed. and with a preface by V. L. Popov | 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 algebra \({\mathcal D}(R)\) of all differential operators on a commutative affine \({\mathbb{C}}\)-algebra R is known to be very well-behaved when R is the coordinate ring \({\mathcal O}(X)\) of a nonsingular affine variety X - in particular, \({\mathcal D}(R)\) is then a simple noetherian domain and an affine \({\mathbb{C}}\)-algebra of finite Gelfand-Kirillov dimension (see e.g. [J.-E. Björk, Rings of Differential Operators (1979; Zbl 0499.13009)] or [\textit{J. C. McConnell} and \textit{J. C. Robson}, Noncommutative Noetherian Rings (1987; Zbl 0644.16008)]). However, in the singular case these nice properties are often lost (see e.g. [\textit{I. N. Bernstein}, \textit{I. M. Gel'fand} and \textit{S. I. Gel'fand}, Usp. Mat. Nauk 27, 185-190 (1972; Zbl 0253.58009)]), although \({\mathcal D}({\mathcal O}(X))\) is always an affine noetherian \({\mathbb{C}}\)-algebra when X is an affine curve (see [\textit{J. L. Muhasky}, Trans. Am. Math. Soc. 307, 705-723 (1988; Zbl 0668.16007)], and [\textit{S. P. Smith} and \textit{J. T. Stafford}, Proc. Lond. Math. Soc., III. Ser. 56, 229-259 (1988; Zbl 0672.14017)]). Moreover, \({\mathcal D}(R)\) is known to be nice when R is the ring of invariants for a finite group of automorphisms of \({\mathcal O}({\mathbb{C}}^ n)\) [\textit{T. Levasseur}, in Lect. Notes Math. 867, 157-173 (1981; Zbl 0507.14012)], when R is the ring of invariants for the diagonal action of a torus on a polynomial ring [\textit{I. M. Musson}, Trans. Am. Math. Soc. 303, 805-827 (1987; Zbl 0628.13019)], and when \(R={\mathcal O}(X)\) for X a quadratic cone in \({\mathbb{C}}^ n\) [\textit{T. Levasseur}, \textit{S. P. Smith} and \textit{J. T. Stafford}, J. Algebra 116, 480-501 (1988; Zbl 0656.17009)].
In this paper, the authors prove that for several classical rings of invariants, \({\mathcal D}(R)\) is a simple noetherian ring. They analyze the following four situations, taking R to be the ring of invariants for the standard action of the given group on the coordinate ring of the given variety: (A) GL(k,\({\mathbb{C}})\) acting on \(M_{p,k}({\mathbb{C}})\times M_{k,q}({\mathbb{C}})\), (B) O(k,\({\mathbb{C}})\) acting on \(M_{k,n}({\mathbb{C}})\), (C) Sp(2k,\({\mathbb{C}})\) acting on \(M_{2k,n}({\mathbb{C}})\), and (D) SO(k,\({\mathbb{C}})\) acting on \(M_{n,k}({\mathbb{C}})\). In each of the first three cases, R is isomorphic to the coordinate ring of an affine variety: (A) the variety of \(p\times q\) matrices of rank at most k (which is singular if \(p\geq q>k)\), (B) the variety of symmetric \(n\times n\) matrices of rank at most k (singular if \(k<n)\), (C) the variety of antisymmetric \(n\times n\) matrices of rank at most 2k (singular if 2k\(\leq n-2).\)
In cases (A), (B), (C), \({\mathcal D}(R)\) is shown to be a simple noetherian domain by proving that it is isomorphic to a suitable factor ring of an enveloping algebra U(\({\mathfrak g})\), where (A) \({\mathfrak g}={\mathfrak gl}(p+q)\), (B) \({\mathfrak g}={\mathfrak sp}(2n)\), (C) \({\mathfrak g}={\mathfrak so}(2n)\). In case (D) (with \(k\leq n)\), \({\mathcal D}(R)\) is shown to be a simple noetherian ring finitely generated as a module over a simple factor ring of U(\({\mathfrak sp}(2n)).\)
To support the line of approach, the paper contains much work on enveloping algebras of semisimple Lie algebras and on rings of \({\mathfrak k}\)-finite vectors. An appendix contains a proof for (a generalization of) Gabber's Lemma: If \({\mathfrak g}\) is a finite-dimensional Lie algebra, M is a finitely generated s-homogeneous (in the sense of GK-dimension) left U(\({\mathfrak g})\)-module, and E is an essential extension of M, then the set of finitely generated U(\({\mathfrak g})\)-modules \(M'\) such that \(M\subseteq M'\subseteq E\) and \(GK\dim (M'/M)\leq s-2\) has a unique maximal element. differential operators; commutative affine \({\mathbb{C}}\)-algebra; coordinate ring; nonsingular affine variety; simple noetherian domain; Gelfand- Kirillov dimension; ring of invariants; group of automorphisms; simple noetherian ring; variety of symmetric n\(\times n\) matrices; simple factor ring; enveloping algebras; semisimple Lie algebras Levasseur, T.; Stafford, J. T., Rings of differential operators on classical rings of invariants, Mem. Amer. Math. Soc., 412, pp., (1989) Noetherian rings and modules (associative rings and algebras), Group actions on varieties or schemes (quotients), Universal enveloping (super)algebras, Infinite-dimensional simple rings (except as in 16Kxx), Determinantal varieties, Automorphisms and endomorphisms, Valuations, completions, formal power series and related constructions (associative rings and algebras), Simple, semisimple, reductive (super)algebras, Modules of differentials, Geometric invariant theory, Sheaves of differential operators and their modules, \(D\)-modules Rings of differential operators on classical rings of 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 See the extensive review by \textit{S. I. Gel'fand} in Zbl 0618.22011 of the first edition (1986). 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, 1988). 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 Let \(A\) be a finite-dimensional associative algebra over an algebraically closed field \(k\). Denote by \(\mathfrak{Aut}(A)\) the affine group scheme of automorphisms of \(A\) and by \(\mathfrak{Inn}(A)\) the smooth characteristic subgroup of inner automorphisms. Then \(\mathfrak{Out}(A)=\mathfrak{Aut}(A)/\mathfrak{Inn}(A)\) is the affine group scheme of outer automorphisms. By the theorem of Cartier, \(\mathfrak{Aut}(A)\) is always reduced if \(\text{char}(k)=0\), but this is not the case if \(\text{char}(k)=p>0\). The authors prove that the group schemes \(\mathfrak{Aut}(A)\) and \(\mathfrak{Out}(A)\) are not reduced if \(A\) admits a \(\mathbb{Z}_p\)-grading which can not be lifted to a \(\mathbb{Z}\)-grading. The proof is based on the following result: an infinitesimal multiplicative subgroup \(\mathfrak{M}\) of a smooth algebraic group \(\mathfrak{G}\) over an algebraically closed field \(k\) of characteristic \(p>0\) is contained in a maximal torus of \(\mathfrak{G}\).
This applies in particular when \(A\) is the group algebra of a \(p\)-group. Another example, where the above result shows that \(\mathfrak{Aut}(A)\) is not reduced, is illustrated by the direct calculation of the automorphism group. finite-dimensional associative algebras; affine group schemes; automorphism groups; inner automorphisms Group schemes, Group rings of finite groups and their modules (group-theoretic aspects), Automorphisms and endomorphisms Non-reduced automorphism 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 \(X\) be a complex algebraic variety with a \(\Gamma\)-action, where \(\Gamma\) is a finite group. Denote by \(H^i(X)\) the complex de Rham cohomology of \(X\) and by \(H^i_c(X)\) the complex cohomology with compact supports, both regarded as \(\Gamma\)-modules. It is known that each cohomology space \(H^i_c(X)\) has a \(\Gamma\)-invariant `weight filtration' whose graded quotients \(Gr_mH^i_c(X)\) give rise to a `weight \(m\)' equivariant Euler characteristic
\[
E^\Gamma_{m,c}(X):=\sum_j(-1)^j Gr_m H^j_c(X)\in R(\Gamma).
\]
Here \(R(\Gamma)\) denotes the Grothendieck ring of \(\Gamma\).
The following polynomials are defined (for \(g\in\Gamma\)):
\[
P^\Gamma_X(t):=\sum_j H^j(X)t^j,\quad Q^\Gamma_{X,c}(t):=\sum_m E^\Gamma_{m,c}(X)t^m,
\]
\[
P^\Gamma_X(g,t):=\sum_j\text{trace}(g,H^j(X))t^j,\quad Q^\Gamma_{X,c}(t):=\sum_m\text{trace}(g,E^\Gamma_{m,c}(X))t^m.
\]
If \(\Gamma=1\) the superscript is dropped and the modules are replaced by their dimensions.
The purpose of the paper is, to compute these polynomials in the case where \(X=G_{rs}\), the set of regular semisimple elements of the complex connected reductive algebraic group \(G\), or \(X={\mathfrak G}_{rs}\) the corresponding set in the Lie algebra \(\mathfrak G\) of \(G\).
This problem can be reduced to analogous \(W\)-equivariant problems concerning respectively maximal tori or toral algebras \(T_{rs}\) and \({\mathcal T}_{rs}\), and \(W=N_G(T)/T\) the corresponding Weyl group.
In Section 4, explicit formulae for the \(P_X(t)\) and \(Q_{X,c}(t)\), \(X\in\{G_{rs},{\mathfrak G}_{rs}\}\) are given in terms of certain polynomials \(P_{T_{rs}}(w,t)\) (in the group case) and \(P_{M_W}(w,t)\) (in the Lie algebra case), which depend on elements \(w\in W\). The polynomials \(P_{M_W}(w,t)\) are explicitly known (see the references in the paper for exceptional groups and groups of type \(A\) and \(B\) as well as the paper [\textit{P. Fleischmann} and \textit{I. Janiszczak}, Manuscr. Math. 72, No. 4, 375-403 (1991; Zbl 0790.52006)] for type \(D\)).
On applying his results to the variety of regular semisimple elements in the Lie algebra, \({\mathfrak G}L_n(\mathbb{C})\), the author obtains stability results for the Betti numbers of this variety.
In the last section the previous results are applied, with the help of \(\ell\)-adic cohomology, to regular semisimple varieties of algebraic groups over finite fields and their Lie algebras. Under certain technical assumptions, explicit formulae are derived for the number of \(\mathbb{F}_q\)-rational points in the regular semisimple variety of the Lie algebras of the general linear group or Lie algebras of type \(B_n\), \(C_n\). Notice that the corresponding formula in Proposition (8.9) for \(B_n\), \(C_n\) contains a slight misprint. It should read
\[
q^{n^2}\prod^n_{k=1} (q^{2k}-1)\sum_{\substack{\lambda=(\lambda^+,\lambda^-)\\ \lambda^+=(i^{m_i}),\;\lambda^-=(j^{n_j})}} \prod_i {q^+_i(-q^{-1})\choose m_i}(q^i-1)^{-m_i}\times\prod_j {q^-_j(-q^{-1})\choose n_j}(q^j+1)^{-n_j}.
\]
[For a different approach to these formulae see also: \textit{P. Fleischmann} and \textit{I. Janiszczak}, J. Algebra 155, No. 2, 482-528 (1993; Zbl 0809.20007); \textit{P. Fleischmann}, Finite Fields Appl. 4, No. 2, 113-139 (1998)]. Poincaré polynomials; complex algebraic varieties; complex de Rham cohomology; Euler characteristic; Grothendieck rings; regular semisimple elements; complex connected reductive algebraic groups; Lie algebras; maximal tori; toral algebras; \(\ell\)-adic cohomology; numbers of rational points G. I. Lehrer, The cohomology of the regular semisimple variety, J. Algebra 199(2) (1998), 666\Ndash689. \small\texttt DOI: 10.1006/jabr.1997.7195. Cohomology theory for linear algebraic groups, Linear algebraic groups over the reals, the complexes, the quaternions, Linear algebraic groups over finite fields, Cohomology of Lie (super)algebras, Classical real and complex (co)homology in algebraic geometry The cohomology of the regular semisimple variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper develops a theory of ``\(S_2\)-ification'' in order to address a conjecture of Lusztig concerning `special pieces'. First we look at the ``\(S_2\)-ification'', where \(S_2\) refers to one of Serre's conditions for normality. Let \(X\) be a scheme of finite type over a Noetherian base scheme \(S\) admitting a dualizing complex, and let \(U\subset X\) be an open set whose complement has codimension at least 2. One assumes an affine group scheme \(G\) to be acting on everything. The authors extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves [not to be confused with ordinary constructible perverse sheaves] by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on \(U\) to perverse sheaves on \(X\) may be defined for a much broader class of perversities than has previously been known. One also introduces a derived category version of the coherent intermediate extension functor.
Under suitable hypotheses, one introduces a construction (called ``\(S_2\)-extension'') in terms of perverse coherent sheaves of algebras on \(X\) that takes a finite morphism to \(U\) and extends it in a canonical way to a finite morphism to \(X\). More specifically, one uses that the coherent intermediate extension (for the ``\(S_2\)-perversity'') of a sheaf of algebras is again a sheaf of algebras. Thus a new scheme may be constructed as the total Spec of a coherent intermediate extension of the push forward to \(U\) of a structure sheaf. In particular, this construction gives a canonical ``\(S_2\)-ification'' of appropriate \(X\). The construction also has applications to the ``Macaulayfication'' problem, and it is particularly well-behaved when \(X\) is Gorenstein.
We now turn to the main goal. It is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. One uses \(S_2\)-extension to give a uniform construction of the desired variety. Because of its construction the variety must be the correct one. To prove the conjecture in full one must still show the variety is smooth. Partial results in this direction are given. perverse coherent sheaves; special pieces in unipotent varieties; Macaulayfication; schemes of finite type; affine group schemes; intersection cohomology functors Achar, P; Sage, D, Perverse coherent sheaves and the geometry of special pieces in the unipotent variety, Adv. Math., 220, 1265-1296, (2009) Representation theory for linear algebraic groups, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Group schemes, Cohomology theory for linear algebraic groups, Derived categories, triangulated categories Perverse coherent sheaves and the geometry of special pieces in the unipotent variety. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Harder, Langlands and Rapoport had proved the Tate conjecture for algebraic cycles for the non-CM-submotives of \(H^ 2\) of Hilbert modular surfaces [\textit{G. Harder}, \textit{R. P. Langlands}, \textit{M. Rapoport}, J. Reine Angew. Math. 366, 53--120 (1986; Zbl 0575.14004)].
The authors consider Hilbert modular forms of CM-type and prove the Tate conjecture for the CM-submotives in \(H^ 2\). This gives the full Tate conjecture asserting the algebraicity of all Tate cycles on Hilbert modular surfaces over an arbitrary number field. The authors also relate, in the CM case, the number of independent divisor classes to the order of pole of the \(L\)-function of the surface at the edge of convergence.
The results have interesting corollaries about the Picard group of the surface. There exist non-trivial divisors defined over metabelian fields which are not in the linear span of the Hirzebruch-Zagier cycles, canonical divisors and the cycles coming from the desingularization of the surface. The methods are quite different from the methods of Harder, Langlands and Rapoport. Hodge cycles; CM-motives; Tate conjecture for algebraic cycles; Hilbert modular surfaces; Hilbert modular forms of CM-type; Picard group V. K. Murty, D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), no. 2, 319--345. Cycles and subschemes, Special surfaces, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Generalizations (algebraic spaces, stacks), Picard groups Period relations and the Tate conjecture for Hilbert modular 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 very long paper the authors study double affine Hecke algebras (DAHA) at roots of unity and their relations with deformed Hilbert schemes. One of their main goals is to compare the spherical DAHA with the algebra of global section of a sheaf of quantum differential operators on the deformed Hilbert scheme. They prove that their categories of finitely generated modules are derived equivalent to some category of coherent sheaves. double affine Hecke algebras; Hilbert schemes; quantum differential operators; quantum moment maps; quantum reduction; categories of coherent sheaves M. Varagnolo and E. Vasserot, Double affine Hecke algebras at roots of unity, \textit{Represent.} \textit{Theory}, 14 (2010), 510--600.Zbl 1280.20005 MR 2672950 Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Quantum groups (quantized enveloping algebras) and related deformations, Module categories in associative algebras Double affine Hecke algebras at roots of unity. | 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 nontrivial finite subgroup of \(SL_n (\mathbb{C})\). Suppose that the quotient singularity \(\mathbb{C}^n/G\) has a crepant resolution \(\pi:X\to \mathbb{C}^n/G\) (i.e. \(K_X= {\mathcal C}_X)\). There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of \(X\) and the representations (or conjugacy classes) of \(G\) with a ``certain compatibility'' between the intersection product and the tensor product. The purpose of this paper is to give more precise formulation of the conjecture when \(X\) can be given as a certain variety associated with the Hilbert scheme of points in \(\mathbb{C}^n\). We give the proof of this new conjecture for an abelian subgroup \(G\) of \(SL_3(\mathbb{C})\). group action; homology group; quotient singularity; crepant resolution; McKay correspondence; Grothendieck group; intersection product; Hilbert scheme of points Ito, Y., Nakajima, H.: McKay correspondence and Hilbert schemes in dimension three. Topology, 39, 1155--1191 (2000) Applications of methods of algebraic \(K\)-theory in algebraic geometry, Group actions on varieties or schemes (quotients), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, \(K\)-theory of schemes McKay correspondence and Hilbert schemes in dimension three | 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 Xu and Yau have related hypersurface singularities to finite and infinite dimensional representations of Lie algebras. This in turn led Yau to a conjecture about the \(sl(2,\mathbb{C})\) action on formal power series rings in \(n\) variables by derivations preserving an \(m\)-adic filtration. The conjecture states which representations can appear when the action is restricted to a certain subspace, \(I(f)\). The conjecture is known to be true for \(n<5\). This paper verifies the conjecture for \(n=6\) in an explicit manner. hypersurface singularities; formal power series; representations of Lie algebras Yu, Y., On Jacobian ideals invariant by reducible \textit{sl} (2; \textit{C}) action, Trans. Amer. Math. Soc., 348, 2759-2791, (1996) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Singularities in algebraic geometry, Simple, semisimple, reductive (super)algebras On Jacobian ideals invariant by a reducible \(sl(2,\mathbb{C})\) action | 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 a simple Lie algebra \(L\). Then \([L,L]=L\) (note that for associative \(PI\)-algebra \(A\) equality \([A,A]=A\) never holds [\textit{A. Ya. Belov}, Sib. Mat. Zh. 44, No. 6, 1239--1254 (2003; Zbl 1054.16015); translation in Sib. Math. J. 44, No. 6, 969--980 (2003)]. In analogy to the group case, the authors deal with notion of {\em commutator with} i.e. minimal \(n\) such that any element \(x\in L\) can be presented as a sum \[x=H_n=\sum_{i=1}^n [a_i,b_i].\]
This can be viewed as a question of possible values of polynomial \(H_n\) on the algebra \(L\) (see review of [the reviewer et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 16, Paper 071, 61 p. (2020; Zbl 1459.16012)])
For finite dimensional simple Lie algebras over algebraically closed field of characteristic zero commutator width equal 1, for many other fields including reals the answer is still unknown.
The paper deals with Lie algebras of (sometimes symplectic) vector fields on algebraic varieties mostly with curves and Danilevsky surfaces. Algebraic geometry properties of these varieties cause some width consequences. Note that Yu.P. Razmyslov established restitution of algebraic variety \(M\) by the Lie algebra of vector fields on \(M\) (I recommend the very deep book [\textit{Yu. P. Razmyslov}, Identities of algebras and their representations. Providence, RI: American Mathematical Society (1994; Zbl 0827.17001)]) In non-rational case for curves, the authors proved that commutator width is at least \(2\). For Danilevsky surfaces some relations with Jacobian Conjecture type questions are established.
The authors discuss some open problems including possible commutator width. Can it be arbitrary large for vector field algebras on algebraic varieties? simple Lie algebras; Lie algebras of algebraic; symplectic and Hamiltonian vector fields; smooth affine curves; Danielewski surfaces; locally nilpotent derivations Lie algebras of vector fields and related (super) algebras, Classification of affine varieties Bracket width of simple Lie algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper determines a large class of examples of a certain invariant Hilbert scheme introduced in [\textit{V.~Alexeev} and \textit{M.~Brion}, J. Algebr. Geom. 14, 83--117 (2005; Zbl 1081.14005)] by relating them to (strict) wonderful varieties, a class of algebraic varieties, equipped with an action of a linear semisimple group and satisfying axioms inspired by the well known compactifications of symmetric spaces of \textit{C. De Concini} and \textit{C. Procesi} [in: Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. Early in the paper, the authors nicely illustrate their main results with a few explicit examples.
We set up some notation. Let \(G\) be a complex connected semisimple algebraic group and let \(\{\lambda_1, \ldots, \lambda_s\}\) be linearly independent dominant weights (after the choice of a Borel subgroup and a maximal torus in it), such that the monoid \(\Gamma\) they generate in the full monoid of dominant weights \(\Lambda^+\) satisfies the following saturation condition due to \textit{D.~Panyushev} [Ann. Inst. Fourier 47, No. 4, 985--1011 (1997; Zbl 0878.14008)]: \(\mathbb{Z}\Gamma \cap \Lambda^+ = \Gamma\). Here \(\mathbb{Z}\Gamma\) stands for the subgroup generated by \(\Gamma\) in the character group of the maximal torus.
The authors show that the invariant Hilbert scheme \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is isomorphic to an affine space (with a specific linear action by the adjoint torus \(T_{\mathrm{ad}}\) of \(G\)). By definition, this Hilbert scheme parametrizes the \textit{nondegenerate} closed \(G\)-subvarieties \(Y\) of \(V:=V(\lambda_1) \oplus \ldots \oplus V(\lambda_s)\) with the property that as \(G\)-modules
\[
\mathbb{C}[Y] \cong \bigoplus_{\lambda \in \Gamma} V(\lambda)^*.
\]
Here \(V(\lambda)^*\) is the dual of the \(G\)-module associated to the dominant weight \(\lambda\) and a subvariety of \(V\) is called nondegenerate if its projection to every isotypical component of \(V\) is nontrivial.
The proof, which uses wonderful varieties, roughly runs as follows. First the authors determine the tangent space \(T_{X_0}\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) to \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) at its `most degenerate point' \(X_0\). This \(X_0\) is the affine multicone of the title. To prove that \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is smooth they find another closed point \(X_1\) on \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) and, using a result from [\textit{V.~Alexeev} and \textit{M.~Brion}, loc. cit.], they determine the dimension of the (open) \(T_{\mathrm{ad}}\)-orbit on \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) through \(X_1\). As this dimension agrees with that of \(T_{X_0}\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) it follows that \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is smooth and, again by [\textit{V.~Alexeev} and \textit{M.~Brion}, loc. cit.], that it is isomorphic to an affine space.
To find the closed point \(X_1\), the authors proceed as follows, generalizing an idea exploited by \textit{S. Jansou} [J. Algebra 306, No. 2, 461--493 (2006; Zbl 1117.14005)]. They first associate a certain combinatorial datum \((S^p(\Gamma),\Sigma(\Gamma))\), introduced by \textit{D. Luna} [Hautes Études Sci. Publ. Math. 91, 161--226 (2001, Zbl 1085.14039)] and called \textit{spherical system}, to \(\Gamma\). (Actually, Luna's spherical systems are more general. Because \(\Gamma\) satisfies Panyushev's saturation condition the spherical systems considered here are \textit{strict}; cf. the appendix of [\textit{D.~Luna}, J. Algebra 313, No. 1, 292--319 (2007; Zbl 1116.22006)]).
Next, they argue that there exists a wonderful variety \(X\) \textit{with} spherical system \((S^p(\Gamma),\Sigma(\Gamma))\), thus confirming a conjecture of Luna [loc.~cit.] for the case of strict spherical systems and strict wonderful varieties. The main part of the proof of the existence of \(X\) is not given in the paper under review; instead the reader is referred to a preprint version [\url{arXiv: math.AG/0603690}]. More recently, the authors expanded this existence proof in [Classification of strict wonderful varieties, \url{arxiv:0806.2263}].
Continuing with the sketch of the proof, the authors also show that \(X\) embeds \(G\)-equivariantly into \(\mathbb{P}(V(\lambda_1)) \times \ldots \times \mathbb{P}(V(\lambda_s))\). The desired subvariety \(X_1\) of \(V\), finally, is the \(G\)-orbit closure of a vector \(v \in V\) which lies in the affine multicone \(\widetilde{X}\) over \(X\) and whose image \([v]\) in \(X\) belongs to the open \(G\)-orbit of \(X\) (wonderful varieties have a unique open orbit). This finishes the overview of the proof.
Furthermore, the authors establish that the universal family over \(\mathrm{Hilb}^G_{\underline{\lambda}}(V)\) is given by the quotient morphism \(\mathrm{Spec}(\mathbb{C}[\pi^{-1}(X)]) \to \pi^{-1}(X)/G\), where \(\pi: \widetilde{X} \dashrightarrow X\) is the natural (rational) map. linear algebraic groups; multicones over flag varieties; spherical varieties; wonderful varieties; equivariant deformations; invariant Hilbert schemes Bravi, P; Cupit-Foutou, S, \textit{equivariant deformations of the affine multicone over a ag variety}, Adv. Math., 217, 2800-2821, (2008) Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Fine and coarse moduli spaces, Representation theory for linear algebraic groups Equivariant deformations of the affine multicone over a flag variety | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Recently, the Riemann-Hilbert correspondence was generalized to the context of holonomic \(D\)-modules by A. D'Agnolo and M. Kashiwara. Namely, they proved that their enhanced de Rham functor induces a fully faithful embedding of the derived category of cohomologically holonomic complexes of \(D\)-modules into the derived category of complexes of cohomologically \(\mathbb{R}\)-constructible enhanced ind-sheaves.
In this series of papers, we study a condition when a complex of \(\mathbb{R}\)-constructible enhanced ind-sheaves \(K\) is induced by a cohomologically holonomic complex of \(D\)-modules. It is our goal to characterize such \(K\) in terms of the restriction of \(K\) to holomorphic curves. In this paper (part I), as a preliminary, we shall study some issues for multi-sets of subanalytic functions. subanalytic functions; rectilinearization; complex blowup; distribution of enhancement factors; holonomic \(D\)-modules; Riemann-Hilbert correspondence Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Sheaves of differential operators and their modules, \(D\)-modules Curve test for enhanced ind-sheaves and holonomic \(D\)-modules. 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 purpose of this paper is to form the problem of the Fourier transforms of relatively invariant functions defined over the finite fields \(\mathbb{F}_q\) and give a complete answer to the problem. Here \(\mathbb{F}_q\) is the finite field with \(q=p^m\) elements where the prime number \(p\) is sufficiently large. This is the ``finite field''-version of Sato's fundamental theorem on the Fourier transforms of relatively invariant functions on prehomogeneous vector spaces. The authors of this paper give an explicit formula of the Fourier transform. According to their theorem (Theorem A1 and A2), it is determined essentially by the \(b\)-function of the relative invariant. However, one undetermined constant \(\kappa^\vee(v)\) remains in their formula. The value of \(\kappa^\vee(v)\) is +1 or --1, but may be undecided. In the later Theorem B and Theorem C, they give a formula to determine it. The determination of \(\kappa^\vee(v)\) seems to be a big contribution to the formula of Fourier transform.
Unfortunately, the paper is full of typographical errors. This paper will be republished in the same journal. The readers should skip the first publication of the article. Gauss sum; Fourier transforms of relatively invariant functions; finite fields; Sato's fundamental theorem; prehomogeneous vector spaces Denef, J.; Gyoja, A.: Character sums associated to prehomogeneous vector spaces. Compositio math. 113, 273-346 (1998) Other character sums and Gauss sums, Homogeneous spaces and generalizations, Gauss and Kloosterman sums; generalizations, Finite ground fields in algebraic geometry Character sums associated to prehomogeneous vector spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a connected semi-simple algebraic group over an algebraically closed field \(K\) of characteristic zero or ``good'' prime; \({\mathfrak g}\) be its Lie algebra; \(D\) be either a torus over \(K\) or a finite cyclic group whose order is not a multiple of the characteristic \(p\) of \(K\); \(X(D)=\Hom(D,K^{\times})\) the character module of \(D\). Assume \(D\) acts morphically on \(G\) by algebraic group automorphisms. Let \(G(0)\) be the identity component of the group of \(D\)-fixed elements of \(G\). Then \(G(0)\) is reductive and its adjoint action on \({\mathfrak g}\) preserves \({\mathfrak g}(\lambda)=\{A\in {\mathfrak g}\); \(d\cdot A=\lambda (d)A\), \(d\in D\}\) for any \(\lambda\in X(D).\)
In this paper the author shows that for a given \(\lambda\in X(D)\), the nilpotent \(G(0)\)-orbits in \({\mathfrak g}(\lambda)\) are naturally parametrized by a finite set of weighted Dynkin diagrams ``independently'' of the characteristic \(p\) of \(K\). In particular, the number of nilpotent orbits in \({\mathfrak g}(\lambda)\) is finite and independent of \(p\). Further, he shows that from the weighted Dynkin diagram associated with a nilpotent \(G(0)\)-orbit in \({\mathfrak g}(\lambda)\), one can extract the following information: structure of a reductive part of the stabilizer \(Z_{G(0)}(N)\) of a nilpotent element \(N\) of \({\mathfrak g}(\lambda)\), the dimension of a nilpotent \(G(0)\)-orbit \(o\), an explicit representation of \(o\), the weighted Dynkin diagram associated with the nilpotent orbit \(G\cdot N\) in \({\mathfrak g}.\)
Note that if \(D\) is a torus and \(\lambda\neq 0\), then the pair \((G(0),{\mathfrak g}(\lambda))\) is a prehomogeneous vector space with a finite number of orbits. Thus in this case the result may be considered as complements to the results of \textit{M. Sato} and \textit{T. Kimura} [Nagoya Math. J. 65, 1- 155 (1977; Zbl 0321.14030)] and \textit{T. Kimura}, \textit{S. Kasai} and \textit{O. Yasukura} [Am. J. Math. 108, 643-691 (1986; Zbl 0604.20044)]. connected semi-simple algebraic group; Lie algebra; torus; character module; identity component; group of D-fixed elements; adjoint action; weighted Dynkin diagrams; number of nilpotent orbits; prehomogeneous vector space N. Kawanaka, Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra, J. Fac. Sci. Univ. Tokyo Section IA Math. 34(3) (1987), 573--597. Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Simple, semisimple, reductive (super)algebras, Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Graded Lie (super)algebras, Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Orbits and stabilizers of nilpotent elements of a graded semisimple Lie algebra | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Gegenstand der Abhandlung ist die Untersuchung der charakteristischen Singularitäten der Modulargleichung
\[
\text{(1)} \quad F(p,q,1)=0,
\]
(wo \(q\) das Modularquadrat einer gegebenen elliptischen Function, \(p\) das Quadrat des transformirten Moduls für eine primäre Transformation ungrader Ordnung \(N\),) un die Untersuchung der in linearen Coordinaten ausgedrückten Gleichung
\[
F(\alpha,\beta,\gamma)=0
\]
der Modularcurve \(C\), welche durch die Substitution \(p=\frac\alpha\gamma, q=\frac\beta\gamma\) mit der Gleichung 1) zusammenhängt. \(P,Q,R\) sind die Spitzen des Dreiecks \(\alpha\beta\gamma\); \(S\) ist der Punkt \(\alpha=\beta=\gamma;p\) und \(q\) sind die Parameter zweier Linienbüschel \(\alpha-p\gamma,\beta-q\gamma\), zwischen deren Strahlen die durch Gleichung (1) ausgedrückte Beziehung besteht: die Modularcurve \(C\) ist der Ort der Durchschnittspunkte entsprechender Straheln beider Büschel. Die Methode der Untersuchung ist schon in früheren Arbeiten von dem Herrn Verfasser befolgt, z. B. in einem der Pariser Akademie überreichten Mémoire: ``Sur les équations modulaires'', das in den Atti a. Acc. d. Lincei (3) I. 1877 gedruckt wurde. Zuerst werden hier die Singularitäten für den Fall untersucht, wo \(N\) keinen quadratischen Theiler hat; dann folgt der entgegengesetzte Fall. Für die Charakteristiken und Singularitäten der Modularcurve \(C\) ergeben sich folgende Resultate. Es sind \(g\) und \(g'\) conjugirte Theiler von \(N\); \(h^2\) ist das grösste in \(N\) enthaltene Quadrat; \(\eta\) der grösste gemeinsame Theiler von \(g\) und \(g'\); \(f(\eta)\) die Anzahl der Zahlen, die \(\leqq\eta\) und relativ prim zu \(\eta; f'(g)\) und \(f'(g')\) sind definirt durch die Gleichung
\[
\frac{f'(g)}g=\frac{f(\eta)}\eta=\frac{f'(g')}{g'};
\]
und es ist
\[
2\nu=\sum f(\eta),\quad A+B=\sum f'(g),\quad A_2+B_2=(A+B)^2,
\]
wo die Summe \(\sum\) sich auf alle Theiler \(g\) von \(N\) erstreckt, wo ferner \(A\) alle \(f'(g)\) umfasst, für welche \(g>\sqrt N\), und zugleich, wenn \(N=\vartheta^2\), den Werth \(\frac12\vartheta'=\frac12 f(\vartheta)\), und wo \(A_2\) alle Terme von \(\sum f'(g_1)\cdot\sum f'(g_2)\), in denen \(g_1g_2>N\), und die Hälfte aller Terme, wofür \(g_1g_2=N\) ist, enthält. Die Definitionen von \(B\) und \(B_1\) ergeben sich analog aus denen von \(A\) und \(A_1\). Ferner bezeichnen \(m,n,K,J,D, T, H,\) resp. die Ordnung, die Klasse, die Spitzen- und Flexions-Singularität, die Discriminanten-Ordnung und Klasse und das Geschlecht der Curve. Die Charakteristiken der Curve genügen dann folgenden Gleichungen:
\[
\begin{matrix} \r &\l \\ m &=2A,\\ n &=3A-B-\vartheta',\\ H &=\frac12(A+B)-3\nu+1,\\ K &=2(A+B)-6\nu+\vartheta',\\ J &=5A-B-6\nu-2\vartheta',\\ J-K &=3A-3B-3\vartheta',\\ D &=4A^2-5A+b+\vartheta',\\ T &=(3A-B-\vartheta')^2-5A+B+\vartheta',\\ T-D &=(3A-B-\vartheta')^2-4A^2. \end{matrix}
\]
Bezeichnen ferner die Symbole \((XXY)\) oder \((YXX)\) einen Zweig, oder ein Aggregat von Zweigen, die die Linie \(XY\) im Punkte \(X\) berühren; die Symbole \(O(XXY), C(XXY), K(XXY), J(XXY), D(XXY), T(XXY)\) die entsprechenden oben angegebenen Charakteristiken der Zweige \((XXY)\), so hat man:
\[
\begin{matrix} \l\\ O(PPQ)=A-B;\\ C(PPQ)=B-\frac12\vartheta',\\ K(PPQ)=A-B-\nu+\frac12\vartheta';\\ J(PPQ)=B-\nu;\\ D(PPQ)=2A^2-A_2-A+\frac12\vartheta';\\ T(PPQ)=B_2-B^2_a\vartheta'B+\frac12\vartheta'+\frac14\vartheta'';\end{matrix}
\]
etc. Auf gleiche Weise werden die Charakteristiker von \((PRR), (QRR), (PSS), (QSS)\) etc. und die Symbole \(O(X), K(X), D(X), C(XY), J(XY), T(XY)\) etc. behandelt, welche eine ähnliche Bedeutung für die Zweige haben, die durch einen gegebenen Punkt \(X\) gehen, oder eine gegebene Linie \(XY\) berühren. Im Folgenden werden die 6 Modularcurven discutirt, welche den 6 Transformationen
\[
x,\quad 1-x,\quad \frac1x, \quad \frac1{1-x}, \quad \frac x{x-1}, \quad \frac{x-1}x
\]
entsprechen, für welche die Modulargleichung ungeändert bleibt. Singularities; elliptic functions; modular curves; characteristic of a curve; modular equations Curves in algebraic geometry, Singularities in algebraic geometry, Elliptic functions and integrals On the singularities of the modular equations and 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 [For the entire collection see Zbl 0745.00066.]
Let \(k\) be a field of characteristic zero and \({\mathcal O} = k[[x]]\) the ring of formal power series in one variable. This paper is concerned with simple modules over the ring of differential operators \({\mathcal D} = {\mathcal D}({\mathcal O}) = k[[x]][\partial]\). In this one-dimensional case the simple modules are holonomic. Those with regular singularities are \(\mathcal O\), \(K/ {\mathcal O}\), and \(Kx^ \alpha\), for \(\alpha \in k \setminus \mathbb{Z}\) (where \(K\) denotes the field of fractions of \(\mathcal O\)). The main effort of this paper is to describe the simple \(\mathcal D\)-modules with irregular singularities. The authors achieve this by a rather elegant descent procedure. These modules are finite-dimensional \(K\)-vector spaces. A result of the second author states that \(\partial\) has an eigenvector over a finite field extension of \(K\). The authors use this to give an extremely explicit description of the simple \(\mathcal D\)-modules. For each finite-dimensional \(K\)-vector space \(M\) they describe all the actions of \(\partial\) on \(M\) that make \(M\) into a simple \(\mathcal D\)-module with irregular singularities. They also extend this classification to \({\mathcal D}(A)\), the ring of differential operators on a subalgebra \(A\) of \(\mathcal O\) with finite codimension, using Smith and Stafford's Morita equivalence between \({\mathcal D}(A)\) and \(\mathcal D\). holonomic modules; ring of formal power series; simple modules; ring of differential operators; regular singularities; irregular singularities; eigenvector; finite field extension; actions; Morita equivalence Den Essen, A. Van; Levelt, A.: An explicit description of all simple K[[X]][\partial]-modules. (1992) Rings of differential operators (associative algebraic aspects), Simple and semisimple modules, primitive rings and ideals in associative algebras, Commutative rings of differential operators and their modules, Valuations, completions, formal power series and related constructions (associative rings and algebras), Singularities of curves, local rings An explicit description of all simple \(k[[x]][\partial]\)-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 Abelian conformal field theory, that is the theory of free fermions over a compact Riemann surface, has been studied, since the late 1980s, from several points of view. The first rigorous mathematical foundation of abelian conformal field theory was provided by the work of \textit{N. Kawamoto}, \textit{Y. Namikawa}, \textit{A. Tsuchiya} and \textit{Y. Yamada} [Commun. Math. Phys. 116, No. 2, 247-308 (1988; Zbl 0648.35080)], Y. Shimizu and K. Ueno. Their approach uses the framework of Sato's universal Grassmannian manifold, the Krichever correspondence between data on Riemann surfaces and soliton equations, complex cobordism theory, formal groups, and other well-developed theories. In his recent paper ``On conformal field theory'' [in: Vector bundles in algebraic geometry, Proc. 1993 Durham Symp., Lond. Math. Soc. Lect. Note Ser. 208, 283-345 (1995; Zbl 0846.17027)] \textit{K. Ueno} proposed another approach to abelian conformal field theory. Namely, taking the vertex operator algebra constructed from the Heisenberg algebra as a gauge group, and applying the basic constructions in the nonabelian conformal theory to this situation, the relationship between conformal blocks and theta functions of higher level gives another mathematically rigorous description of the physicists' operator formalism in the fermionic theory.
The aim of the present paper is to give a geometric interpretation of one of Ueno's results stated in this context. More precisely, Ueno had described the spaces of vacua (conformal blocks) by imposing an extra gauge condition expressed by vertex operators of even level \(M\) among the Fock spaces. His main result, in this regard, consisted then in establishing an isomorphism between the space of conformal blocks and the space of \(M\)th-order theta functions on the Jacobian of the underlying pointed stable curve \(C\).
In the paper under review, the author points out that Ueno's result can be extended to the case of level \(M = 1\), and he gives another description of the space of conformal blocks in this particular odd-level case.
The basic framework used here is, on the one hand, the Beilinson-Bernstein theory of localizations for representations of certain infinite-dimensional Lie algebras associated with special ``dressed'' moduli spaces and, on the other hand, a version of Skornyakov's theory of the \(\pi\)-Picard group of locally free sheaves on supercurves (for \(N=1\) or \(N=2)\) with symmetry group \(\pi\). The respective general theories of Beilinson-Bernstein [cf. \textit{A. A. Beilinson} and \textit{V. V. Schechtman}, Commun. Math. Phys. 118, No. 4, 651-701 (1988; Zbl 0665.17010)] and of Skornyakov-Manin [cf. \textit{Yu. I. Manin}, Topics in non-commutative geometry (1991; Zbl 0724.17007)] are tailored to the particular context of abelian conformal field theory, and this program forms the main body of the paper. The central results are then the following theorems:
(1) The \(\pi\)-Picard functor for a proper smooth supercurve of dimension \(N = 1\) or \(N = 2\) is representable by a smooth superscheme, the so-called \(\pi\)-Picard variety.
(2) The space of (Ueno's) conformal blocks equals a fiber of the Beilinson-Bernstein localization of the Fock representation on the \(\pi\)-Picard scheme of the respective supercurve.
(3) In the case of level \(M = 1\), the space of conformal blocks is canonically isomorphic to the space of theta functions on the Jacobian of the underlying pointed stable curve \(C\).
As for related and further results, the author refers to two forthcoming papers entitled ``Moduli of \(N = 1\) stable superconformal curves and abelian conformal field theory'' and ``Moduli of \(N = 2\) superconformal curves''. super Riemann surfaces; dressed moduli spaces; Picard group; Picard functor; Picard variety; abelian conformal field theory; vertex operator algebra; Heisenberg algebra; conformal blocks; theta functions of higher level; infinite-dimensional Lie algebras; supercurves; Beilinson-Bernstein localization; Fock representation Virasoro and related algebras, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Riemann surfaces, Algebraic functions and function fields in algebraic geometry, Theta functions and abelian varieties, Families, moduli of curves (algebraic) Abelian conformal field theory and \(N=2\) supercurves | 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 0566.14001. quotients by vector fields; characteristic p; discriminantal; locus; semi-simple derivations; quotient; singularities; Cohen-Macaulay singularities; p-radical descent; class groups of normal domains Aramova, A., Avramov, L.: Singularities of quotients by vector fields in characteristicp. Math. Ann.273, 629--645 (1986) Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Finite ground fields in algebraic geometry, Geometric invariant theory Singularities of quotients by vector fields in characteristic p | 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 \(\mathfrak{gl}(n)\), 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. These results are joint work with Oleg Sheinman [Russ. Math. Surv. 63, No. 4, 727--766 (2008); translation from Usp. Mat. Nauk 63, No. 4, 131--172 (2008; Zbl 1204.17016)]. Lie algebras of current type; local cocycles; central extensions; Krichever-Novikov type algebras; Tyurin parameters Applications of Lie algebras and superalgebras to integrable systems, Relationships between algebraic curves and integrable systems, Infinite-dimensional Lie (super)algebras, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 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 Let \(G\) be an algebraic group scheme over an algebraically closed field \(k\) of characteristic 0. Let \(K=k[[t]]\). Let \(UG\) be the kernel of \(G(K)\to G(k)\). The author looks at the question of recovering \(G\) from \(UG\). It is shown that the related question for a Lie algebra can be answered when the Lie algebra is semi-simple. As a consequence, an answer is derived when \(G\) is semi-simple and a linear representation of \(G\) over \(k\) is specified. semisimple algebraic groups; prounipotent groups; algebraic group schemes; Lie algebras; linear representations Andy R. Magid, Prounipotent prolongation of algebraic groups, Recent progress in algebra (Taejon/Seoul, 1997) Contemp. Math., vol. 224, Amer. Math. Soc., Providence, RI, 1999, pp. 169 -- 187. Linear algebraic groups over the reals, the complexes, the quaternions, Group schemes, Lie algebras of Lie groups Prounipotent prolongation 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 Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups. contractions; Lie algebras; affine algebraic groups; affine group schemes Burde, Dietrich, Contractions of Lie algebras and algebraic groups, Arch. Math. (Brno), 43, 5, 321-332, (2007) Group schemes, Applications of Lie (super)algebras to physics, etc., Lie algebras and Lie superalgebras, Linear algebraic groups and related topics, Finite-dimensional groups and algebras motivated by physics and their representations Contractions of Lie algebras and 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 \(G\) be a simple algebraic group and \(P\) a parabolic subgroup of \(G\). The group \(P\) acts on the Lie algebra \({\mathfrak p}_u\) of its unipotent radical \(P_u\) via the adjoint action. The modality of this action, \(\text{mod}(P:{\mathfrak p}_u)\), is the maximal number of parameters upon which a family of \(P\)-orbits on \({\mathfrak p}_u\) depends. More generally, we also consider the modality of the action of \(P\) on an invariant subspace \(\mathfrak n\) of \({\mathfrak p}_u\), that is \(\text{mod}(P:{\mathfrak n})\). In this note we describe an algorithmic procedure, called MOP, which allows one to determine upper bounds for \(\text{mod}(P:{\mathfrak n})\).
The classification of the parabolic subgroups \(P\) of exceptional groups with a finite number of orbits on \({\mathfrak p}_u\) was achieved with the aid of MOP. We describe the results of this classification in detail in this paper. In view of the results of \textit{L. Hille} and \textit{G. Röhrle} [Transform. Groups 4, No. 1, 35-52 (1999; Zbl 0924.20035)], this completes the classification of parabolic subgroups of all reductive algebraic groups with this finiteness property.
Besides this result we present other applications of MOP, and illustrate an example. linear algebraic groups; Lie algebras; modality of parabolic subgroups Jürgens, U.; Röhrle, G.: MOP - algorithmic modality analysis for parabolic group actions. Experimental math. 11, No. no. 1, 57-67 (2002) Linear algebraic groups over arbitrary fields, Lie algebras of linear algebraic groups, Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Symbolic computation and algebraic computation MOP -- algorithmic modality analysis for parabolic 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 The authors consider flat families of embedded reduced curves on a smooth surface \(S\) such that for each member \(C\) of the family the number of singular points of \(C\) and for each singular point \(x\in C\) the analytic (resp. the equisingular) type of \((C,x)\) is fixed, then they obtain a locally closed subscheme \(H_S^{ea}\) (resp. \(H_S^{es}\)) of the Hilbert scheme \(H_S\) of \(S\). They study local properties of \(H_S^{ea}\) (resp. \(H_S^{es})\) and give conditions on the smoothness of these subschemes. They improve previous results for curves in \(\mathbb{P}^2\) and give answer to some existing problem of curves of given degree having a fixed number of singularities of given analytic type. equianalytic singularity; flat families of embedded reduced curves; number of singular points; Hilbert scheme; singularities of given analytic type Greuel G.-M., Lossen C.,Equianalytic and equisingular families of curves on surfaces, Manuscripta Math.,91 (1996), 323--342. Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Singularities of curves, local rings, Equisingularity (topological and analytic) Equianalytic and equisingular families of curves on 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 extensive paper the rough classification in the sense of Enriques-Kodaira is done for the Hilbert modular surfaces \(X_0 ({\mathfrak n})\) of geometric genus \(p_g \leq 1\) associated to the subgroup \(\Gamma_0 ({\mathfrak n})\) of all matrices \(\left( \begin{smallmatrix} a & b \\ c & d \end{smallmatrix} \right)\) of the full Hilbert modular group such that \(c \equiv 0 ({\mathfrak n})\). Here \({\mathfrak n}\) is an integral ideal in the ring of integers of a real quadratic number field of class-number 1, i.e. \(\mathbb{Q} (\sqrt p)\), \(p\) prime \(\equiv 1(4)\) or \(p = 2\). Since the geometric genus equals the dimension of the cusp-forms of weight 2, \textit{H. Saito}'s formula [J. Math. Kyoto Univ. 24, 285-303 (1984; Zbl 0547.10027)] first yields \(p = 2\), 5, 13, 17 in which cases the ideals in question are determined as well as their norms, the number of elliptic fixed points [\textit{A. Prestel}, Math. Ann. 177, 181-209 (1968; Zbl 0159.113)] and the order of the fundamental group. Finally the classification is achieved following \textit{F. Hirzebruch}'s and \textit{A. Van de Ven}'s method [Invent. Math. 23, 1-29 (1974; Zbl 0296.14020)] in studying line configurations on the surface.
Contrary to the case of principal congruence subgroups where each surface of geometric genus 1 is a \(K3\) surface [\textit{G. van der Geer}, Topology 18, 29-39 (1979; Zbl 0418.14021)], in the present situation both honestly elliptic surfaces and surfaces of general type with geometric genus 1 do appear. classification of Hilbert modular surfaces; geometric genus; elliptic surfaces; surfaces of general type Families, moduli, classification: algebraic theory, Modular and Shimura varieties, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Arithmetic ground fields for surfaces or higher-dimensional varieties, Special surfaces Hilbert modular surfaces with \(p_g\leq 1\) | 0 |
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