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The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a finite group of automorphisms of a non-singular three-dimensional complex variety \(M\), whose canonical bundle \(\omega_M\) is locally trivial as a \(G\)-sheaf. We prove that the Hilbert scheme \(Y=G \text{-Hilb }{M}\) parametrising \(G\)-clusters in \(M\) is a crepant resolution of \(X=M/G\) and that there is a derived equivalence (Fourier-Mukai transform) between coherent sheaves on \(Y\) and coherent \(G\)-sheaves on \(M\). This identifies the K-theory of \(Y\) with the equivariant K-theory of \(M\), and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible. quotient singularities; McKay correspondence; derived categories; group of automorphisms; three-dimensional complex variety; Hilbert scheme; crepant resolution; Fourier-Mukai transform; equivariant K-theory T.~Bridgeland, A.~King, and M.~Reid. Mukai implies McKay: the McKay correspondence as an equivalence of derived categories. \(ArXiv Mathematics e-prints\), August 1999. Automorphisms of surfaces and higher-dimensional varieties, Derived categories, triangulated categories, Global theory and resolution of singularities (algebro-geometric aspects), Equivariant \(K\)-theory, Group actions on varieties or schemes (quotients), \(3\)-folds The McKay correspondence as an equivalence of derived categories | 1 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Gamma\) be a Fuchsian group of genus at least 2 (at least 3 if \(\Gamma\) is non-oriented). We study the spaces of homomorphisms from \(\Gamma\) to finite simple groups \(G\), and derive a number of applications concerning random generation and representation varieties. Precise asymptotic estimates for \(|\Hom(\Gamma,G)|\) are given, implying in particular that as the rank of \(G\) tends to infinity, this is of the form \(|G|^{\mu(\Gamma)+1+o(1)}\), where \(\mu(\Gamma)\) is the measure of \(\Gamma\). We then prove that a randomly chosen homomorphism from \(\Gamma\) to \(G\) is surjective with probability tending to 1 as \(|G|\to\infty\). Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties \(\Hom(\Gamma,\overline G)\), where \(\overline G\) is \(\text{GL}_n(K)\) or a simple algebraic group over \(K\), an algebraically closed field of arbitrary characteristic. A key ingredient of our approach is character theory, involving the study of the `zeta function' \(\zeta^G(s)=\sum\chi(1)^{-s}\), where the sum is over all irreducible complex characters \(\chi\) of \(G\). Fuchsian groups; spaces of homomorphisms; finite simple groups; random generation; asymptotic estimates; randomly chosen homomorphisms; Lang-Weil estimates; dimensions of representation varieties; simple algebraic groups; zeta functions; irreducible complex characters M. W. Liebeck and A. Shalev, 'Fuchsian groups, finite simple groups and representation varieties', \textit{Invent. Math.}159 (2005) 317-367. Fuchsian groups and their generalizations (group-theoretic aspects), Simple groups: alternating groups and groups of Lie type, Probabilistic methods in group theory, Other Dirichlet series and zeta functions, Representations of finite groups of Lie type, Group actions on varieties or schemes (quotients), Linear algebraic groups over arbitrary fields, Simple groups, Arithmetic and combinatorial problems involving abstract finite groups Fuchsian groups, finite simple groups and representation varieties. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The paper deals with finite-dimensional algebras \(\Sigma\) over an algebraically closed field \(k\) whose derived category \({\mathcal D}^b(\text{mod }\Sigma)\) of the category \(\text{mod }\Sigma\) of finite-dimensional modules over \(\Sigma\) is equivalent to the derived category \({\mathcal D}^b(\text{coh }\mathbb{X})\) of the category of coherent sheaves on a weighted projective line \(\mathbb{X}\) in the sense of \textit{W. Geigle} and \textit{H. Lenzing} [in: Singularities, representations of algebras, and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)].
To an almost concealed-canonical algebra, which by definition is an algebra of the form \(\text{End}(T)\), where \(T\) is a tilting sheaf in \(\text{coh }\mathbb{X}\) [see \textit{H. Lenzing} and the reviewer in: Representation theory of algebras, CMS Conf. Proc. 18, 455-473 (1996; Zbl 0863.16013)], there is associated the Tits quiver whose vertices are the indecomposable direct summands \(P_m\) of \(T\). Moreover, the Tits quiver has integer-valued arrows \(P_m@>-d_{m,l}>>P_l\) where \(d_{m,l}\) denotes the Euler form \(\langle S_l,S_m\rangle\) applied to the corresponding simple \(\Sigma\)-modules.
The author shows that the rank function \(rk\colon K_0(\mathbb{X})\to\mathbb{Z}\) satisfies an additive property which is expressed in terms of the bigraph associated to the Tits quiver. Further, the author proves that the result extends to the representation-infinite quasitilted algebras of canonical type in the sense of \textit{H. Lenzing} and \textit{A. Skowroński} [Colloq. Math. 71, No. 2, 161-181 (1996; Zbl 0870.16007)]. Finally, it is shown that the rank additivity does not generalize to the case of algebras \(\Sigma\) which are given by tilting complexes in \({\mathcal D}^b(\text{coh }\mathbb{X})\). finite-dimensional algebras; derived categories; finite-dimensional modules; categories of coherent sheaves; weighted projective lines; almost concealed-canonical algebras; tilting sheaves; Tits quivers; indecomposable direct summands; Euler forms; rank functions; representation-infinite quasitilted algebras; rank additivity; tilting complexes Thomas Hübner, Rank additivity for quasi-tilted algebras of canonical type, Colloq. Math. 75 (1998), no. 2, 183 -- 193. Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Derived categories, triangulated categories, Vector bundles on curves and their moduli Rank additivity for quasi-tilted algebras of canonical 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 In recent years many new (co)homology theories of (non)commutative (operator) algebras have been discovered: The homological \(K\)-functor Ext of \textit{L. G. Brown}, \textit{R. G. Douglas} and \textit{P. A. Fillmore} [see Bull. Am. Math. Soc. 79, 973--978 (1973; Zbl 0277.46052)] having after all some interesting application in the structure theory of \(C^*\)-algebras [see the reviewer, Funkts. Anal. Prilozh. 9, No. 1, 63--64 (1975; Zbl 0347.46068)], Kasparov's \(KK\)-functor [see \textit{G. G. Kasparov}, Izv. Akad. Nauk SSSR, Ser. Mat. 44, 571--636 (1980; Zbl 0448.46051)], Connes' cyclic homology \(HC\)-functor [see \textit{M. Karoubi}, C. R. Acad. Sci., Paris, Sér. I 297, 381--384, 447--450, 513--516, 557--560 (1983; Zbl 0528.18008, Zbl 0528.18009, Zbl 0528.18010, Zbl 0532.18009)], and the Gel'fand-Fuks cohomology theory of infinite-dimensional Lie algebras. The most important point in these developments is replacing the structural function sheaf by an arbitrary (non)commutative algebra. The paper under review is a new account of this fact, where the algebra of functions on a manifold is replaced by a (non)commutative algebra.
One considers for each algebra \(A\) over a field \(K\) of characteristic zero the standard bimodule resolution
\[
R_* :\quad... \to A\otimes A\otimes A\to^{\delta}A\otimes A\to^{\delta}A\to 0
\]
with the differential \(\delta (a_ 1\otimes...\otimes a_ k)=\sum^{k-1}_{i=1}(-1)^ ia_ i\otimes...\otimes a_ ia_{i+1}\otimes...ca_ k\), the standard complex \(C_*(A)\) of the Lie algebra \(\mathfrak{gl}_{\infty}(A)\) of infinite matrices of finite type over \(A\) with coefficients in the unit module, and finally a complex homomorphism
\[
\phi: C_*(A)\to (R_*,\delta),\quad \phi ((m_ 1\otimes a_ 1)\wedge...\wedge (m_ k\otimes a_ k))=\sum_{i}\text{sgn}(i)\text{tr}(m_{i_ 1}...m_{i_ k}) a_{i_ 1}\otimes...\otimes a_{i_ k}.
\]
The image of \(\phi\) is the subcomplex \(\bar R_*\subset R_*\) consisting of chains that are invariant under the transfer map \(t\), \(t(a_ 1\otimes...\otimes a_ k)=(- 1)^{k-1}a_ k\otimes a_ 1\otimes...\otimes a_{k-1}\). Via \(\phi\), the additive \(K\)-functor group
\[
K^+_*(A)=\lim_{\leftarrow} H_*(\mathfrak{gl}_{\infty}(A),K)
\]
is isomorphic [\textit{J.-L. Loday} and \textit{D. Quillen}, C. R. Acad. Sci. Paris, Sér. I 296, 295--297 (1983; Zbl 0536.17006); \textit{B. L. Tsygan}, Usp. Mat. Nauk 38, No. 2 (230), 217--218 (1983; Zbl 0518.17002)] to Connes' cyclic cohomology group \(HC^*(A)=H_*(\bar R_*,\delta).\)
The authors show that \(K^+_*(A)\) are the higher derived functors of the functor \(K^+_ 1\) in the category of algebras:
\[
K^+_ 1(R)\simeq R/[R,R],\quad H_*(R/[R,R])\simeq \lim_{\leftarrow} H_*(\mathfrak{gl}_{\infty}(A),\mathfrak{gl}_{\infty}(K);K).
\]
On the category of arrows \(A\to^{f}B\) one defines (Th. 1) the relative additive \(K^+\)-functor \(K^+_*(B,A)\) as the derived functors of \(K^+_ 0(B,A)=B/([B,B]+\text{Im}\, f)\). One has (Th. 2) an exact sequence
\[
... \to K^+_{i+1}(C,B) \to K^+_ i(B,A) \to K^+_ i\quad (C,A) \to K^+_ i(C,B) \to...
\]
associated with the pair of arrows \(A\to^{f}B\to^{g}C.\)
The authors propose also (in Section 2) a construction independent from the Connes-Karoubi one of the Chern characters \(Ch: K_ n(A)\to K^+_{n+2i+1}(A)\), \(i\in\mathbb Z_+\). Then, the Bott periodicity morphism \(K^+_ i(A)\to K^+_{i-2}(A)\) exists, and hence, the groups \(K^+_ i\) with i of fixed parity form an inverse system.
The authors define the cohomology of an algebra \(A\) as
\[
H^{\text{odd}}(A)=\lim_{\leftarrow}K^+_{2i}(A) \oplus R^ 1\lim_{\leftarrow}K^+_{2i+1}(A),\quad H^{\text{ev}}(A)=\lim_{\leftarrow}K^+_{2i+1}(A) \oplus R^ 1\lim_{\leftarrow}K^+_{2i\quad}(A).
\]
In the case where \(A\) is commutative (Section 3), \(H^{\text{odd}}(A)+H^{\text{ev}}(A)\) is isomorphic to the crystalline cohomology of the spectrum of A and there exists a morphism
\[
K^+_ n(A)\to \oplus_{N}H^*(X_{\text{cris}}, \mathcal O_ X/J^ N).
\]
If \(R\) is a free commutative differential graded algebra for which there exists an epimorphic quasiisomorphism \(R\to A\) the authors consider the de Rham complex \(R: R\to^{d}\Omega^ 1_ R\quad \to \quad...\) and prove that \(K^+_*(A,K)\simeq H_*(\Omega^*_ R/(K+d\Omega^*_ R))\). This just coincides with Connes' result in the case where \(\text{Spec}\,A\) is a nonsingular variety. cohomology theories of noncommutative operator algebras; Lie; algebra of infinite matrices of finite type; homological K-functor; \(C^*\)-algebras; Kasparov's KK-functor; cyclic homology; Gel'fand-Fuks cohomology theory of infinite-dimensional Lie; algebras; additive K-functor; derived functors; Chern characters; Bott periodicity; crystalline cohomology; differential graded algebra; de Rham complex; Gel'fand-Fuks cohomology theory of infinite-dimensional Lie algebras Feĭgin, Boris; Tsygan, Boris, Additive \(K\)-theory and crystalline cohomology, Funktsional. Anal. i Prilozhen., 19, 2, 52-62, 96, (1985) \(K\)-theory and operator algebras, Ext and \(K\)-homology, Kasparov theory (\(KK\)-theory), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Categories, functors in functional analysis, Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), \(p\)-adic cohomology, crystalline cohomology, Cohomology of Lie (super)algebras, \(K\)-theory and operator algebras (including cyclic theory), Continuous cohomology of Lie groups, Resolutions; derived functors (category-theoretic aspects), Topological \(K\)-theory, Homology of classifying spaces and characteristic classes in algebraic topology, Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity, Graded rings and modules (associative rings and algebras), Classifying spaces for foliations; Gelfand-Fuks cohomology Additive \(K\)-theory and crystalline cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 simply connected split semisimple algebraic group of rank \(r\) over a field \(\mathbb K\) of characteristic zero and let \(B\) denote the standard Borel subgroup of \(G\) corresponding to the positive roots. Moreover, let \(\mathfrak g\) denote the Lie algebra of \(G\), let \(q\) be an element in \(\mathbb K\) that is transcendental over \(\mathbb Q\) and let \(\mathcal U_q(\mathfrak g)\) denote the quantized universal enveloping algebra of \(\mathfrak g\) at \(q\) with standard generators \(X_1^\pm,\dots,X_r^\pm\) and \(K_1^{\pm 1},\dots,K_r^{\pm 1}\). For \(I\subseteq\{1,\dots,r\}\) denote by \(P_I\supseteq B\) the standard parabolic subgroup of \(G\) corresponding to \(I\). Finally, let \(R_q[G/P_I]\) denote the quantized coordinate ring of the partial flag variety \(G/P_I\). The ring \(R_q[G/P_I]\) is a deformation of the coordinate ring of the multicone over \(G/P_I\) and is invariant under the conjugation action of the group-like elements \(H:=\langle K_1^{\pm 1},\dots,K_r^{\pm 1}\rangle\) of \(\mathcal U_q(\mathfrak g)\).
The goal of the paper under review is to study the \(H\)-invariant prime ideals of \(R_q[G/P_I]\) that do not contain the augmentation ideal. The main result is a bijection between these invariant prime ideals and certain pairs of Weyl group elements. In particular, all these invariant prime ideals are completely prime. Moreover, the author conjectures that the inclusion of invariant prime ideals is reflected in the Bruhat order of the components of the parametrizing pairs. The proof of the main result uses techniques of \textit{A. Joseph} [Quantum groups and their primitive ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3.\ Folge. 29. Berlin: Springer-Verlag (1995; Zbl 0808.17004)] similar to those used in establishing a natural bijection between primitive ideals in \(R_q[\mathrm{SL}(n)]\) and symplectic leaves of the associated Poisson group \(\mathrm{SL}(n,\mathbb C)\) due to \textit{T. J. Hodges} and \textit{T. Levasseur} [J.\ Algebra 168, No.\ 2, 455-468 (1994; Zbl 0814.17012)] as well as in \textit{M. Gorelik}'s investigation of the spectra of quantum Bruhat cell translates [J.\ Algebra 227, No.\ 1, 211-253 (2000; Zbl 1038.17006)]. As a consequence one obtains a stratification of the \(H\)-invariant prime spectrum of \(R_q[G/P_I]\) and then the strata are related to \(H\)-invariant prime ideals of the algebras investigated by the author in a previous paper [Proc.\ Lond.\ Math.\ Soc.\ (3) 101, No.\ 2, 454-476 (2010; Zbl 1229.17020)]. Analogous results are also obtained for quantum deformations of the coordinate rings of the cones \(\mathrm{Spec}\left(\bigoplus_{n\in\mathbb Z_{\geq 0}}H^0(G/P_I,\mathcal L_{n\lambda})\right)\) over \(G/P_I\) for certain dominant weights. quantum partial flag varieties; invariant prime ideals; completely prime ideals; stratifications; simply connected split semisimple algebraic groups; Lie algebras of algebraic groups; quantized universal enveloping algebras; quantized coordinate rings Milen Yakimov, A classification of \(H\)-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1249-1261. Ring-theoretic aspects of quantum groups, Quantum groups (quantized enveloping algebras) and related deformations, Lie algebras of linear algebraic groups, Ideals in associative algebras, Linear algebraic groups over the reals, the complexes, the quaternions, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations A classification of \(H\)-primes of quantum partial flag varieties. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X_{m,n}=M_n(\mathbb{C})^{\oplus m}\) be the affine space of \(m\)-tuples of \(n\times n\) matrices with the action of \(\text{GL}_n\) given by simultaneous conjugation, where \(m\geq 2\) and one may replace \(\mathbb{C}\) by any algebraically closed field of characteristic 0. The problem of classifying the orbits of \(X_{m,n}\) is considered as one of the most ``hopeless'' problems in algebra. The first approximation to this problem is the description of the quotient variety \(V_{m,n}\) with coordinate ring coinciding with the ring of invariant polynomial functions \(\mathbb{C}[V_{m,n}]=\mathbb{C}[X_{m,n}]^{\text{GL}_n}\), because the points of \(V_{m,n}\) classify the closed orbits in \(X_{m,n}\).
In this important paper the author outlines a procedure which allows to study better the highly singular variety \(V_{m,n}\). It involves various classical and recent techniques. In particular, the method relies on joint work of the author and Procesi on étale local structure of matrix invariants and on the recent results of the author on the nullcone of representations of quivers. As an illustration of the power of his method the author works out the case \(n=2\) and any \(m\) recovering the known results for \(m=2\) and gives a short account on the results for \(m\)-tuples of \(3\times 3\) matrices. affine spaces of tuples of matrices; simultaneous conjugation of tuples of matrices; matrix invariants; representations of quivers; actions of general linear groups; orbits; rings of invariant functions; singular varieties Trace rings and invariant theory (associative rings and algebras), Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients) Orbits of matrix tuples | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Lie theory, in our contemporary understanding, is one of the central areas in mathematics and mathematical physics. Initiated by Sophus Lie's pioneering program of studying continuous transformation groups (Lie groups) in a systematic way, begun in the second half of the 19th century, Lie theory has rapidly grown into one of the most important, active, propelling and ubiquitous topics of modern mathematics in the 20th century. In fact, Lie groups, Lie algebras, linear algebraic groups, their representation theory, and their invariant theory have become part of the very foundations of modern mathematics, involving a virtually unique blend of algebraic, analytic, geometric, topological and combinatorial aspects and methods. Also, Lie theory has utmost significant applications in modern theoretical physics, especially in quantum theory and relativity theory.
Accordingly, there is a large number of textbooks and monographs on topics in Lie theory, which generally differ with regard to their viewpoint, purpose, coverage, methodology, level, and prerequisites.
The textbook under review, presented by one of the most renowned researchers and teachers in various fields of algebra and geometry, provides another introduction to the subject in a very original and uniquely multifarous way. Namely, in contrast to the majority of the majority existing introductory texts on Lie groups, the author has tried to present the various different aspects of Lie theory as a whole and in a unified manner, thereby combining the general theory with many concrete examples and applications. Simultaneously, the exposition is really meant to be introductory, with the prerequisites kept to a minimum and with a wealth of related background material provided throughout the text. Due to the author's expertise, invariant theory and its relations to Lie theory has been chosen as the guiding pedagogical principle for this comprehensive account of the subject, which appears to be fairly unique in the relevant textbook literature.
After about 25 years of development from lecture notes and courses taught at various universities worldwide, the book under review represents the author's approach to the subject in its complete and final form, thereby covering all the essential material in a single new primer of Lie theory.
The text consists of fifteen chapters, each of which is subdivided into several sections and subsections. The chapters are grouped in such a way that they represent the many different topics treated in the book, some of which can be taken as the subject of an entire, largely independent graduate course.
Chapter 1 gives a first introduction to group actions, orbit spaces, invariants, equivariant maps, and group representations, together with many classical examples. Chapter 2 treats, in this context, symmetric functions, resultants, discriminants, Bézoutians, Schur functions, and the Cauchy formulas as further classical background material. Chapter 3 touches upon another classical topic, namely the invariant theory of algebraic forms à la \textit{A. Capelli} (1902). This material is presented in modern form and includes such basics like the polarization process, the Aronhold method, the Clebsch-Gordan formula, the Capelli identity, covariants, and the Gayley \(\Omega\)-process. In Chapter 4, this invariant-theoretic approach is taken as a pretext for introducing Lie groups and Lie algebras in a systematic way, together with their first fundamental structure theorems. Chapter 5 develops the necessary tensor calculus from scratch and uses this crucial framework to discuss the Clifford algebra, spin groups, some basic constructions in representation theory, the universal enveloping algebra, and free algebras in the sequel.
Chapter 6 provides a short introduction to semisimple algebras, together with an explanation of the related general methods of noncommutative algebra, including matrices over division rings, semisimple modules, Reynolds operators, the double centralizer theorem, Wedderburn's theorem, primitive idempotents, and other related concepts.
Chapter 7 introduces algebraic groups, with a special emphasis on linearly reductive groups and Borel subgroups. This chapter requires some basic knowledge of the underlying elementary algebraic geometry, which the author freely refers to as for additional reading. The material presented here is to stress the parallel theory of reductive algebraic and compact (complex) Lie groups, on the one hand, and to prepare the ground for the general theory of group representations developed in the following chapters, on the other hand.
Chapter 8 begins the study of the representation theory of various groups with extra structure. This chapter essentially focuses on characters, matrix coefficients, the Peter-Weyl theorem, Weyl's ``unitary trick'', and representations of linear reductive groups. Hopf algebras, Hopf ideals, and the Tannaka-Krein duality are briefly introduced at the end of this chapter, mainly in order to describe the link between reductive groups and compact Lie groups more closely later on.
Chapter 9 returns to invariant theory and is titled ``Tensor Symmetry''. Among the topics briefly touched upon here are the First Fundamental Theorem (FFT) of invariant theory for the linear group, Young symmetrizers and Young diagrams, representations of linear groups, polynomial functors and Schur functors, branching rules, and (semi-) standard diagrams.
Chapter 10 deals with the structure and classification of semisimple Lie algebras and their representations via root systems, decomposition methods, dual Hopf algebras, and Cartan-Weyl theory in general. This chapter also contains the corresponding theory of adjoint and simply connected algebraic groups and their compact (Lie) forms.
Chapter 11 offers a deeper study of the relationship between invariants and the representation theory of classical groups (à la H. Weyl), culminating in the corresponding First and Second Fundamental Theorems, spin representations, and Weyl's character formula.
The last four chapters are meant to complement the theory developed so far. Chapter 12 explains some combinatorial aspects of representation theory via the theory of tableaux after Robinson-Schensted, Knuth, Schützenberger, and Littlewood-Richardson, whereas Chapter 13 briefly discusses standard monomials, Grassmannians, flag varieties, Schubert calculus, and further combinatorial tools in the study of invariants and representations of classical groups. Chapter 14 offers a glimpse into geometric invariant theory à la Hilbert-Mumford, and Chapter 15 returns to the classical invariant theory of binary forms in its modern setting (via Hilbert series) and in its computational aspects.
All in all, this unique textbook, combing invariant theory and Lie theory in a very natural way, refers to virtually every (classical and modern) aspect of this deep interrelation, though sometimes in a more concise and informal style. Such a broad panoramic view to the subject is certainly a novelty in the relevant textbook literature and cannot be found anywhere else. The author's effort at providing such a rich source of insight must be seen as being highly welcome, rewarding, valuable and utmost masterly likewise. Lie groups; Lie algebras; algebraic groups; invariant theory; semisimple algebras; representation theory; algebraic forms; tableaux; Schur functions; Hopf algebras C. Procesi, \textit{Lie Groups}, Universitext, Springer, New York, 2007. Research exposition (monographs, survey articles) pertaining to topological groups, General properties and structure of complex Lie groups, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Geometric invariant theory, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Representation theory for linear algebraic groups, Actions of groups on commutative rings; invariant theory Lie groups. An approach through invariants and representations | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(E\) denote an elementary Abelian \(p\)-group of rank \(n\), and let \(k\) denote an algebraically closed field of characteristic \(p>0\). The representation theory of elementary Abelian \(p\)-groups has played a key role in understanding the modular representations of more general finite groups, particularly through the use of cohomological support varieties. In the case of elementary Abelian groups, the Avrunin-Scott Theorem identifies cohomological support varieties with rank varieties which are defined via cyclic shifted subgroups of \(kE\). A cyclic shifted subgroup of \(kE\) is actually a subalgebra of \(kE\) and isomorphic to \(k[t]/(t^p)\). Subalgebras of the form \(k[t]/(t^p)\) have played a key role over the past thirty years in the development of the modular representation theory of various structures. This culminated in a sense with Friedlander and Pevtsova's introduction of the notion of \(\pi\)-points (appropriate flat maps from \(k[t]/(t^p)\) to \(kG\)) to develop a general theory for an arbitrary finite group scheme \(G\).
This work takes a renewed (and quite fruitful) look at the representation theory of elementary Abelian groups by considering the more general notion of a rank \(r\) shifted subgroup \(k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\) for \(1\leq r\leq n\). Let \(V\subset\text{Rad}(kE)\) be an \(n\)-dimensional subspace chosen so that the composite \(V\to\text{Rad}(kE)\to\text{Rad}(kE)/\text{Rad}^2(kE)\) is an isomorphism and consider the variety of Grassmannians \(\text{Grass}(r,V)\) of \(r\)-planes in \(V\). Associated to any \(U\in\text{Grass}(r,V)\) is an algebra \(C(U)\simeq k[t_1,\dots,t_r]/(t_1^p,\dots,t_r^p)\) and an associated flat map \(C(U)\to kE\). For a finite dimensional \(kE\)-module \(M\), the \(r\)-rank variety of \(M\) is defined to be the subset \(\text{Grass}(r,V)_M\) of \(\text{Grass}(r,V)\) consisting of those \(U\) for which the restriction of \(M\) to \(C(U)\) is not free. The variety \(\text{Grass}(r,V)_M\) is shown to be closed in \(\text{Grass}(r,V)\) and essentially dependent only on \(M\) not the choice of \(V\). Unlike the \(r=1\) case, when \(r=2\), every closed subvariety of \(\text{Grass}(2,V)\) cannot be realized as some \(\text{Grass}(2,V)_M\).
For a given \(U\in\text{Grass}(r,V)\) and \(kE\)-module \(M\), one can consider the radical (or socle) of \(M\) upon restriction to \(C(U)\). More generally, one can consider higher radicals (or socles). This can be used to extend Friedlander and Pevtsova's notion of generalized support varieties (for \(r=1\)) to higher \(r\). Specifically, the authors define non-maximal \(r\)-radical (and \(r\)-socle) support varieties. Some computations of such are given including an appendix by the first author that involves some computer calculations.
Further, the authors introduce an analogue of modules of constant Jordan type: modules of constant \(r\)-radical (or \(r\)-socle) type. Again, this is done in full generality for higher radicals (or socles). Having constant radical (or socle) type means that the dimension of the radical (or socle) of \(M\) restricted to \(C(U)\) is independent of the choice of \(U\in\text{Grass}(r,V)\). The authors present a number of examples of such modules and also investigate when the Carlson \(L_\zeta\) modules have such type.
In the second part of the paper, the authors use modules of constant \(r\)-radical or \(r\)-socle type to construct algebraic vector bundles on \(\text{Grass}(r,V)\). Again, numerous examples are given as well as examples of how various standard bundles can be realized by these constructions. elementary Abelian \(p\)-groups; Grassmannians; algebraic vector bundles; rank varieties; support varieties; modules of constant Jordan type; modules of constant radical type; modules of constant socle type; group algebras; cyclic shifted subgroups; finite group schemes Carlson, J. F.; Friedlander, E. M.; Pevtsova, J., Representations of elementary abelian \(p\)-groups and bundles on Grassmannians, Adv. Math., 229, 5, 2985-3051, (2012) Modular representations and characters, Grassmannians, Schubert varieties, flag manifolds, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group rings of finite groups and their modules (group-theoretic aspects), Finite abelian groups, Group schemes, Torsion groups, primary groups and generalized primary groups Representations of elementary Abelian \(p\)-groups and bundles on Grassmannians. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(A\) be an algebra over a field \(k\) and let \(D^b(A)\) be the derived category of bounded complexes of \(A\)-modules. Then, the group \(D\text{Pic}_k(A)\) is defined as the group of isomorphism classes of self-equivalences of \(D^b(A)\) given by the derived tensor product of a complex of \(A\otimes_kA^{\text{op}}\)-bimodules. Observe that this group is denoted by \(\text{TrPic}_k(A)\) by Rouquier and the reviewer [CMS Conf. Proc. 18, 721--749 (1996; Zbl 0855.16015)]. In the paper under review the case of a finite dimensional hereditary \(k\)-algebra \(A\) and algebraically closed field \(k\) is studied.
Let \(\Gamma^{\text{irr}}\) be the union of the non regular components of the Auslander-Reiten quiver of \(D^b(A)\). The group of automorphisms modulo inner automorphisms of \(A\) is an algebraic group and the identity component \(\text{Out}^\circ_k(A)\) of this group is a normal subgroup of \(D\text{PiC}_k(A)\). The main result of the paper under review is the following. The group \(D\text{Pic}_k(A)\) is a semidirect product of \(\text{Out}^\circ_k(A)\) acted upon by the subgroup of those automorphisms \(\Gamma^{\text{irr}}\) which commute with the action of Auslander-Reiten translation and degree shift on \(\Gamma^{\text{irr}}\). Finally, the authors study the case of finite representation type and the case of tame representation type in complete detail. equivalences of derived categories; derived categories of bounded complexes; self-equivalences; finite dimensional hereditary algebras; Auslander-Reiten quivers; finite representation type; tame representation type; Auslander-Reiten translations J. Miyachi and A. Yekutieli, ''Derived Picard groups of finite-dimensional hereditary algebras,'' Compositio Math., vol. 129, iss. 3, pp. 341-368, 2001. Representations of quivers and partially ordered sets, Derived categories, triangulated categories, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Representation type (finite, tame, wild, etc.) of associative algebras, Grothendieck groups, \(K\)-theory, etc., Picard groups Derived Picard groups of finite-dimensional hereditary algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This is the second in a series of papers which give an explicit description of the reconstruction algebra as a quiver with relations; these algebras arise naturally as geometric generalizations of preprojective algebras of extended Dynkin diagrams [cf. Trans. Am. Math. Soc. 363, No. 6, 3101-3132 (2011; Zbl 1270.16022)]. This paper deals with dihedral groups \(G=\mathbb D_{n,q}\) for which all special CM modules have rank one, and we show that all but four of the relations on such a reconstruction algebra are given simply as the relations arising from a reconstruction algebra of type \(A\). As a corollary, the reconstruction algebra reduces the problem of explicitly understanding the minimal resolution (= G-Hilb) to the same level of difficulty as the toric case. reconstruction algebras; quivers with relations; noncommutative resolutions; CM-modules; surface singularities; Cohen-Macaulay singularities; labelled Dynkin diagrams; resolutions of singularities Wemyss, M., Reconstruction algebras of type \textit{D} (I), J. Algebra, 356, 158-194, (2012) Representations of quivers and partially ordered sets, Rings arising from noncommutative algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Cohen-Macaulay modules, Syzygies, resolutions, complexes in associative algebras Reconstruction algebras of type \(D\). 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 \textit{K. Pommerening} showed [in Math. Z. 176, 359-374 (1981; Zbl 0438.14010)], that if \(G=SL_ n\) and H is a subgroup of G which is obtained from the unipotent radical of a parabolic subgroup by removing any set of simple roots then the algebra of regular functions on G which are H-invariant with respect to the action of H on G by right translations is finitely generated. In the paper under review Pommerening's theorem is partially generalized to arbitrary semisimple groups G (though for \(G=SL_ n\) the results are weaker than Pommerening's). finite generation; algebra of invariants; unipotent radical; parabolic subgroup; simple roots; algebra of regular functions; action; semisimple groups Representation theory for linear algebraic groups, Linear algebraic groups over arbitrary fields, Vector and tensor algebra, theory of invariants, Geometric invariant theory, Group actions on varieties or schemes (quotients) On a problem of Pommerening in invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) be a Chevalley-Demazure group scheme. It is well-known that, as a representable covariant functor from the category of commutative rings to the category of groups, \(G\) is uniquely determined by the semisimple complex Lie group \(G (\mathbb{C})\). The author generalizes this result for simple Chevalley-Demazure group schemes, as well as for absolutely almost simple algebraic groups, by replacing the complex field \(\mathbb{C}\) by an integral domain containing an infinite field. representable covariant functor; category of commutative rings; category of groups; Lie group; simple Chevalley-Demazure group schemes; absolutely almost simple algebraic groups Yu Chen, ''Isomorphic Chevalley groups over integral domains,'' \textit{Rend. Sem. Mat. Univ. Padova}, \textbf{92}, 231-237 (1994). Group schemes, Linear algebraic groups over adèles and other rings and schemes, Integral domains, Linear algebraic groups over arbitrary fields Isomorphic Chevalley groups over integral domains | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(A\) be an associative \(K\)-algebra of \(K\)-dimension \(d<\infty\) with a unit, and let \(\text{Mod}(A,n)\) be the affine variety of left \(A\)-module structures on \(K^n\). Let us fix an \(n(d-1)\)-dimensional \(A\)-module \(L\), and denote by \({\mathcal L}_A(n)\) the set of isomorphism classes \([L]\) of \(n(d-1)\)-dimensional submodules of \(A^n\). Let us consider the set \(\text{Mod}(A,n)_L\) of points \(x\in\text{Mod}(A,n)\) such that there is a short exact sequence \(0\to L\to A^n\to K_x\to 0\), where \(K_x\) denotes the module corresponding to \(x\). -- First the author proves that \(\text{Mod}(A,n)_L\) is a smooth irreducible locally closed subset of \(\text{Mod}(A,n)\) whose dimension is equal to \(\dim\Hom_A(L,A^n)-\dim\text{End}_A(L)\). Further, suppose that \(A\) is a subfinite algebra, that is, every projective \(A\)-module has only finitely many isomorphism classes of submodules. Then it is proved that the dimension of \(\text{Mod}(A,n)\) is equal to \(\max_{[L]\in{\mathcal L}_A(n)}\{\dim\Hom_A(L,A^n)-\dim\text{End}_A(L)\}\). The author also determines lower and upper bounds for the number of irreducible components of the varieties of modules \(\text{Mod}(A,n)\), describes some other useful properties of this variety, and considers interesting applications and examples. finite-dimensional algebras; associative algebras; varieties of modules; degenerations of modules; Auslander-Reiten theory; Dynkin quivers Richmond, N.J.: A stratification for varieties of modules. Bull. Lond. Math. Soc. 33(5), 565--577 (2001) Representations of quivers and partially ordered sets, Special varieties, Finite rings and finite-dimensional associative algebras, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers A stratification for varieties of modules. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We derive a root test for degenerations as described in the title. In the case of Dynkin quivers this leads to a conceptual proof of the fact that the codimension of a minimal disjoint degeneration is always one. For Euclidean quivers, it enables us to show a periodic behaviour. This reduces the classification of all these degenerations to a finite problem that we have solved with the aid of a computer. It turns out that the codimensions are bounded by two. Somewhat surprisingly, the regular and preinjective modules play an essential role in our proofs. finite-dimensional algebras; finite-dimensional modules; degenerations of modules; preprojective representations; tame Dynkin quivers Bongartz K. and Fritzsche T. (2003). On minimal disjoint degenerations for preprojective representations of quivers. Math. Comput. 72: 2013--2042 Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients) On minimal disjoint degenerations for preprojective representations of quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Chevalley groups; algebraic group; Kac-Moody-algebra; deformation; simple elliptic singularities; quotient of complex simple Lie group Singularities of surfaces or higher-dimensional varieties, Linear algebraic groups over global fields and their integers, Group actions on varieties or schemes (quotients), Deformations of singularities, Analysis on real and complex Lie groups Chevalley groups over \(\mathbb C((t))\) and deformations of simply elliptic 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 If G is a small finite subgroup of \(SL_ 2({\mathbb{C}})\), McKay computed that the associated diagram to the table of multiplication of representations of G is the extended Dynkin diagram of G. This was explained geometrically later on by Gonzalez-Sprinberg, Verdier, Artin, Knörrer and myself: if (X,0) is the corresponding rational double point, \(\pi: \tilde X\to X\) the minimal desingularization, the first Chern class of \(\pi^*(M)/torsion\), where M is the reflexive hull of the flat module on X-0 associated to an irreducible representation of G, is dual to one and only one exceptional divisor of \(\pi\). The multiplication is computable with Chern classes.
If G now is a small finite subgroup of \(GL_ 2({\mathbb{C}})\), the McKay correspondence does not hold: to one possible Chern class there might correspond several reflexive modules. The author constructs for each exceptional divisor \(E_ i\) a module \(M_ i\) whose corresponding first Chern class is dual to \(E_ i\). He shows how to modify the multiplication. Shortly, if \(M=M_ i\), it behaves as in the rational double point case, if \(M\neq M_ i\), it does not. The formulae are given. As a corollary, the Chern character does not determine M. Further the author gives a complete description in the cyclic case (also through an approach based on invariant theory) and computes some examples of first Chern classes in the general case. two-dimensional quotient singularities; Dynkin diagram; McKay correspondence; reflexive modules; Chern class Wunram, J.: Reflexive Moduln auf zweidimensionalen Quotientensingularitäten. Dissertation Fachber. Math. Univ. Hamburg (1986) Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Characteristic classes and numbers in differential topology Reflexive Moduln auf zweidimensionalen Quotientensingularitäten. (Reflexive modules on two dimensional quotient singularities) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The concept of Beauville surfaces was introduced by F. Catanese in 1997, and in 2000 their global rigidity was shown by \textit{F. Catanese} [Am. J. Math. 122, No. 1, 1--44, (2000; Zbl 0983.14013)]. In 2005, \textit{I. Bauer} et al. [Mediterr. J. Math. 3, No. 2, 121--146 (2006; Zbl 1167.14300)] first systematically studied Beauville surfaces, they constructed many new examples, and presented a lot of conjectures. Quite a number of these conjectures have been solved in the meantime and nowadays there exists a substantial literature on Beauville surfaces.
A Beauville surface \(S\) is a surface isogenous to a higher product of curves, i.e. \(S = (C_1 \times C_2)/G\) is a quotient of a product of two smooth curves \(C_1\) and \(C_2\) of genera at least two, modulo a free action of a finite group \(G\), which is rigid, i.e. it has no non-trivial deformations. Beauville surfaces are either of \textit{mixed} or \textit{unmixed} type, according respectively as the \(G\)-action exchanges the two factors (then \(C_1\cong C_2\)) or \(G\) acts diagonally on the product \(C_1\times C_2\). There is a pure group theoretical condition which characterizes the groups of Beauville surfaces: the existence of a ``Beauville structure''. In the paper under review the author focus on the unmixed case: a finite group \(G\) admits an unmixed Beauville structure if there exist two pairs of generators \((x_1 ,y_1)\) and \((x_2 ,y_2 )\) for \(G\) such that \(\Sigma(x_1 ,y_1)\cap \Sigma(x_2 ,y_2 )=\{1_G\}\), where, \(\Sigma(x,y)\) (for \(x,y\in G\)) is the union of conjugacy classes of all powers of \(x\), all powers of \(y\), and all powers of \(xy\). Moreover the triple \((\mathrm{ord}(x_i), \mathrm{ord}(y_i), \mathrm{ord}(x_iy_i))\) is called the type of \(T_i:=(x_i,y_i)\).
In the paper under review, the authors prove that several finite simple groups admit an unmixed Beauville structure. In particular they show that \(\mathrm{PSL}(2,p^e)\) and some other families of finite simple groups of Lie type of low Lie rank (e.g. Suzuki groups, Ree groups, etc.) admit an unmixed Beauville structure. Moreover for the alternating and symmetric groups they prove the stronger result that almost all of these groups admit an unmixed Beauville structure with fixed types. In particular these last result completely solves Conjecture 7.18 of Bauer, Catanese and Grunewald [Zbl 1167.14300]. Beauville surfaces; finite simple groups; surfaces of general type DOI: 10.1007/s00229-013-0607-0 Families, moduli, classification: algebraic theory, Surfaces of general type, Simple groups: alternating groups and groups of Lie type, Fuchsian groups and their generalizations (group-theoretic aspects), Riemann surfaces New Beauville surfaces and finite simple 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 \(R\) be an arbitrary finite commutative ring with identity. Denote by \(G\) a Chevalley-Demazure group scheme associated with a connected complex simple Lie group \(G_C\) and a faithful representation of \(G_C\). Let \(G(R)\) be a Chevalley group over the ring \(R\). The aim of this paper is to calculate the order of the finite group \(G(R)\). finite commutative rings; Chevalley-Demazure group schemes; connected complex simple Lie groups; Chevalley groups; orders Other matrix groups over rings, Group schemes Orders of Chevalley groups over finite commutative rings | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities \textit{C. De Concini} and \textit{C. Procesi} [Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)] invented the \textit{Wonderful Compactification} of a complex semisimple adjoint group \(G\). This compactificaton is a smooth projective variety containing \(G\) as a dense open subvariety and where the boundary is a normal crossing divisor with structure determined by the root datum of \(G\). This compactification is claimed to have many applications, particularly in spherical geometry.
In the present article, the author consider the loop group \(LG\) of the group \(G\) and construct an analogue of the wonderful compactification. The loop group can be defined as the group of maps from a punctured formal disc to the group \(G\). The points are the \(\mathbb C((z))\)-points in \(G\), that is \(G(\mathbb C((z))\), where \(C((z))\) is the field of formal Laurent series. An embedding \(X^{\mathrm{aff}}\) is constructed which is not compact, but which have nice properties justifying it as a loop group analogue of the wonderful compactification.
The main application of the theory developed in this work concerns the moduli stack \(\mathcal M_G(C)\) of \(G\)-bundles on a family of nodal curves \(C\). This is not a compact stack, and so \(X^{\mathrm{aff}}\) is used as a kind of compactification. The compactification of the coarse moduli space of \(G\)-bundles on \(C\), is the original goal of the wonderful compactification. There exists compactification for semistable groups, but the author claims that non of these are leading to a satisfactory construction of a compact moduli space for bundles over families of nodal curves.
This article shows that a compactification \(\overline G\) of \(G\) is insufficient to compactify \(\mathcal M_G(C)\) in general, but that the embedding \(X^{\mathrm{aff}}\) gives enough additional data to compactify \(\mathcal M_G(C)\).
\textit{E. Frenkel} et al. [Adv. Math. 288, 201--239 (2016; Zbl 1342.14111)] use the compactification of the moduli of \(\mathbb C^\times\)-bundles to define Gromov-Witten invariants for \([\mathrm{pt}/\mathbb C^\times]\). Their index theorem suggest that similar invariants can be defined on a completion of \(\mathcal M_G(C)\). Then the parallels between \(X^{\mathrm{aff}}\) and the wonderful compactification \(\overline{G_{\mathrm{ad}}}\) suggest that also other constructions related to \(\overline{G_{\mathrm{ad}}}\) have loop analogues.
Vinberg has given an alternative construction of \(\overline{G_{\mathrm{ad}}}\) using a monoid \(S_G\) called the Vinberg monoid, the construction was generalized by Thaddeus and Martens to provide stacky compactifications for any split reductive group. The main result of this article makes it relevant to ask if there is a loop group analogue of the Vinberg monoid.
In geometric representation theory, the wonderful compactification is used e.g. to define the Harish-Chandra transform defined on the category of \(D\)-modules on \(G\), and the results of this article indicates that similar constructions exists for \(D\)-modules on the loop group LG. The wonderful compactification generalizes to elliptic Springer theory by giving an analogue of Lustig's character sheaves for loop groups. This is originally used to give a geometric construction of character sheaves for \(p\)-adic groups. In the present work, the intention is to study sheaves on conjugacy classes in \(LG\). Applying results from \textit{V. Baranovsky} and \textit{V. Ginzburg} [Int. Math. Res. Not. 1996, No. 15, 733--751 (1996; Zbl 0992.20034)], this can be studied by sheaves on the moduli space \(M_{G,E}\) of semistable bundles on an elliptic curve.
Because conjugacy classes in \(LG\) are \(\Delta(LG)\) orbits in \(LG\), the author investigates if \(\Delta(\mathbb C^\times\ltimes LG)\) orbits in \(X^{\mathrm{aff}}\) have a modular interpretation like the interpretation given by Baranovsky and Ginsburg. If this was true, then it can be used to formulate a theory of character sheaves for loop groups.
The author points out that \(LG\) is an ind-scheme rather than a scheme. In this article, the objects under study is thus the category of \textit{ind-schemes}, and the reader is supposed to have good knowledge of such, together with their stack theory. Then \(LG\) is studied through its representations, and it is proved that \(LG\) has a class of projective representations behaving in many ways like the finite dimensional representations of a semisimple group. These are the \textit{honest representations} of a central extension \(\widetilde{LG}\) of \(LG\) by \(\mathbb C^\times\). Such representations are infinite dimensional, and one introduce another \(\mathbb C^\times\) and puts \(G^{\mathrm{aff}}=\mathbb C^\times\ltimes\widetilde{LG}\) to decompose the representation into a direct sum of finite dimensional weight spaces for a maximal torus in \(G^{\mathrm{aff}}(\mathbb C)\). The author uses the representation theory of \(G^{\mathrm{aff}}(\mathbb C)=\mathbb C^\times\ltimes\widetilde{LG}(\mathbb C)\) to replace representation theory of a finite dimensional semisimple group. \(G^{\mathrm{aff}}(\mathbb C)\) is called a Kac-Moody group and is associated to an affine Dynkin diagram in the same way a semisimple group is associated to a Dynkin Diagram.
The wonderful compactification of \(G\) is the compactification of \(G_{\mathrm{ad}}=G/Z(G)\) given by choosing a regular dominant weight \(\lambda\), letting \(V(\lambda)\) be the associated highest weight representation of \(G\), and defining \(\overline{G_{\mathrm{ad}}}=\overline{G\times G[\mathrm{id}]}\subset\mathbb P \mathrm{End}(V(\lambda)).\) The analogue construction is then given by letting \(\lambda\) be a dominant weight \(\underline{\lambda}\) of \(G^{\mathrm{aff}}\) and using the associated representation \(V(\underline{\lambda})\) to construct an ind-scheme \(\mathbb P \mathrm{End}^{\mathrm{ind}}(V(\underline{\lambda}))\) to obtain \(X^{\mathrm{aff}}=\overline{G^{\mathrm{aff}}\times G^{\mathrm{aff}}[\mathrm{id}]}\subset\mathbb P \mathrm{End}^{\mathrm{ind}}(V(\underline{\lambda})).\)
The main result in the article states that when \(G\) is a simple, connected and and simply connected group over \(\mathbb C\) with maximal torus \(T\), the ind-scheme \(X^{\mathrm{aff}}\) contains \(G^{\mathrm{aff}}_{\mathrm{ad}}\) as a dense open sub-ind scheme. The main theorem also contains statements of the properties of \(X^{\mathrm{aff}}\), the boundary of the wonderful compactification of \(LG\), and the properties of the maximal torus. Also, the maximal tori are studied by toric varieties, leading to an explicit description of the orbits of the group-action.
The article contains a very nice study of compactifications, and introduce new tools to study such. It is not very self-contained, as stacks, ind-schemes, and Lie-algebra actions is a basis for the article, making it much deeper than it seems. However, taking this basis into account, the article is very well written and give a framework for wonderful compactifications. loop groups; affine Lie algebras; moduli of \(G\) bundles on curves; embeddings of reductive groups; representation theory; spherical varieties; wonderful compactification; torus group; Harish-Chandra transform; character sheaves; ind-scheme; compactification; flag varieties; divisors in ind-schemes Solis, P., A wonderful embedding of the loop group Compactifications; symmetric and spherical varieties, Representations of Lie and linear algebraic groups over real fields: analytic methods A wonderful embedding of the loop group | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Since Mumford's Geometric Invariant Theory quotients of actions of reductive groups and moduli spaces of vector bundles have become a major tool in classification theory. On the other hand representation theory of finite dimensional algebras and quivers provides a systematic study of classification problems which appear in many fields of mathematics. The basic work of \textit{A. D. King} [Q. J. Math., Oxf. II. Ser. 45, No. 180, 515-530 (1994; Zbl 0837.16005)] connects both fields: he shows the existence of the moduli space of representations over a finite dimensional algebra.
The author studies a class of moduli spaces which are closely related to preprojective algebras. After reviewing King's results he introduces framed moduli spaces and studies their properties, in particular their Grothendieck groups and their Chow rings. This is done by using a method of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [J. Reine Angew. Math. 441, 33-44 (1993; Zbl 0814.14003)], and leads to an explicit description of generators for both groups as the classes of tautological bundles. Moreover these moduli spaces have the remarkable property, that the Chow ring and the singular cohomology ring coincide.
Further, the author describes the cotangent bundle as an open subvariety of the moduli space of representations of the corresponding preprojective algebra. Then he applies his results to simple singularities. He recalls McKay's construction and relates the moduli space of the preprojective algebra of a tame quiver to a quotient singularity of the form \({\mathcal C}/\Gamma\), where \(\Gamma\) is a finite subgroup of \(\text{SU}(2)\). Moreover he obtains a description of the exceptional set of the minimal resolution in terms of the quiver.
Finally, framed moduli are related to Kac-Moody algebras. A description of those algebras is given in terms of convolution algebras [\textit{V. Ginzburg}, C. R. Acad. Sci., Paris, Sér. I 312, No. 12, 907-912 (1991; Zbl 0749.17009)]. These convolution algebras are obtained from Lagrangian subvarieties of the product of framed moduli spaces of a fixed affine quiver with fixed frame, but various dimension vectors. Moreover those moduli spaces are ALE spaces [\textit{P. Kronheimer, H. Nakajima}, Math. Ann. 288, No. 2, 263-307 (1990; Zbl 0694.53025)], which are of interest in symplectic geometry. moduli spaces of vector bundles; finite dimensional algebras; quivers; Grothendieck groups; Chow rings; framed moduli; Kac-Moody algebras; convolution algebras Nakajima, H; Bautista, R (ed.); Martínez-Villa, R (ed.); Pena, JA (ed.), Varieties associated with quivers, No. 19, 139-157, (1996), Providence Representations of quivers and partially ordered sets, Families, moduli, classification: algebraic theory, Universal enveloping (super)algebras Varieties associated with quivers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(B_ 0\) be a local singularity of dimension \(d\). The paper under review considers the problem due to Lech, whether for every deformation \((A, {\mathfrak m}) \to (B, {\mathfrak m})\) of \(B_ 0\) the inequality \(H_ A^{d+1} \leq H^ 1_ B\) between the Hilbert functions is true, and gives a positive answer in the case that the formal versal deformation of \(B_ 0\) is a base change of an algebraic family \((R,M) \to (S,N)\), where \(R\) is regular and \(\dim S = \dim R + d\). So one should lift versal deformations in that way. There are obstructions against this in certain second Harrison cohomology groups. The author generalizes results of \textit{B. Herzog} [Manuscripta Math. 68, No. 4, 351-371 (1990; Zbl 0709.13008)]. Lech conjecture; deformations of local singularities; Hilbert functions; Harrison cohomology groups Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Multiplicity theory and related topics, Singularities in algebraic geometry Lech's conjecture on deformations of singularities and second Harrison cohomology | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \((Q,\sigma)\) be a symmetric quiver, where \(Q=(Q_0,Q_1)\) is a finite quiver without oriented cycles and \(\sigma\) is a contravariant involution on \(Q_0\sqcup Q_1\). The involution allows us to define a nondegenerate bilinear form \(\langle\;,\;\rangle\) on a representation \(V\) of \(Q\). We shall call the representation orthogonal if \(\langle\;,\;\rangle\) is symmetric, and symplectic if \(\langle\;,\;\rangle\) is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type \(c^V\) and, in the case when the matrix defining \(c^V\) is skew-symmetric, by the Pfaffians \(pf^V\). In particular we give an explicit finite set of generators. symmetric quivers of finite type; representations of quivers; rings of semi-invariants; actions of products of classical groups; Coxeter functors; Pfaffians; Schur modules; generic decompositions; bilinear forms Aragona, R.: Semi-invariants of Symmetric Quivers. PhD thesis, arXiv:1006.4378v1 [math. RT] (2009) Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Quadratic and bilinear forms, inner products, Geometric invariant theory, Vector and tensor algebra, theory of invariants Semi-invariants of symmetric quivers of finite type. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Hirzebruch's resolution of cusps; Atiyah-Singer invariant; alpha-invariant; Hilbert's modular surface; resolution of singularities Meyer, W.; Sczech, R.: Über eine topologische und zahlentheoretische anwendung von hirzebruchs spitzenauflösung. Math. ann. 240, 69-96 (1979) Families, moduli, classification: algebraic theory, Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Special surfaces, Global ground fields in algebraic geometry, Groups acting on specific manifolds Über eine topologische und zahlentheoretische Anwendung von Hirzebruchs Spitzenauflösung | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 cohomology of the Hilbert scheme of points \(\mathrm{Hilb}_n(S)\) of non-singular quasi-projective surfaces \(S\) has been described in terms of vertex operators on \(\mathcal{F}:=\bigoplus_nH^*(\mathrm{Hilb}_n(S,\mathbb{Q}))\) introduced by Nakajima and Grojnowski. In this paper the authors introduce another natural set of vertex operators on \(\mathcal{F}\) that depends on a line bundle \(\mathcal{L}\) over \(S\). These operators are defined in terms of the Chern classes of the alternating Ext-groups between two ideal sheaves twisted by \(\mathcal{L}\), and can be characterized by their commutation relations with the Nakajima operators. Moreover, the authors give an explicit formula for them in terms of the latter.
For the \((\mathbb{C}^*)^2\)--equivariant cohomology of \(\mathrm{Hilb}_n(\mathbb{C}^2)\) the trace of these vertex operators becomes the Nekrasov instanton partition function with matter in the adjoint representation. Equivariant localization reduces the above mentioned formula to a Pieri-type formula for the Jack symmetric functions. Hilbert scheme of points; Nakajima operators; Ext-groups; Nekrasov instanton partition function; Jack symmetric functions E. Carlsson and A. Okounkov, \textit{Exts and Vertex Operators}, arXiv:0801.2565. Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Exts and vertex operators | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \({\mathcal C}=mod(R)\), where R is a finite dimensional K-algebra. We recall that a map \(f: X\to Y\) in \({\mathcal C}\) is called irreducible if f can not be performed in the form \(f=gh\), where h is not a splittable monomorphism and g is not a splittable epimorphism. The aim of this paper is to study irreducible maps in \({\mathcal C}\) in terms of the radical rad \({\mathcal C}\) of \({\mathcal C}\). We recall that given X and Y in \({\mathcal C}\) we have
\[
rad(X,Y)=\{f: X\to Y| \quad 1_ X-gf\quad is\quad invertible\quad for\quad all\quad g: Y\to X\}.
\]
It is proved that if \(X=C_ 1\oplus..\oplus C_ n\) and \(C_ 1=...=C_ n=C\), Y are indecomposable then \(f=(f_ 1,...,f_ n): X\to Y\), with \(f_ i: C_ i\to Y\), is irreducible iff \(\bar f_ 1,...,\bar f_ n\in rad(C,Y)/rad^ 2(C,Y)\) are linearly independent over \(K_ C=End(C)/rad End(C).\) Similarly almost split sequences in \({\mathcal C}\) are characterized.
Let I(X,Y) be the set of all \(\bar g\in rad(X,Y)/rad^ 2(X,Y)\), where \(g: X\to Y\) runs through all irreducible maps, and let \(K^*_ C\) be the group of units in \(K_ C\). The author studies the algebraic variety I(X,Y) with an obvious action of the algebraic group \(G_{XY}=K^*_ X\times K^*_ Y\). It is proved that I(X,Y) is an open subvariety of \(rad(X,Y)/rad^ 2(X,Y)\) and if R is of finite representation type then I(X,Y) has finitely many \(G_{XY}\)-orbits. It is concluded that \(\dim_ KI(X,Y)\leq 1\) for X,Y indecomposable, provided K is algebraically closed. Moreover, if \(0\to X\to Z\to Y\to 0\) is an almost split sequence in C and \(Z=Z_ 1^{n_ 1}\oplus..\oplus Z_ t^{n_ t}\), where \(Z_ 1,...,Z_ t\) are pairwise nonisomorphic and indecomposable, then \(n_ 1,...,n_ t\leq 3\) and if \(n_ i\geq 2\) then \(n_ j=1\) for \(j\neq i.\)
The author also studies \({\mathcal C}\) modulo \(rad^{\infty}(C)=\cap^{\infty}_{n=1}rad^ n(C)\) and the graph \(\Gamma_ R\) of \({\mathcal C}\) whose points are isoclasses of indecomposables in \({\mathcal C}\) and there is an arrow [X]\(\to [Y]\) in \(\Gamma_ R\) iff I(X,Y)\(\neq 0\). The shapes of \(\Gamma_ R\) are presented for hereditary algebras R of finite representation type corresponding to Dynkin diagrams with fixed orientations.
Part of results in this paper was announced by the author [in Bull. Am. Math. Soc., New. Ser. 2, 177-180 (1980; Zbl 0428.16029)]. Part of them was also presented by other authors [see \textit{C. M. Ringel} in Lect. Notes Math. 831, 104-136, 137-287 (1980; Zbl 0444.16019 and Zbl 0448.16019)]. The diagram \(\Gamma_ R\) was introduced earlier by \textit{M. Auslander} [in Lect. Notes Pure Appl. Math. 37, 245-327 (1978; Zbl 0404.16007)]. radical of a category; group action on varieties; finite dimensional K- algebra; irreducible maps; almost split sequences; finite representation type; hereditary algebras; Dynkin diagrams Bautista, R, Irreducible morphisms and the radical of a category, An. Inst. Mat. Univ. Nac. Autónoma México, 22, 83-135, (1982) Representation theory of associative rings and algebras, Category-theoretic methods and results in associative algebras (except as in 16D90), Group actions on varieties or schemes (quotients), Finite rings and finite-dimensional associative algebras Irreducible morphisms and the radical of a category | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The notion of an invariant is certainly one of the most general and fundamental concepts in mathematics, since the construction of invariants is inevitably required whenever it comes to classify mathematical objects of some well-defined class. Actually, the origins of invariant theory can be traced back to the early 19th century, in particular to the systematic study of quadratic forms by Lagrange, Gauss, Cauchy, and Jacobi, on the one hand, and to the development of projective geometry by Poncelet, Möbius, Chasles, Steiner, Plücker, and Schläfli, on the other hand. Invariant theory as such arose in the middle of the 19th century, when the study of the general problem of constructing invariante of systems of forms of arbitrary degree was initiated, in close connection with the theory of determinants. At this stage, the most propelling contributions to the creation of invariant theory were made by G. Boole, O. Hesse, G. Salmon, J. Sylvester, A. Cayley, Ch. Hermite, G. Eisenstein, and other great mathematicians of that time. Then the baton was passed to S. Aronhold, F. Brioschi, A. Clebsch, P. Gordan, A. Capelli, S. Lie, and F. Klein, whose efforts were concentrated on the development of formal algebraic methods for constructing invariants and finding, in various concrete cases, a finite generating set and defining relations for a respective algebra of invariants. This first period in the history of invariant theory, the so-called ``naive period'' (or ``pre-Hilbert period''), culminated with the discovery of the ``symbolic method'' which in theory allowed the computation of all respective invariants by an almost purely mechanical process. Complete results, however, were obtained only in a few simple cases, mainly for binary forms, as the actual computations by means of the symbolic method led to enormous labor, disproportionate with the interest of the outcome, especially in a period when all calculations had to be done by hand. Nevertheless, invariant theory was considered to be of great significance, during that period, because it was understood as being nearly identical to projective geometry. This point of view was most consistently and clearly expounded in F. Klein's famous ``Erlanger Programm'' in 1872.
Anyway, with regard to the limit of applicability of the symbolic method in invariant theory, the next problem was the search for ``fundamental systems'' of invariants, i.e., finite sets of invariants such that any invariant would be a polynomial expression in the fundamental invariants. The existence of such fundamental systems was proved, under certain assumptions, by D. Hilbert in 1890, in an ingenious paper which made him famous overnight and which may be considered as the first paper in modern algebra, due to its conceptual approach and methods. But Hilbert's spectacular results also spelled the doom of invariant theory in those days, which was left with no more big problems to solve and soon sank into oblivion. Well, invariant theory has been pronounced dead several times, in the sequel, and like the phoenix it has been again and again rising from its ashes. The first revival was prompted by the works of I. Schur, H. Weyl, and B. Cartan on the global theory of semi-simple Lie groups and their representations in the 1930's, from which it was realized that classical invariant theory was really a special case of that new theory.
But again a seeming lack of challenging problems was probably the reason why important new developments did not occur after the publication of \textit{H. Weyl}'s famous book ``The classical groups, their invariants and representations'' (2nd edition Princeton, N.J. 1946; reprint 1997). This second period of stagnation of invariant theory ended abruptly in the late 1950's, when M. Nagata constructed a counter-example to the 14th Hilbert problem, and the third golden age of invariant theory dawned with the publication of \textit{M. Mumford}'s epoch-making book ``Geometric invariant theory'' (1965; Zbl 0147.39304). In fact, Mumford realized that invariant theory provided him with some of the tools he needed for his solution of the ``moduli problem'' for algebraic curves and principally polarised abelian varieties, and that some essential ideas and techniques for constructing algebraic orbit spaces (geometric quotients) lay buried in D. Hilbert's brilliant (and long forgotten) papers on invariant theory from the 1890's. In his book ``Geometric invariant theory'', D. Mumford (loc. cit.) has radically modernized and greatly generalized Hilbert's original ideas, using the language of modern algebraic geometry, as well as important results by Chevalley, Nagata, Grothendieck, Iwahori, Tate, Tits, and himself. In the past forty years, invariant theory, especially geometric invariant theory, has established itself as a central topic of algebraic geometry and general classification theory in mathematics.
Thus invariant theory has a very long, fascinating and somewhat unique history, which has seen alternating periods of flourishing and stagnation, changes in its formulation, and varying fields of application. In these days, invariant theory stands there as an important part of the general theory of (algebraic) transformation groups, and as more momentous than ever in mathematics and physics.
The main purpose of the book under review is to give a concise and self-contained exposition of the main ideas of classical and modern invariant theory, with a special emphasis on the more recent (algebro-) geometric aspects of the subject, but basically following the historical path. According to the troubled history of invariant theory, the comparatively low number of experts and active researchers in the field, at least so in the ``post-Hilbert period'', and due to the fact that invariant theory has not been a preferential teaching subject at universities, during the last century, there are not many accessible introductory textbooks covering invariant theory in its different aspects. Most classical texts do not cover the basics of Mumford's geometric invariant theory, of course, and Mumford's pioneering text ``GIT'' is just too advanced and difficult for beginners. Among the more recent texts on invariant theory introducing various aspects of the subject, there are just the books: ``Invariant theory, old and new'' by \textit{J. A. Dieudonné} and \textit{J. B. Carrel} from 1971 (New York and London 1971; Zbl 0258.14011), ``Geometrische Methoden in der Invariantentheorie'' by \textit{H. Kraft} (Aspects Math. D1, Braunschweig und Wiesbaden, 1984; Zbl 0569.14003), ``Algebraic transformation groups'' by \textit{H. Kraft}, \textit{P. Slodowy} and \textit{T. A. Springer} (DMV Seminar, 13; 1989; Zbl 0682.00008), and the encyclopedic survey ``Invariant theory'' by \textit{E. G. Vinberg} and \textit{V. L. Popov} [in: Algebraic geometry, IV, Encycl. Math. Sci. 55, 123-278 (1994); translation from Itogy Nauki Tekh., Ser. Sovrem., Probl. Mat., Fundam. Napravleniya 55, 137-309 (1989; Zbl 0789.14008)].
Now there is also the book under review, based on several graduate courses taught by the author in the 1990's at Seoul National University, the University of Michigan, and at Harvard University. As the author points out in the preface to his book, this text is literally intended to motivate and prepare a beginner to study modern invariant theory more thoroughly, especially with a view towards algebraic geometry (moduli problems) and differential geometry (Lie groups). The essential novelty offered by this text, among the very few comparable introductions to modern invariant theory, is the large number of examples and applications. Also, for the first time in an introductory text of geometric invariant theory, the author has included a thorough study of linearizations of group actions on varieties, stability properties and geometric quotients, and torus actions on affine spaces. A particular feature is given by the numerous concrete examples from classical projective geometry, providing a beautiful illustration of the abstract methods in geometric invariant theory and linking the different aspects of the whole subject.
As to the contents of the book, the text is subdivided into twelve chapters which lead the reader systematically from classical invariant theory to the fundamental concepts of modern geometric invariant theory, basically along the historical line of development described above. Here is a list of the single chapters, with keywords indicating their contents:
1. The symbolic method (first examples, the polarization and restitution process, bracket functions);
2. The first fundamental theorem (Cayley's omega-process, Grassmann varieties, the straightening algorithm);
3. Reductive algebraic groups (the Gordan-Hilbert theorem, H. Weyl's unitary trick, affine algebraic groups, Nagata's theorem);
4. Hilbert's fourteenth problem (the problem, Weitzenböck's theorem, Nagata's counter-example to Hilbert's 14th problem);
5. Algebra of covariants (examples of covariants, covariants of an action, linear representations of reductive groups, dominant weights, the Cayley-Sylvester formula, standard tableaux);
6. Quotients (categorical and geometric quotients, rational quotients, Rosenlicht's theorem, examples);
7. Linearizations of actions (linearized line bundles, existence of linearizations, the linearization of a group action);
8. Stability (stable points with respect to a linearized bundle, quotients as quasi-projective varieties, examples of quotients);
9. Numerical criterion of stability (Hilbert-Mumford stability and Kempf stability);
10. Projective hypersurfaces (invariants of non-singular projective hypersurfaces, binary forms, plane cubics and cubic surfaces);
11. Configurations of linear subspaces (diagonal actions of \(SL_{n+1}\) on products of Grassmannians, stable configurations, configurations of points in projective space, configurations of points in projective 3-space);
12. Toric varieties (torus actions on affine space, fans, toric varieties, examples).
Each chapter comes with its individual bibliographical notes for further reading, followed by a set of (quite challenging) exercises. Although many of the exercises are highly demanding, no hints for solution are offered, which might turn out to be pretty hard for beginners in the field. On the other hand, there are those many beautiful examples scattered throughout the text, helping the reader to grasp the fascination of (old and new) invariant theory.
The exposition is very systematic, lucid and sufficiently detailed. It certainly transpires the author's passion, versatility, all-round knowledge in mathematics, and mastery as both an active researcher and devoted teacher. It is very gratifying to see this beautiful, modern and still down-to-earth introduction to classical and contemporary invariant theory at the public's disposal, after a long period of craving for suitable standard texts in this literarily somewhat neglected field.
The book assumes only minimal prerequisites for graduate students: a basic knowledge of algebraic varieties, a profound knowledge of multilinear algebra, and some rudiments of the theory of linear representations of groups. Everything else, including the necessary facts from the theory of linear algebraic groups, is sufficiently explained in the course of the text. In this regard, the present book is really nearly self-contained.
Finally, as to the merely computational aspects of invariant theory, it should be remarked that those are not particularly stressed in this approach. The interested reader might find it very useful to consult the recent book ``Computational invariant theory'' by \textit{H. Derksen} and \textit{G. Kemper} [Encyclopaedia of Math. Sci., Invariant theory and algebraic transformation groups, 130 (1) (2002; Zbl 1011.13003)] for additional and parallel reading. In fact, the development of computational commutative algebra has provided another decisive impulse to revive the classical (algorithmic) methods of invariant theory, with numerous important applications in various fields, including the invariant theory of reductive groups and the construction of moduli spaces in geometry, and these new aspects of constructive invariant theory and its applications are comprehensively expounded in that just as recent reference book. invariant theory; geometric invariant theory; geometric quotients; orbit spaces; Hilbert's fourteenth problem; Lie groups; GIT; configurations of linear subspaces; linearizations of actions; classification theory; stability [7] I. Dolgachev, \(Lectures on invariant theory\), Cambridge University Press, (2003). &MR 20 | &Zbl 1023. Actions of groups on commutative rings; invariant theory, Vector and tensor algebra, theory of invariants, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Geometric invariant theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra Lectures on invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This expository paper describes the geometries associated to simle Lie groups, following J. Tits' theory of buildings and shadows. First the definitions of ''building'', ''apartment'', ''Dynkin-diagram'' are given and motivated. Then the author describes the buildings and the associated geometries for the classical groups and for \(G_ 2\). In the last two chapters ''shadows'' are defined in order to investigate the geometries of the exceptional groups. The shadow geometry for \(E_ 8\) is explicitly derived. geometries of exceptional groups; simle Lie groups; buildings; shadows; Dynkin-diagram Betty Salzberg, Buildings and shadows, Aequationes Math. 25 (1982), no. 1, 1 -- 20. Linear algebraic groups over finite fields, Simple groups: alternating groups and groups of Lie type, Other finite incidence structures (geometric aspects), Classical groups (algebro-geometric aspects) Buildings and shadows | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 scheme defined over an algebraically closed field \(k\) of characteristic \(p\) and let \(\mathcal D_X\) be the sheaf of \(k\)-linear differential operators on \(X\). The scheme \(X\) is called \(D\)-affine if every \(\mathcal D_X\)-module \(M\) is generated over \(\mathcal D_X\) by its global sections and \(H^i(X,M) = 0\) for all \(i > 0\). First, the author recalls some well known facts. For example, every flag variety in characteristic zero is \(D\)-affine (see [\textit{A. Beilinson} and \textit{J. Bernstein}, C. R. Acad. Sci., Paris, Sér. I 292, 15--18 (1981; Zbl 0476.14019)]). However, in positive characteristic it is not true (see [\textit{M. Kashiwara} and \textit{N. Lauritzen}, C. R., Math., Acad. Sci. Paris 335, No. 12, 993--996 (2002; Zbl 1016.14009)]), although some flag varieties are still \(D\)-affine. Next, any smooth \(D\)-affine projective toric variety is a product of projective spaces (see [\textit{J. F. Thomsen}, Bull. Lond. Math. Soc. 29, No. 3, 317--321 (1997; Zbl 0881.14020)]), etc. In the paper under review the author describes some further results concerning the classification of smooth projective \(D\)-affine varieties over fields of any characteristic. In particular, he proves that any smooth projective \(D\)-affine variety is algebraically simply connected and its image under a fibration is \(D\)-affine as well. In zero characteristic such \(D\)-affine varieties are, in fact, uniruled. Moreover, assuming \(p=0\) or \(p>7\), he shows that a smooth projective surface is \(D\)-affine if and only if it is isomorphic to either \(\mathbb P^2\) or \(\mathbb P^1\times \mathbb P^1\). Some results are also obtained for three-folds \(D\)-affine varieties. In positive characteristic case the author applies his own modified version of the generic semipositivity theorem due to \textit{Y. Miyaoka} [Proc. Symp. Pure Math. 46, No. 1, 245--268 (1987; Zbl 0659.14008)]. smooth projective varieties; flag varieties; Miyaoka's semipositivity theorem; cotangent bundle; rational surfaces; divisorial contractions; fibrations; crystalline differential operators; étale fundamental group; semistability; reflexives sheaves; semipositive sheaves; uniruled varieties; Riemann-Hilbert correspondence; stable Higgs bundle; Chern classes; flat connections; Artin's criterion of contractibility; Kodaira dimension; Hirzebruch surface; canonical divisor; surfaces of general type; Barlow's surfaces; del Pezzo surfaces; Fano three-folds Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules On smooth projective D-affine varieties | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(X\) be an algebraic variety defined over a field \(K\) and denote by \(X(K)/R\) the set of \(K\)-rational points of \(X\) modulo rational equivalence as originally defined by \textit{Yu. I. Manin} [Cubic forms. Algebra, geometry, arithmetic. Transl. from the Russian by M. Hazewinkel. 2nd ed. North-Holland Mathematical Library, Vol. 4. Amsterdam-New York-Oxford: North-Holland (1986; Zbl 0582.14010)]. In this paper, the authors consider absolutely simple adjoint linear algebraic groups \(G\). By Weil's classification, these groups are of type \(A_n\), \(B_n\), \(C_n\) or \(D_n\) (non-trialitarian in case \(D_4\)). Now \(G(K)/R\) is naturally equipped with a group structure and one would like to know if this group is trivial. \textit{A. S. Merkurjev} [Publ. Math., Inst. Hautes Étud. Sci. 84, 189-213 (1996; Zbl 0884.20029)] showed that this is so if \(G\) is of inner type \(A_n\) or of type \(B_n\), but that triviality generally fails for type \(D_n\) (\(n\geq 3\)).
In the present paper, the authors consider the situation where \(K\) is a function field of a smooth geometrically integral curve over a nondyadic local field. They show that triviality holds over such fields if the absolutely simple adjoint group is of type \(^2A_n^*\) (i.e. where the underlying central simple algebra has square-free index), \(C_n\) and \(D_n\) (non-trialitarian in case \(D_4\)). The main ingredients in the proofs are Merkurjev's description of \(G(K)/R\) as the quotient group of similitudes of a certain Hermitian form [loc. cit.], the fact that the \(u\)-invariant of such a field \(K\) is at most \(8\) due to \textit{R. Parimala} and \textit{V. Suresh} [Ann. Math. (2) 172, No. 2, 1391-1405 (2010; Zbl 1208.11053)], the first author's own classification results for Hermitian forms over algebras with involution [J. Algebra 385, 294-313 (2013; Zbl 1292.11056)], and a theorem proved also in the present paper and of interest in its own right, namely, that the Rost invariant of \(SL_1(A)\) for a central simple algebra \(A\) of index at most \(4\) over such a field \(K\) has trivial kernel. algebraic groups; adjoint groups; R-equivalence; nondyadic local fields; function fields of curves; algebras with involution; Hermitian forms; Rost invariant R. Preeti and A. Soman, Adjoint groups over \Bbb Q_{\?}(\?) and R-equivalence, J. Pure Appl. Algebra 219 (2015), no. 9, 4254 -- 4264. Linear algebraic groups over local fields and their integers, Quadratic forms over general fields, Bilinear and Hermitian forms, Classical groups, Galois cohomology of linear algebraic groups, Rational points, Other nonalgebraically closed ground fields in algebraic geometry, Finite-dimensional division rings, Rings with involution; Lie, Jordan and other nonassociative structures Adjoint groups over \(\mathbb Q_p(X)\) and R-equivalence. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Dimer models are introduced by string theorists to study four-dimensional \(N=1\) superconformal field theories. See, e.g., a review by \textit{K. D. Kennaway} [Int. J. Mod. Phys. A 22, No. 18, 2977-3038 (2007; Zbl 1141.81328)] and references therein for a physical background. A dimer model is a bipartite graph on a real two-torus which encodes the information of a quiver with relations. A typical example of such a quiver is the McKay quiver determined by a finite Abelian subgroup \(G\) of \(\text{SL}(3,\mathbb{C})\) [see \textit{M. Reid}, ``McKay correspondence'', \url{arXiv:alg-geom/9702016v3}, \textit{K. Ueda} and \textit{M. Yamazaki}, Commun. Math. Phys. 301, No. 3, 723-747 (2011; Zbl 1211.81090)]. In this case, the moduli space of representations of the McKay quiver (for the dimension vector \((1,1,\dots,1)\)) coincides with the moduli space of \(G\)-constellations considered by \textit{A. Craw} and \textit{A. Ishii} [Duke Math. J. 124, No. 2, 259-307 (2004; Zbl 1082.14009)]. For a generic choice of a stability parameter \(\theta\), the moduli space of \(G\)-constellations is a crepant resolution of the quotient singularity \(\mathbb{C}^3/G\) and the derived category of coherent sheaves on the moduli space is equivalent to the derived category of finitely generated modules over the path algebra of the McKay quiver. It is expected that these kinds of statements can be generalized to the case of dimer models that are `consistent' in the physics context, which should be called `brane tilings'. In this note, we discuss a slightly weaker notion of non-degenerate dimer models, which is strong enough to ensure that the moduli space is a crepant resolution of the three-dimensional toric singularity determined by the Newton polygon of the characteristic polynomial (see Theorem 6.4). We expect that one has to impose further conditions to prove the derived equivalence.
For the proof, we use a generalization of the description of a torus-fixed point on the moduli space in terms of a choice of a covering by hexagons of the fundamental region of a real 2-torus due to \textit{I. Nakamura} [J. Algebr. Geom. 10, No. 4, 757-779 (2001; Zbl 1104.14003)]. Many of the arguments are similar to those of \textit{A. Ishii} [in Clay Mathematics Proceedings 3, 227-237 (2005; Zbl 1156.14308)]. There is also a physics paper by \textit{S. Franco} and \textit{D. Vegh} [Moduli spaces of gauge theories from dimer models: proof of the correspondence, \url{arXiv:hep-th/0601063v2}] which deals with the relation between brane tilings and moduli spaces. dimer models; superconformal field theories; bipartite graphs; quivers with relations; McKay quivers; moduli spaces; representations of quivers; crepant resolutions; quotient singularities A. Ishii and K. Ueda, \textit{On moduli spaces of quiver representations associated with dimer models}, arXiv:0710.1898. Representations of quivers and partially ordered sets, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, String and superstring theories; other extended objects (e.g., branes) in quantum field theory On moduli spaces of quiver representations associated with dimer models. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(K/F\) be an abelian Galois extension of fields. The group \(\text{Dec}(K/F)\) is the subgroup of the relative Brauer group \(\text{Br}(K/F)\) generated by the relative Brauer groups \(\text{Br}(L/F)\) of all the cyclic extensions \(L/F\) contained in \(K\). This group was introduced by the reviewer [J. Algebra 70, 420-436 (1980; Zbl 0473.16004)] in relation with the construction of indecomposable division algebras of prime exponent. If \(K/F\) is elementary abelian of degree 4, then it is known that \(\text{Dec}(K/F)\) is the 2-torsion subgroup in \(\text{Br}(K/F)\).
In the present paper, the author explicitly constructs for each prime \(p\) and each integer \(n\geq 1\) (\(n \geq 2\) if \(p = 2\)) a field \(F\), an abelian extension \(K/F\) with Galois group \(({\mathbb{Z}}/p^ n \mathbb{Z})\times (\mathbb{Z}/p\mathbb{Z})\) and a central simple algebra \(A\) of exponent \(p\) split by \(K\) whose Brauer class is not in \(\text{Dec}(K/F)\). The base field \(F\) is a rational function field in three variables over a field of characteristic zero containing sufficiently many roots of unity. The methods of proof are essentially valuation-theoretic. abelian Galois extensions; relative Brauer groups; cyclic extensions; indecomposable division algebras of prime exponent; central simple algebras; Brauer class; rational function fields Finite-dimensional division rings, Equations in general fields, Valued fields, Brauer groups of schemes, Separable extensions, Galois theory Dec groups for arbitrarily high exponents | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 0679.00007.]
Hilbert's fourteenth problem is reformulated in several versions, e.g.
(Popov-Pommerening conjecture:) If \(k\) is a field, \(G\) a reductive group acting rationally on a finitely generated \(k\)-algebra \(A\) and \(H\) a subgroup of \(G\) normalized by a maximal torus of \(G\), then \(A^ H\) is a finitely generated \(k\)-algebra; or
(Conjecture 2.13:) If \(G\) is a connected and simply connected simple algebraic group over \(k\) with root system \(\Phi\), \(H=\prod_{\alpha\in S}U_ \alpha\), where \(S\) is a quasiclosed subset of \(\Phi^ +\) (relative to some ordering), then \(k[G]^ H\) is a finitely generated \(k\)-subalgebra.
The known results on the Popov-Pommerening conjecture are presented and it is shown to be affirmative in the case when \(G\) is a reductive group of type \(B_ 2\).
[See also the attached article by \textit{D. L. Wehlau}, ibid. 221-228 (1989; Zbl 0746.20023)]. finite generation of invariant algebra; Hilbert's fourteenth problem; Popov-Pommerening conjecture Tan, L., \textit{some recent developments in the Popov-pommerening conjecture}, Group actions and invariant theory, 207-220, (1989), American Mathematical Society, Providence, RI Geometric invariant theory, Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory Some recent developments in the Popov-Pommerening conjecture | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection Zbl 0743.00050.]
This survey paper consists of three parts and an extensive bibliography. The first part is a general introduction to the theory of singularities of complex analytic spaces, which centers on the many characterisations of the \(ADE\)-singularities. The by now classical highlights are treated: the connection with Lie groups, quotient singularities and the platonic solids, the resolution graph and the Dynkin diagram, simpleness in the sense of Arnol'd. The author also covers the characterisation of the \(ADE\)-singularities by finite representation type. In many places references to newer and further developments are given.
The last two chapters introduce more advanced topics of current research in a careful discussion of problems and questions. The first one is the deformation space of rational surface singularities. The last chapter concerns moduli spaces for singularities and modules over local rings, a subject to which the author has contributed substantially. Whereas in the global case moduli spaces are constructed with Geometric Invariant Theory, and the groups appearing are reductive, one deals here with unipotent groups. simple singularities; deformation theory; bibliography; platonic solids; Dynkin diagram; moduli spaces Greuel, G.-M., Deformation und klassifikation von singularitäten und moduln, 177-238, (1992), Stuttgart Local complex singularities, Complex surface and hypersurface singularities, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients) Deformation and classification of singularities and 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 thesis studies the split octonion algebra and its automorphism group. This group is a Lie group of type \(G_2\). One object of the thesis is to look at methods introduced by \textit{K. A. Behrend} [Math. Ann. 301, No. 2, 281--305 (1995; Zbl 0813.20052)], \textit{G. Harder} and \textit{U. Stuhler} [Canonical parabolic subgroups of Arakelov group schemes. Preprint, 2003, \url{http://www.unimath.gwdg.de/stuhler/arakelov.pdf}] in the special case of this group. In this context, the various systems of norms on the octonions are of importance, which form either symmetric rooms or Bruhat-Tits buildings. Thereby, the algebra structure of the octonions is very helpful for the analyses. A by-product is that an invariant flag of the octonions under the Borel group almost completely consists of subalgebras. Finally it is shown that the complementary polyhedra introduced by Behrend degenerate to a point when one looks at the standard apartment. It is possible to describe their location in terms of the grades (as Arakelov bundles) of the first two spaces of the invariant flag. split octonion algebra; automorphism group; Lie group of type \(G_2\); symmetric rooms; Bruhat-Tits buildings; standard apartment; Arakelov bundles; invariant flag Composition algebras, Group schemes, Root systems, Groups with a \(BN\)-pair; buildings, Representation theory for linear algebraic groups, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) Octonions and reduction theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We discuss how the classification of finite simple groups is used and mention some specific applications to various other fields of mathematics as well as to group theory. finite groups; finite simple groups; applications of simple groups; Brauer groups; Riemann surfaces; polynomials; function fields Guralnick, Robert, Applications of the classification of finite simple groups.Proceedings of the International Congress of Mathematicians---Seoul 2014. Vol. II, 163-177, (2014), Kyung Moon Sa, Seoul Finite simple groups and their classification, Primitive groups, Coverings of curves, fundamental group, Algebraic field extensions Applications of the classification of finite simple 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 subject of this note is the Vassiliev-Goodwillie-Weiss spectral sequence. This objects calculates an approximation of the homology of the space of knots. It has been proved by \textit{P. Lambrechts} et al. in [Geom. Topol. 14, No. 4, 2151--2187 (2010; Zbl 1222.57020)] that this spectral sequence degenerates at the \(E_2\)-page when working with rational coefficients. Besides, \textit{I. Volić} in [Compos. Math. 142, No. 1, 222--250 (2006; Zbl 1094.57017)] has shown that the knot invariant given by the \(E_2\)-page of this spectral sequence is the universal finite type invariant (or more precisely the associated graded one given by the filtration according to the grade). It has been conjectured that these two results remain true when working with integer coefficients.
In this note we are interested specifically in the first conjecture (to know whether the spectral sequence degenerates with integer coefficients). In joint work with Pedro Boavida de Brito we have introduced a new tool to study this problem. We construct a non-trivial action of the absolute Galois group of \(\mathbb Q\) on this spectral sequence. This action gives us information on the differentials. In this way we can show that a given differential is zero if we ignore the torsion for the first small integers (see Theorem 5.2 for the precise result).
\dots
Let us give some details on the contents of this note. The first section gives a quick introduction to the Goodwillie-Weiss embedding calculus. The second is a specialization of this theory to the case of embeddings from \(\mathbb R\) to \(\mathbb R^d\) following the work of \textit{D. P. Sinha} [J. Am. Math. Soc. 19, No. 2, 461--486 (2006; Zbl 1112.57004)]. The third section is an introduction to the Grothendieck-Teichmüller group and to the absolute Galois group of \(\mathbb Q\) and to their actions on the profinite completions of the pure braid groups. In the fourth section we recall the theory of completion of spaces at a prime number. This is an analog in homotopy theory of the completion of groups. Finally in the fifth section, we can give the main results together with a sketch of the proof. Note that only the results of this final part are original. They are due to Pedro Boavida de Brito and the author. A complete proof will be published soon. Vassiliev-Goodwillie-Weiss spectral sequence; space of knots; \(E_2\)-term; universal finite type invariant; Vassiliev invariant; integer coefficients; action; Galois group; differentials; completion Finite-type and quantum invariants, topological quantum field theories (TQFT), Spectral sequences in algebraic topology, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Galois group and knot space | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 exposition of history of solution of the complete intersection problem in the invariant theory of finite groups. More precise, let \(G\subset GL_ n(C)\) be a finite linear group, \(S=C[X_ 1,...,X_ n]\) the ring of polynomials in n variables so the action of G is continued to S in the natural manner. Let \(S^ G\) be the invariant ring of G that is \(S^ G=\{f\in S| \quad gf=f\quad for\quad all\quad g\in G\}.\)
Problem. Describe G for which \(S^ G\) is the ring of a complete intersection. complete intersection; invariant theory of finite groups Geometric invariant theory, Complete intersections, History of algebraic geometry, History of mathematics in the 20th century, Linear algebraic groups over the reals, the complexes, the quaternions, Group actions on varieties or schemes (quotients), Polynomial rings and ideals; rings of integer-valued polynomials Anneaux d'invariants de groupes finis intersections complètes. (Complete intersection invariant rings of finite groups) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author considers finite separable coverings of curves \(f\colon X\to Y\) over a field of characteristic \(p\geq 0\). He is interested in describing the possible monodromy groups of these covers if the genus of \(X\) is fixed. There has been much progress on this problem over the past decade for \(p=0\). Recently, \textit{D. Frohard} and \textit{K. Magaard} [Ann. Math. (2) 154, No. 2, 327-345 (2001; Zbl 1004.20001)] completed the final step in resolving the Guralnick-Thompson conjecture posed in [\textit{R. M. Guralnick} and \textit{J. G. Thompson}, J. Algebra 131, No. 1, 303-341 (1990; Zbl 0713.20011)] showing that only finitely many nonabelian simple groups other than alternating groups occur as composition factors for a fixed genus. There is an ongoing project to get a complete list of the monodromy groups of indecomposable rational functions with only tame ramification.
In this paper, the author focuses on the case \(p>0\) and shows that many simple groups do not occur as composition factors for a fixed genus. He also proves a reduction theorem reducing the problem to the case of almost simple groups and obtains some results on bounding the size of automorphism groups of curves for \(p>0\). The author notes that prior to his results there was not a single example of a finite simple group which could be ruled out as a composition factor of the monodromy group of a rational function in any positive characteristic. coverings of curves; monodromy groups; permutation groups; automorphisms of curves; genera; finite simple groups; Guralnick-Thompson conjecture Guralnick, R.: Monodromy groups of coverings of curves. In: Galois Groups and Fundamental Groups. Math. Sci. Res. Inst. Publ., vol. 41, pp. 1--46. Cambridge Univ. Press, Cambridge (2003) Primitive groups, Coverings of curves, fundamental group, Automorphisms of curves, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Compact Riemann surfaces and uniformization, Simple groups: alternating groups and groups of Lie type Monodromy groups of coverings 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 Consider the general linear group \(G=\text{GL}(n,k)\) over a field \(k\) of characteristic \(p>0\) and the monoid scheme \(M\) of \(n\) by \(n\) matrices over \(k\). The rational representations of \(M\) are equivalent to the polynomial representations of \(G\). Further, the homogeneous polynomial representations of degree \(d\) are equivalent to the representations of the Schur algebra \(S(n,d)\). More generally, the representation theory of \(G\) has often been studied by considering the representation theory of the group scheme \(G_rT=(F^r)^{-1}(T)\) where \(F\) is the Frobenius morphism and \(T\) is a maximal torus in \(G\). These two ideas were combined in work of \textit{S. R. Doty, D. K. Nakano}, and \textit{K. M. Peters} [Proc. Lond. Math. Soc., III. Ser. 72, No. 3, 588-612 (1996; Zbl 0856.20025)] who considered the monoid scheme \(M_rD\) for the submonoid \(D\) of diagonal matrices. The relationship between \(M_rD\) and \(G_rT\) is analogous to that between \(M\) and \(G\). They study the polynomial representation theory of \(G_rT\) and are led to define certain infinitesimal Schur algebras \(S(n,d)_r\) which play a similar role.
In later work of \textit{S. R. Doty, D. K. Nakano}, and \textit{K. M. Peters} [Contemp. Math. 194, 57-67 (1996; Zbl 0856.20026)], they determine the blocks of the algebras \(S(2,d)_1\). In the work under review, the author continues the investigation of the blocks of \(S(n,d)_r\). He formulates a general conjecture relating a block for \(S(n,d)_r\) to the corresponding block for \(G_rT\) extending the known result for \(S(2,d)_1\). One containment of the conjecture is proved in general and the reverse containment is shown to hold when \(n=2\) (and any \(r\)).
The author then considers the quantum general linear group \(q\text{-GL}(n,k)\) introduced by \textit{R. Dipper} and \textit{S. Donkin} [Proc. Lond. Math. Soc., III. Ser. 63, No. 1, 165-211 (1991; Zbl 0734.20018)] and introduces the notion of a quantum Schur algebra. He develops some basic representation theory of these algebras as done for infinitesimal Schur algebras and once again identifies the blocks in the case \(n=2\). infinitesimal Schur algebras; quantum Schur algebras; blocks; polynomial representations; Frobenius kernels; general linear groups; monoids of matrices; monoid schemes; rational representations Cox A.G.: On the blocks of the infinitesimal Schur algebras. Quart. J. Math 51, 39--56 (2000) Representation theory for linear algebraic groups, Representation of semigroups; actions of semigroups on sets, Group schemes, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations On the blocks of the infinitesimal Schur 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 aim of the book is to make an attempt to expose a new unifying approach to the study of numerous quantized algebras. This approach is based on investigations of noncommutative schemes in categories. The point of departure in the first chapter of the book is the notion of a left spectrum \(\text{Spec}_l R\) of an associatve ring \(R\) with unity. It consists of all left ideals \(P\) in \(R\) with the following property. For any element \(x\in R\setminus P\) there exists a finitely generated additive subgroup \(H\) in \(R\) such that \(zHx\subseteq P\) where \(z\in R\) implies \(z\in P\). The set \(\text{Spec}_l R\) is nonempty since it contains the set \(\text{Max}_l R\) of all maximal left ideals and the set \(\text{Spec}_l' R\) of all completely prime left ideals. If \(P,P'\in \text{Spec}_l R\) then \(P\leq P'\) if there exists a finitely generated additive subgroup \(H\) in \(R\) such that \(zH\subseteq P\) implies \(z\in P'\). There exists a topology \(\tau_*\) on \(\text{Spec}_l\) with the base \(\{\bigcup_{P'\geq P} P'\mid P\in \text{Spec}_l R\}\). There exists a topology \(\tau^*\) on \(\text{Spec}_l R\) determined by the base of closed subsets of the form \(\{p\in \text{Spec}_l R\mid p\leq M\}\) where \(M\) runs through the set of all proper ideals in \(R\). Flat localizations of Abelian categories are discussed. A stability theorem with respect to localizations is proved for the left spectrum. Special categories of rings are selected. (Pre)images of morphisms in these categories preserve preorder \(\leq\). It is shown that the intersection of all elements of \(\text{Spec}_l R\) is the Levitzki radical of \(R\). In particular the topological space \(\text{Spec}_l R\) has a base of quasi-compact open sets. Structure presheaves of modules over rings, noncommutative quasi-affine schemes and projective spectra are introduced.
In the second chapter all main `small' quantized rings are introduced: the quantum plane \(k_q[X, Y]\), the algebra of \(q\)-differential operators \(D_{q,h}\), quantum Heisenberg and Weyl algebra \(W_{q, 1}\), the quantum envelope \(U_q(sl(2))\), the coordinate ring \(M_q(2)\) of \(2\times 2\) matrices, the coordinate ring \(A(SL_q(2))\), twisted \(SL(2)\) group, \(W_v(sl(2))\) of Woronovicz. The goal of chapter 2 is to develop the representation theory of these algebras. It is shown that all these algebras are specializations of the hyperbolic ring \(A\{\theta, \zeta\}\) which is generated by a commutative ring \(A\) with fixed \(\theta\in \Aut A\), \(\zeta\in A\), and by elements \(x\), \(y\) with defining relations
\[
xa= \theta(a) x,\quad ay= y\theta(a),\quad a\in A;\quad xy= \zeta,\quad yx= \theta^{- 1}(\zeta).
\]
It is worth to mention that the theory of these and even more general classes of algebras under the name of generalized Weyl algebras is developed by \textit{V. V. Bavula} [Generalized Weyl algebras (Bielefeld, preprint, 1994)]. Unfortunately, it is not mentioned in the book.
In section 1 of chapter 2 the left spectrum of a skew polynomial ring \(A[X, \theta]\) is studied. These results are applied to the quantum plane. Section 3 contains an almost complete description of \(\text{Spec}_l R\) of a hyperbolic ring \(R\). This description is applied to the rings mentioned above.
Chapter 3 is devoted to the categorical point of view on geometrical objects. The author introduces the notion of the spectrum of an Abelian category, studies its behaviour with respect to localizations, Serre subcategories, Grothendieck categories, local Abelian categories, localizations at points, topologies on categories.
Chapter 4 contains generalization of the notion of a hyperbolic ring. Namely the author introduces the notion of a hyperbolic category over an Abelian category. The main result of the chapter is the description of a hyperbolic category which is naturally related to the spectrum of the underlying Abelian category. This result generalizes similar results from chapter 2.
Chapter 5 is concerned with skew PBW monads in a monoidal category \(A\). A skew PBW monad is a generalization to monads of the notion of a hyperbolic ring. Other examples of skew PBW monads are related to Kac-Moody and Virasoro Lie algebras. The author studies semigroup-graded monads \(F\) and their spectra. The main result of the chapter shows that all points of \(F\)-modules which `grow up' over a given point of \(\text{Spec } A\) can be represented by a graded module with the grading associated to this point. The chapter ends with considerations of quasi-holonomic modules, \(F\)-comodules, spectra of Weyl algebra and their quantizations, representations of Kac-Moody algebra, two-parameter deformations of the algebras \(M(2)\), \(GL(2)\).
In chapter 6 the author surveys major approaches to noncommutative spectral theory such as injective spectrum by Gabriel, Goldman's spectrum of a ring, affine scheme by F. Van Oystaeyen and A. Verschoren, Cohn's affine scheme. It is shown that these spectra can be deduced from the case considered in the book. The chapter ends with exposing properties of Gabriel-Krull dimension and a calculations of dimensions of some spectra.
The last chapter 7 is concerned with the projective spectrum of graded monads. Affine and projective fibres, blowing up and related topics are considered. flat localizations of Abelian categories; structure presheaves of modules; quantized algebras; noncommutative schemes in categories; left spectrum; maximal left ideals; completely prime left ideals; categories of rings; Levitzki radical; quasi-affine schemes; projective spectra; quantized rings; quantum planes; algebra of \(q\)-differential operators; Weyl algebras; quantum envelopes; coordinate rings; generalized Weyl algebras; skew polynomial rings; Serre subcategories; Grothendieck categories; hyperbolic rings; skew PBW monads; monoidal category; Kac-Moody and Virasoro Lie algebras; semigroup-graded monads; Gabriel-Krull dimension Rosenberg, A.L.: Algebraic Geometry Representations of Quantized Algebras. Kluwer Academic Publishers, Dordrecht, Boston London (1995) Research exposition (monographs, survey articles) pertaining to associative rings and algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras, Quantum groups (quantized enveloping algebras) and related deformations, Noncommutative algebraic geometry, Torsion theories; radicals on module categories (associative algebraic aspects), Rings of differential operators (associative algebraic aspects), Local categories and functors, Abelian categories, Grothendieck categories, Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, ``Super'' (or ``skew'') structure, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Abstract manifolds and fiber bundles (category-theoretic aspects) Noncommutative algebraic geometry and representations of quantized 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 \({\mathfrak g}\) be a semisimple complex Lie algebra with a real form \({\mathfrak g}_{\mathbb R}\) and let \(G\), \(G_{\mathbb R}\) be the corresponding Lie groups. Assume that \(G_{\mathbb R}\) is connected and contains a compact maximal torus \(T\) with Lie algebra \({\mathfrak t}\). Let \({\mathfrak b}\) be a Borel subalgebra of \({\mathfrak g}\) containing a Cartan subalgebra which is the complexification of \({\mathfrak t}\) and let \(B\) be the corresponding Borel subgroup of \(G\). Let \(K_{\mathbb R}\) be the unique connected maximal compact subgroup of \(G_{\mathbb R}\) containing \(T\) and let \(K\) be the complexification of \(K_{\mathbb R}\). Write \(B_K:=K\cap B\), \(Z:=K/B_K\) and \(D:=G_{\mathbb R}/T\) (with an appropriate complex structure).
For any anti-dominant weight \(\mu\), there is an associated homogeneous line bundle \(L_\mu\) over \(D\). The Harish-Chandra module \(V^\mu\) can be realised as \(V^\mu:=H^d(D,L_\mu)\), where \(d\) is the dimension of \(Z\). The completion \(\widehat{V}^\mu\) is obtained by restricting \(H^d(D,L_\mu)\) to the formal neighbourhood of \(Z\) in \(D\). The modules \(V^\mu\) and \(\widehat{V}^\mu\) satisfy a linear PDE system. It is the object of this paper to investigate the differential relations satisfied by \(V^\mu\) and \(\widehat{V}^\mu\) in the context of the complex geometry of flag domains and the related correspondence and cycle spaces.
There are three filtrations of a certain cochain complex \((C^\bullet({\mathfrak n},\widehat{{\mathcal O}G}_{\mathcal W})_{-\mu},d)\); in fact, there are four filtrations in all, since the first has a variant which turns out to be more important. Each of these filtrations leads to a spectral sequence abutting to the completion \(\widehat{H}^q(D,L_\mu)\) of \(H^q(D,L_\mu)\) along \(Z\). The first filtration (and its variant) arise from a naïve consideration of the order of vanishing along the inverse image of \(Z\), the second gives rise to the Hochschild-Serre spectral sequence and the third is an amalgam of the other two. There are important compatibilities among the filtrations. Sections 3 to 6 of the paper are concerned with the filtrations, the associated spectral sequences and the relationships between them.
In Sections 7 and 8, the authors make the assumption that \(H^q(Z,\wedge^pN_{Z/D}^*\otimes L_\mu)=0\) for \(0\leq q\leq d-1\) and all \(p\geq0\). This assumption is satisfied in particular if \(\mu\) is \(K\)-anti-dominant and sufficiently far from the walls of the Weyl chamber \(-C_K\). They prove in Theorem 7.2 that the tableau \(A\) associated to \(V^\mu\) is \(H^d(Z,N^*_{Z/D}\otimes L_\mu)\) and is involutive. In the same theorem, they also identify the Spencer sequence associated with this tableau and describe the characteristic variety \(\Xi_A\) and the characteristic sheaf. The characteristic module \(M_{\mu,A}\) is also defined in terms of a minimal free resolution. The characteristic sheaf \({\mathcal M}_{\mu,A}\) is then defined in Section 8 as the localisation of \(M_{\mu,A}\). This sheaf has support \(\Xi_A\) and is analysed further in Section 8. Note in particular that \(\Xi_A\) depends only on the Weyl chamber for which \(\mu\) is anti-dominant, while \(M_{\mu,A}\) and \({\mathcal M}_{\mu,A}\) depend on \(\mu\) itself and encode more information than \(\Xi_A\). Section 8 also contains examples which illustrate the geometry behind the construction and the variety of phenomena that arise.
Finally, there is an appendix in which the authors discuss, without formal proofs, the definition of higher symbol maps and characteristic varieties when the vanishing assumption of Sections 7 and 8 does not hold. representations of Lie groups; Harish-Chandra module; PDE; homogeneous space; spectral sequence; correspondence space; characteristic variety Homogeneous spaces and generalizations, Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Semisimple Lie groups and their representations, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.), Homogeneous complex manifolds, Representation theory for linear algebraic groups, Overdetermined systems of PDEs with variable coefficients, Holomorphic fiber spaces, Differential geometry of homogeneous manifolds, Spectral sequences in algebraic topology On the differential equations satisfied by certain Harish-Chandra 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 A symmetric quiver \((Q,\sigma)\) is a finite quiver without oriented cycles \(Q=(Q_0,Q_1)\) equipped with a contravariant involution \(\sigma\) on \(Q_0\sqcup Q_1\). The involution allows us to define a nondegenerate bilinear form \(\langle-,-\rangle_V\) on a representation \(V\) of \(Q\). We shall say that \(V\) is orthogonal if \(\langle-,-\rangle_V\) is symmetric and symplectic if \(\langle-,-\rangle_V\) is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if \((Q,\sigma)\) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type \(c^V\) and, when the matrix defining \(c^V\) is skew-symmetric, by the Pfaffians \(pf^V\). To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector. symmetric quivers of tame type; representations of quivers; rings of semi-invariants; actions of products of classical groups; Coxeter functors; Pfaffians; Schur modules; generic decompositions; bilinear forms Representations of quivers and partially ordered sets, Representation type (finite, tame, wild, etc.) of associative algebras, Quadratic and bilinear forms, inner products, Geometric invariant theory, Vector and tensor algebra, theory of invariants Semi-invariants of symmetric quivers of tame 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 Reprint of the first edition (1984; Zbl 0588.14027). Mordell conjecture; Rational points; Seminar; Bonn/Germany; Wuppertal/Germany; proof of Tate conjecture; proof of Shafarevich conjecture; proof of the Mordell conjecture; logarithmic singularities; compactification of the moduli space of abelian varieties; modular height of an abelian variety; p-divisible groups; intersection theory on arithmetic surfaces; Riemann- Roch theorem; Hodge index theorem; rational points G. FALTINGS - G. WÜSTHOLZ, Rational points, Aspects of Math., Vieweg, 1986. Zbl0636.14019 MR863887 Arithmetic ground fields for abelian varieties, Rational points, Special algebraic curves and curves of low genus, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Conference proceedings and collections of articles, Elliptic curves Rational points. Seminar Bonn/Wuppertal 1983/84. 2nd ed | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 work on the problem of determining which zero dimensional schemes deform to a set of distinct points, i.e. are smoothable. This is a fundamental problem in the theory of Hilbert schemes of points -- and active and exciting area of research in algebraic geometry and commutative algebra.
The authors define a syzygetic invariant which gives insight to the problem mentioned above. Using their invariant the authors are able to deduce several interesting examples including: families which are not smoothable; Hilbert schemes of points which have components intersecting away from the smoothable component; and information about the Hilbert scheme of nine points in five dimensional affine space.
The paper is expertly written and contains necessary background information on Hilbert schemes, inverse systems, and regularity for homogeneous ideals. Several enlightening examples are given including computations of their introduced \(\kappa\)-vector. Theorems establishing necessary, as well as both necessary and sufficient conditions for certain classes of schemes of regularity two to be smoothable are given. Artinian algebras; smoothability; syzygies; deformation theory; punctual Hilbert schemes; Hilbert schemes of points Erman D., Velasco M.: syzygetic approach to the smoothability of 0-schemes of regularity two. Adv. Math. 224(3), 1143--1166 (2010) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Deformations and infinitesimal methods in commutative ring theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative Artinian rings and modules, finite-dimensional algebras A syzygetic approach to the smoothability of zero-dimensional 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 \(G\) be a connected reductive algebraic group over an algebraically closed field \(k\), defined and split over a perfect subfield \(k_ 0\) of \(k\). Fix a split torus \(T\) and a split Borel subgroup \(B\) containing \(T\). The author considers certain subsets of the unipotent radical of \(B\), defined by triples of elements of the Weyl group corresponding to \(T\). He gives an explicit decomposition of these sets as a disjoint union of locally closed \(k_ 0\)-subvarieties of \(G\), which are products of affine spaces with tori. This yields a geometric interpretation of formulas of \textit{N. Kawanaka} [Osaka J. Math. 12, 523-544 (1975; Zbl 0314.20031)] for the structure constants in the Hecke algebra of a finite Chevalley group. connected reductive algebraic groups; split torus; split Borel subgroups; unipotent radical; Weyl groups; products of affine spaces with tori; Hecke algebras; finite Chevalley groups Curtis, C. W.: A further refinement of the Bruhat décomposition. Proc. amer. Math. soc. 102, 37-42 (1988) Linear algebraic groups over arbitrary fields, Classical groups (algebro-geometric aspects), Linear algebraic groups over finite fields, Representation theory for linear algebraic groups A further refinement of the Bruhat decomposition | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(K\) be a field. In algebraic-geometric language, the Hilbert's irreducibility criterion can be expressed as the statement that the affine spaces over \(K\) are not thin. Recall that a subset \(T\) of a \(K\)-variety \(V\) is thin if there exists a proper Zariski-closed subset \(C\) of \(V\), and finitely many \(K\)-varieties \(V_1, \cdots, V_n\) each of dimension dim \(V\) and separable, dominant morphisms \(p_i : V_i \rightarrow V\) of degree \(> 1\) such that \(T \subseteq C(K) \cup \bigcup_{i=1}^n p_i(V_i(K))\).
\noindent In their famous paper [J. Algebra 106, 148--205 (1987; Zbl 0597.14014)] \textit{J. L. Colliot-Thélène} and \textit{J. J. Sansuc}, introduced the notion of a Hilbertian variety -- a \(K\)-variety \(V\) is said to be of Hilbert type if \(V(K)\) is thin. The importance of this definition is that one would have an affirmative answer to Noether's inverse Galois problem if one knew that every unirational variety over \(\mathbb{Q}\) is of Hilbert type. They showed that connected reductive algebraic groups over number fields are of Hilbert type. Recently, \textit{L. Bary-Soroker}, \textit{A. Fehm} and \textit{S. Petersen} [Ann. Inst. Fourier 64, No. 5, 1893--1901 (2014; Zbl 1359.12001)] generalized this and proved:
\noindent Let \(f : V \to S\) be a dominant morphism of \(K\)-varieties. If the set of \(s \in S(K)\) for which the fiber \(f^{-1}(s)\) is a \(K\)-variety of Hilbert type is known to be not thin, then \(V\) is of Hilbert type.
\noindent From this, they also deduced:
\noindent For a linear algebraic group \(G\) over a perfect Hilbertian field \(K\) and a connected algebraic subgroup, the quotient \(G/H\) is of Hilbert type. In particular, every linear algebraic group over a perfect Hilbertian field is of Hilbert type.
\noindent In the paper under review, the author proves that the above result carries over to non-connected subgroups also when \(K\) is a number field. More precisely, he proves:
\noindent Let \(K\) be a number field. Let \(G\) be a connected linear algebraic \(K\)-group, and let \(H \subset G\) be a \(K\)-subgroup, not necessarily connected. Assume that the semisimple group \([G/R_u(G),G/R_u(G)]\) is simply connected and that the kernel Ker \((H \rightarrow H^{\mathrm{mult}})\) is connected and geometrically character-free. Then, the variety \(G/H\) is of Hilbert type.
\noindent He proves a similar result for global fields of positive characteristic under the assumption that \(G\) above is reductive, \(H\) not necessarily smooth but that the kernel above is smooth, connected and semisimple. He proves the above theorem using a local-global principle for certain homogeneous spaces. In addition, he gives another proof using the so-called weak ``weak approximation''. Note that varieties with this property are of Hilbert type and tori have this weak weak approximation property although they don't always have the weak approximation property. variety of Hilbert type; thin sets; weak weak approximation; reductive groups; homogeneous spaces Borovoi, M., Homogeneous spaces of Hilbert type, Int. J. Number Theory, 11, 397-405, (2015) Homogeneous spaces and generalizations, Linear algebraic groups over global fields and their integers, Hilbertian fields; Hilbert's irreducibility theorem Homogeneous spaces of Hilbert 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 We describe the Horrocks-Mumford bundle, null-correlation bundles and Tango bundles in terms of graded modules over the exterior algebra via the Bernstein-Gelfand-Gelfand correspondence. Furthermore we show that properties such as indecomposability and stability of these bundles can be proven purely algebraically. vector bundles on projective spaces; graded modules; exterior algebras; Bernstein-Gelfand-Gelfand correspondence; Horrocks-Mumford bundles; Tango bundles; indecomposability; derived categories of coherent sheaves; finite-dimensional algebras Representations of orders, lattices, algebras over commutative rings, Module categories in associative algebras, Derived categories, triangulated categories, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Classical vector bundles and representations of quivers. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Inspired by algebraic geometry over groups, one of the authors of the paper under review, Boris Plotkin, alone and with collaborators, started about 10 years ago to develop algebraic geometry in a significantly more general setting, over algebras of an arbitrary variety \(\Theta\) of universal algebras. This area holds many similarities to classical algebraic geometry but, instead of closely connected with ideal theory of finitely generated polynomial algebras over fields, is closely associated with congruence theories of finitely generated free algebras of those varieties. The group \(\Aut(\Theta^0)\) of automorphisms of the category \(\Theta^0\) of finitely generated free algebras of \(\Theta\) is of great importance.
In the present paper, the authors define semi-inner automorphisms for the categories of free (semi)modules and free Lie modules. Under natural restrictions on the (semi)ring, all automorphisms of \(\Theta^0\) are semi-inner. This holds for the variety \(_R\mathcal M\) of semimodules over an IBN-semiring \(R\). (IBN means invariant basis number, i.e., if the free objects of finite ranks \(m\) and \(n\) are isomorphic, then \(m=n\).) In particular, this is true if the ring \(R\) is Artinian, Noetherian, PI, if \(R\) is a division semiring, etc. The main results are obtained as consequences of a general approach developed in the setting of semiadditive categorical algebra. The paper concludes with an appendix which provides a new easy-to-prove version of a reduction theorem of Mashevitzky, B. Plotkin, and E. Plotkin, used essentially in the present paper. universal algebraic geometry; free modules over Lie algebras; free semimodules over semirings; semi-inner automorphisms; varieties of universal algebras; congruences of finitely generated free algebras; automorphism groups; free Lie modules Katsov, Y.; Lipyanski, R.; Plotkin, B., Automorphisms of categories of free modules, free semimodules, and free Lie modules, Comm. Algebra, 35, 931-952, (2007) Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.), Rings arising from noncommutative algebraic geometry, Semirings, Module categories in associative algebras, Identities, free Lie (super)algebras, Automorphisms and endomorphisms of algebraic structures, Categories of algebras, Noncommutative algebraic geometry Automorphisms of categories of free modules, free semimodules, and free Lie 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 Surfaces of general type with \(p_g=0\) continue to be the object of intensive study. In [\textit{I. C. Bauer} and \textit{F. Catanese}, in: The Fano conference. Papers of the conference organized to commemorate the 50th anniversary of the death of Gino Fano (1871--1952), Torino, Italy, September 29--October 5, 2002. Torino: Università di Torino, Dipartimento di Matematica. 123--142 (2004; Zbl 1078.14051)], Ingrid Bauer and Fabrizio Catanese studied smooth projective surfaces with \(p_g=q=0\) that are the quotients of a product of curves (isogenous to a product of curves) by a finite abelian group \(G\). They showed that only four abelian groups \(G\) are possible in this case and proved that all the cases occur, giving also a description of the connected components of the corresponding moduli space. The present paper computes the integral cohomology groups of these four families. surfaces of general type; \(p_g=0\); homology groups; products of curves; actions of finite abelian group; isotrivial fibrations; surfaces isogenous to a product; fake quadrics; branched coverings; fundamental group Surfaces of general type, Classical real and complex (co)homology in algebraic geometry, Series and lattices of subgroups Homology of some surfaces with \( p_g = q = 0\) isogenous to a product | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In the book under review, the authors provide the very first comprehensive introduction to the theory and rich applications of Cox rings. These rings are really important in the modern algebraic and arithmetic geometry. As it was pointed out by the authors, the whole story came back to the idea of \textit{D. A. Cox} [J. Algebr. Geom. 4, No. 1, 17--50 (1995; Zbl 0846.14032)] around 1995 when he introduced the notion of homogeneous coordinate rings in the context of toric varieties. Five years later, \textit{Y. Hu} and \textit{S. Keel} [Mich. Math. J. 48, 331--348 (2000; Zbl 1077.14554)] observed a fundamental connection between Mori theory and geometric invariant theory (GIT) via Cox rings characterizing the so-called Mori Dream Spaces as a special class of algebraic varieties having finitely generated Cox rings.
Let us recall that the Cox ring of a projective variety \(X\) over an algebraically closed field is defined as
\[
\text{Cox}(X) = \bigoplus_{(n_{1}, \dots, n_{r}) \in \mathbb{Z}^{r}} H^{0}\bigg(X, \mathcal{O}_{X}(L_{1}^{n_{1}} \otimes \dots \otimes L_{r}^{n_{r}})\bigg),
\]
where \((L_{1}, \dots, L_{r})\) is a basis of the \(\mathbb{Z}\)-module \(\text{Pic}(X)\) of classes of invertible sheaves on \(X\) modulo isomorphism under the tensor product, and we have assumed that the linear and numerical equivalence of the group of Cartier divisors on \(X\) are the same.
We have several natural questions and problems that can be addressed in the context of Cox rings, for instance: {\parindent=6mm \begin{itemize}\item[-] find algebraic / geometrical properties that are decoded by Cox rings; \item[-] find conditions which guarantee that Cox rings are finitely generated; \item[-] find generators (and relations between generators) for these rings.
\end{itemize}} These basic three problems can be viewed as a core of the book, and the authors answer on them, successively, developing the whole theory from the basic notions. Of course, the readers are assumed to be familiar with basic concepts of algebraic geometry (for instance those from \textit{R. Hartshorne}'s fundamental textbook [Algebraic geometry. Graduate Texts in Mathematics. 52. New York-Heidelberg-Berlin: Springer-Verlag. (1977; Zbl 0367.14001)]), and they should have some experience with toric varieties, algebraic groups, and finally GIT. Additionally, according to my point of view, knowing some basic aspects of the Minimal Model Programme would be also useful in order to understand geometrical meaning of certain objects better. Let me now present, only briefly, the main topics of the book. I will add some comments about the most interesting (from my subjective point of view) aspects of the theory.
In Chapter 1, the authors introduce the notion of Cox rings via Cox sheaves. Starting from basics devoted to graded algebras, gradings, and quasitorus actions, they define Cox sheaves for normal irreducible varieties with finitely generated class groups, which allows to define Cox rings. In addition, the authors study basic algebraic properties of these rings, namely normality and integrity, and in a concluding section, they look at geometric realizations of Cox sheaves.
In Chapter 2, the authors focus on the case of toric varieties. As a first step, they define toric varieties and present a fundamental Cox's result which tells us (roughly) that Cox rings of toric varieties are polynomial rings with appropriate gradings. After that, they introduce the linear Gale duality and the concept of bunches of cones. Next, they also recall and study the Gelfand-Kapranov-Zelevinsky (GKZ) decompositions with full technical proofs. At the end of the chapter, they also study the so-called good toric quotients and they look at toric varieties from a point of view of bunches of cones. I think that this chapter would be easier to understand (and some technical parts) if the authors would made a use of matroid theory.
In Chapter 3, the authors present a combinatorial approach toward the geometry of varieties with finitely generated Cox rings via GIT. At the very beginning, they provide a short introduction to geometric invariant theory, and after that they focus on the so-called bunched rings. Personally, I really appreciate that the authors provide examples of flag varieties and quotients of quadrics from the point of view of bunched rings -- it was really helpful to understand this approach properly. Finally, they present a short introduction to base loci and cones of divisors and they define Mori Dream Spaces. In particular, they present a very nice result saying that any effective Weyl divisor on a Mori Dream Space admits a unique Zariski decomposition (please be aware of the fact that here one assumes that the positive part of the decomposition is \textit{movable} -- see Definition 3.3.4.6 for details). The last section of this chapter is devoted to \(T\)-varieties of complexity \(1\) via bunched rings.
In Chapter 4, the authors focus on various geometric problems related to Cox rings. They present some finite generation criteria. In particular, they recall a geometric criterion formulated by Hu and Keel which tells us that the finite generation of the Cox ring of an irreducible normal complete variety \(X\) with finitely generated divisor class group is equivalent to the existence of finitely many small \(\mathbb{Q}\)-factorial modifications such that the semiample cone of the target space for each modification is polyhedral and the movable cone of \(X\) is the sum of pull-backs of semiample cones of the target spaces of all small \(\mathbb{Q}\)-factorial modifications. Another criterion tells us that the finite generation of the Cox ring is equivalent to the fact that the movable and the semiample cone coincide and this cone is polyhedral. The authors present the concept of Cox-Nagata rings which are strictly related to Nagata's counterexample to Hilbert's fourteenth problem. They also present a small (but remarkable) observation that the Cox rings of the blow up of the projective plane along \(9\) very general points is not finitely generated, which geometrically can be rephrased saying that for this blow-up the number of negative curves (i.e., reduced and irreducible curves with negative self-intersections) is infinite. Finally, they also recall a very interesting result due to \textit{A.-M. Castravet} and \textit{J. Tevelev} [Compos. Math. 142, No. 6, 1479--1498 (2006; Zbl 1117.14048)] (generalizing a result due to Mukai) which tells us that for the variety \(X_{a,b,c}\) obtained by blowing-up of \((\mathbb{P}^{c-1})^{a-1}\) at \(b+c\) points in (very) general position the condition that the Cox ring of \(X_{a,b,c}\) is finitely generated is equivalent that the following inequality holds: \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} > 1.\) Moreover, the authors study liftability of automorphisms (including automorphisms of Mori Dream Spaces) and another criteria of the finite generation of Cox rings for varieties with torus action and for almost homogeneous varieties.
In Chapter 5 (the most interesting from my point of view), the authors focus on the so-called Mori Dream Surfaces and they provide a bunch of finite-generation criteria for different classes of surfaces. After recalling basics on algebraic surfaces, they provide a first result for rational surfaces which tells us that if \(X\) is a smooth complex rational projective surface with anti-canonical Itaka dimension \(\geq 1\), then \(X\) is a Mori Dream Surface if anti-canonical Itaka dimension is equal to \(2\) or this dimension is equal to \(1\) and a relatively minimal model of the associated elliptic fibration is extremal. This result can be also translated into geometry since \(X\) as above is a Mori Dream Surface if and only if it contains only finitely many \((-1)\) and \((-2)\)-curves. Next, the authors consider the so-called extremal rational elliptic surfaces (actually Jacobian elliptic fibrations which were classified by \textit{R. Miranda} and \textit{U. Persson} [Math. Z. 193, 537--558 (1986; Zbl 0652.14003)]), and for these fibrations they provide a criterion which tells us that for a rational smooth projective surface \(X\) admitting jacobian and relatively minimal elliptic fibration \(\pi\) the fact that \(X\) is a Mori Dream Surface is equivalent that the Mordell-Weil group \(MW(\pi)\) is finite. In the next section, the authors consider \(K3\) surfaces and they present a criterion saying that for a \(K3\) surface \(X\) the finite generation of the Cox ring is equivalent that the automorphism group of \(X\) is finite. After these (somehow) general results, the authors focus on smooth del Pezzo surfaces providing a complete description of their Cox rings. In the last section of this chapter, the authors study Cox rings of Gorenstein log del Pezzo \(\mathbb{K}^{*}\)-surfaces.
In Chapter 6, the authors study arithmetic questions related to Cox rings. Starting with the notion of universal torsors, they study the problem of the existence of rational points on algebraic varieties (Hasse principle, the Brauer-Manin obstructions). In the rest part of this chapter, the authors study Manin's conjecture which (very roughly speaking) predicts the asymptotic behaviour of the number of rational points of bounded anti-canonical height.
At this point I would like to emphasize that I really appreciate the way how the authors wrote this book. It contains a lot of explicit examples and basic constructions. This allow to understand the theory much faster, even if certain parts are really technical, and I think this is the key asset of the book. Moreover, after each chapter, the authors provide a list of exercises (sometimes involving) which helps to study the subject in detail. Of course, one can find some flaws and typos, and there are some ambiguous places which might be better explained, but it does not change the fact that this an extremely interesting and friendly-written introduction to the subject, and it is suitable for advanced graduate students. Cox rings; algebraic varieties; homogeneous spaces; graded algebras and rings; line bundles; toric varieties; geometric invariant theory; actions of groups; algebraic surfaces; Mori Dream Spaces; Zariski decompositions; Manin's conjecture; Hasse principle; Brauer-Manin obstructions; del Pezzo surfaces; \(K3\) surfaces; Enriques surfaces; GKZ decompositions; GALE transformations; flag varieties; combinatorial methods in algebraic geometry Arzhantsev, Ivan; Derenthal, Ulrich; Hausen, Jürgen; Laface, Antonio, Cox rings, Cambridge Studies in Advanced Mathematics 144, viii+530 pp., (2015), Cambridge University Press, Cambridge Research exposition (monographs, survey articles) pertaining to algebraic geometry, Divisors, linear systems, invertible sheaves, Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies Cox 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 We develop a Harder-Narasimhan theory for Kisin modules generalizing a similar theory for finite flat group schemes due to \textit{L. Fargues} [J. Reine Angew. Math. 645, 1--39 (2010; Zbl 1199.14015)]. We prove the tensor product theorem, in other words, that the tensor product of semi-stable objects is again semi-stable. We then apply the tensor product theorem to the study of Kisin varieties for arbitrary connected reductive groups. algebraic groups; deformation theory; finite flat group schemes; geometric invariant theory; \(p\)-adic Hodge theory Galois theory, Geometric invariant theory, Modular and Shimura varieties A Harder-Narasimhan theory for Kisin 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 Let \(A\) be a homogeneous coordinate ring of a quantum projective plane, which is a non-commutative analogue of the polynomial ring in three variables. When \(A\) is a finite module over its center \(Z(A)\), the author defines a scheme \(S=\text{Proj}(Z(A))\) and a sheaf \(\mathcal A\) of \({\mathcal O}_S\)-algebras by \({\mathcal A}(S_{(f)})=A[f^{-1}]\), where \(S_{(f)}\) is the open affine subset of \(S\) for each non-zero homogeneous element \(f\in Z(A)\). The center \(\mathcal Z\) of \(\mathcal A\) is defined by \({\mathcal Z}(S_{(f)})=Z(A[f^{-1}]_0)\) and \(\text{Spec}({\mathcal Z})\) is computed for several families of algebras \(A\). quantum projective planes; homogeneous coordinate rings; graded algebras; Hilbert series; regular algebras; schemes; sheaves of algebras Mori, I, The center of some quantum projective planes, J. Algebra, 204, 15-31, (1998) Graded rings and modules (associative rings and algebras), Associative rings of functions, subdirect products, sheaves of rings, Noncommutative algebraic geometry, Center, normalizer (invariant elements) (associative rings and algebras) The center of some quantum projective planes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 proposes a generalization of the well-known triality between simply-laced Dynkin diagrams, simple Lie algebras, and Kleinian groups (with associated quotient singularities), and provides an account of the results achieved in this direction by the author and others. The exposition is exhaustive while concise, and subtle points and open questions are emphasized throughout the text.
The idea is to attach a surface singularity and a Lie algebra to a regular system of weights, which denotes a system of four integers \(W:=(a,b,c,h)\). A number of requirements is imposed upon this singularity and this algebra, in order to make this a generalization of a usual triality picture described above and also of elliptic Lie algebra theory. A singularity is defined as a zero set of a generic polynomial \(f_W\) of degree \(h\) in a weighted projective space \(P(a:b:c)\). An algebra is constructed through an intermediate step, which includes a construction of the homotopy category of graded matrix factorizations for \(f_W\), establishing a strongly exceptional collection there, and taking its associated quiver to construct a Lie algebra.
On the ``geometrical side'', investigating the singularities obtained from regular weight systems, and their vanishing cycle lattices, the author presents a notion of \(*\)-duality for regular weight systems. On one hand, it is shown to underlie the ``strange'' duality of Arnold; on the other, the author cites \textit{A. Takahashi} [Commun. Math. Phys. 205, No. 3, 571--586 (1999; Zbl 0974.14005)] where it is proven that the \(*\)-duality is equivalent to mirror symmetry for Landau-Ginzburg models. regular system of weights; simple Lie algebras; elliptic Lie algebras; homotopy category of matrix factorizations; vanishing cycles; \(*\)-duality Saito, K.: Towards a categorical construction of Lie algebras, Adv. stud. Pure math. 50, 101-175 (2008) Categories in geometry and topology, Lie algebras and Lie superalgebras, Singularities of surfaces or higher-dimensional varieties, Homology and cohomology theories in algebraic topology, Simple, semisimple, reductive (super)algebras, Derived categories, triangulated categories, Calabi-Yau manifolds (algebro-geometric aspects) Towards a categorical construction on 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 study of finite subgroups of a simple algebraic group \(G\) reduces in a sense to those which are almost simple. If an almost simple subgroup of \(G\) has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of \(G\), then the subgroup is called \textit{non-generic}. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic.
A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups \(\mathrm{Alt}_n\), \(\mathrm{Sym}_n\) for \(n\geq 10\), do not occur as Lie primitive subgroups of an exceptional algebraic group.
A subgroup of \(G\) is called \(G\)-completely reducible if, whenever it lies in a parabolic subgroup of \(G\), it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-\(G\)-completely reducible, non-generic simple subgroups.
As an intermediate result, for each simply connected \(G\) of exceptional type, and each non-generic finite simple group \(H\) which embeds into \(G/Z(G)\), we derive a set of \textit{feasible characters}, which restrict the possible composition factors of \(V\downarrow S\), whenever \(S\) is a subgroup of \(G\) with image \(H\) in \(G/Z(G)\), and \(V\) is either the Lie algebra of \(G\) or a non-trivial Weyl module for \(G\) of least dimension.
This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for \(n\geq 10\), \(\mathrm{Alt}_n\) and \(\mathrm{Sym}_n\), as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type. algebraic groups; exceptional groups; finite simple groups; Lie primitive; subgroup structure; complete reducibility Research exposition (monographs, survey articles) pertaining to group theory, Exceptional groups, Group schemes On non-generic finite subgroups of exceptional 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 \par
\textit{Masayoshi Nagata}, \textit{Hideyasu Sumihiro} and \textit{Masayoshi Miyanishi}, ''History of invariant theory. I.'' (2--12);
\textit{Masayoshi Nagata}, \textit{Hideyasu Sumihiro} and \textit{Masayoshi Miyanishi}, ''History of invariant theory. II. Geometric invariant theory'' (13--75);
\textit{Yōhei Tanaka}, ''Differential invariants and pseudoseries invariants'' (76--95);
\textit{Keiichi Watanabe}, ''On the conditions under which invariant subrings become complete intersections'' (96--117);
\textit{Toshiyuki Tanizaki}, ''On the defining ideals of the closure of conjugacy classes consisting of pseudonilpotent matrices, and representations of Weyl groups'' (118--141);
\textit{Takehiko Miyata}, ''Invariants of finite groups and simple algebras'' (142--163);
\textit{Haruhisa Nakajima}, ''Invariants of finite groups and Hilbert functions'' (164--174);
\textit{Yutaka Hiramine}, ''Permutation groups and invariants - introduction to Wielandt's work'' (175--179);
\textit{Tomoyuki Yoshida}, ''Applications of invariant theory to coding theory'' (180--197);
\textit{Tatsuo Kimura}, ''Prehomogeneous vector spaces and relative invariants'' (198--208);
\textit{Tamaki Yano}, ''Invariants of finite reflection groups and a flat coordinate system for isolated singularities'' (209--235);
\textit{Kenji Ueno}, ''On a certain automorphism on K3 surfaces'' (236--242).
The articles (all in Japanese) of this volume will not be reviewed individually. invariant theory; Proceedings; Symposium; Kyoto; RIMS; geometric invariant theory; Hilbert functions; coding theory; prehomogeneous vector spaces; reflection groups Geometric invariant theory, Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to linear algebra, Proceedings, conferences, collections, etc. pertaining to group theory, Group actions on varieties or schemes (quotients) Invariant theory and related matters. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, March 16--18, 1981 | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 a necessary and sufficient condition for the Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continued fractions. Moreover, we prove that any three dimensional Gorenstein abelian quotient singularity possesses a crepant Fujiki-Oka resolution as a corollary. This alternative proof of existence needs only simple computations compared with the results ever known. crepant resolutions; Fujiki-Oka resolutions; higher dimension; finite groups; abelian groups; Hirzebruch-Jung continued fractions; invariant theory; multidimensional continued fractions; quotient singularities; toric varieties Singularities in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, \(3\)-folds, \(4\)-folds, \(n\)-folds (\(n>4\)), Group actions on varieties or schemes (quotients), Toric varieties, Newton polyhedra, Okounkov bodies, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Crepant property of Fujiki-Oka resolutions for Gorenstein abelian quotient 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 A Mori dream space is a normal projective variety with finitely generated divisor class group and finitely generated Cox ring. This paper deals with computing the automorphism group of such a space via an algorithm. A set of homogeneous generators and relations of the cox ring, together with the class of an ample divisor serves as an input of the algorithm. As an application of their algorithm, the authors derive various theoretical consequences such as a bound for the dimension of the automorphism group as well as practical computations like the classification of the automorphism groups of cubic surfaces with at most ADE singularities for a general choice of parameters. They also note that their approach can be applied to Gröbner basis computations or tropical varieties. group actions on varieties and schemes; actions of groups on commutative rings; invariant theory; automorphisms of surfaces and higher-dimensional varieties 10.1090/mcom/3185 Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory, Automorphisms of surfaces and higher-dimensional varieties, Computational aspects of higher-dimensional varieties Computing automorphisms of Mori dream spaces | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Given a finite Abelian algebraic group \(G=\text{Spec}(A)\) over a field \(k\), one can quickly develop the theory of linear representations of \(G\). In particular, the one-dimensional linear representations form the Cartier dual to \(G\), \(G^*=\Hom(G,\mathbb G_m)\), represented by the linear dual \(A^*\) to \(A\). However, this theory does not hold if \(G\) is infinite -- for example \(\Hom(G,\mathbb G_m)\) is not a scheme, and \(\Hom(A,k)\) is not, in general, a Hopf algebra.
In the work under review, the authors generalize the notions of group scheme and \(k\)-algebras, which allow them to develop a representation theory for Abelian algebraic groups, and in doing so develop of Cartier duality which behaves functorially.
In detail, let \(\mathcal K\) be the identity functor on the category \(\mathcal C\) of commutative \(k\)-algebras. Then a \(\mathcal K\)-algebra is a covariant functor \(\mathcal A\) from \(\mathcal C\) to Abelian groups such that \(\mathcal A(C)\) is a \(C\)-algebra for all objects \(C\) in \(\mathcal C\). One could use the term ``algebra scheme'' in lieu of \(\mathcal K\)-algebra. For a \(\mathcal K\)-algebra \(\mathcal A\) we define \(\text{Spec}(\mathcal A)=\mathcal Hom_{\mathcal K\text{-alg}}(\mathcal A,\mathcal K)\). If \(\mathcal A\) is quasi-coherent, i.e., \(\mathcal A\) is the cokernel of a map \(\mathcal K^n\to\mathcal K^m\), then \(\text{Spec}(\mathcal A)\) and \(\text{Spec}(A)\) agree on points, where \(A\) is the \(k\)-algebra corresponding to \(\mathcal A\), that is, \(\mathcal A(C)=A\otimes_kC\) for all objects \(C\) in \(\mathcal C\). Then \(X:=\text{Spec}(\mathcal A)\) is said to be an affine \(k\)-scheme, agreeing with the usual notion of affine scheme when evaluated at points, and \(X\) is an inductive finite \(k\)-scheme if \(\mathcal A\) is a profinite \(\mathcal K\)-algebra. The dual of \(\mathcal A\) is defined by \(\mathcal A^*(C)=\Hom_{k\text{-alg}}(S^{\cdot}M,C)\), where \(S^{\cdot}M\) is the symmetric algebra of \(M\), and such functions are called distributions. With these generalized definitions, the category of \(G\)-modules, \(G\) finite or inductive finite, is equivalent to the category of \(\mathcal A^*\)-modules, and \(G^*=\Hom(G,\mathbb G_m)\). Abelian algebraic groups; representations of algebraic groups; Cartier duality; profinite algebras; affine schemes Group rings of infinite groups and their modules (group-theoretic aspects), Group schemes, Affine algebraic groups, hyperalgebra constructions, Representation theory for linear algebraic groups Profinite algebras and their 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 Any simple elliptic singularity of type \(\~{D}_5\) can be obtained by taking the intersection of the nilpotent variety and the 4-dimensional ''good slices'' in the semi-simple Lie algebra \(sl\)(2,\(C\))\(sl\)(2,\(C\)). We describe these new slices purely by the structure of the Lie algebra. We also construct the semi-universal deformation spaces of View the MathML sourceView the MathML source-singularities by using the 4-dimensional ''good slices''. simple elliptic singularities; \(\tilde{D}_5\)-singularities; Lie algebras; deformations; Björner-Welker sequence A new construction of \(\tilde{D}_5\)-singularities and generalization of slodowy slices | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(G\) denote either a special orthogonal group or a symplectic group defined over the complex numbers. We prove the following saturation result for \(G\): given dominant weights \(\lambda^1,\dots,\lambda^r\) such that the tensor product \(V_{N\lambda^1}\otimes\cdots\otimes V_{N\lambda^r}\) contains nonzero \(G\)-invariants for some \(N\geqslant 1\), we show that the tensor product \(V_{2\lambda^1}\otimes\cdots\otimes V_{2\lambda^r}\) also contains nonzero \(G\)-invariants. This extends results of Kapovich-Millson and Belkale-Kumar and complements similar results for the general linear group due to Knutson-Tao and Derksen-Weyman. Our techniques involve the invariant theory of quivers equipped with an involution and the generic representation theory of certain quivers with relations. reductive groups; irreducible representations; semi-invariants of quivers; tensor product multiplicities; classical groups; invariant theory Sam, S, Symmetric quivers, invariant theory, and saturation theorems for the classical groups, Adv. Math., 229, 1104-1135, (2012) Representation theory for linear algebraic groups, Representations of quivers and partially ordered sets, Geometric invariant theory, Vector and tensor algebra, theory of invariants, Actions of groups on commutative rings; invariant theory Symmetric quivers, invariant theory, and saturation theorems for the classical groups. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities [For the entire collection see Zbl 0676.00006.]
Let F be an algebraically closed field of characteristic 0, G a reductive linear algebraic group defined over F, V a finite dimensional representation over F which has an element with trivial G-stabilizer, F[V] the symmetric algebra on V and F(V) its field of fractions. Let \(F(V)^ G\) be the field of G-invariant elements of F(V). According to Bogomolov's result \(F(V)^ G\) is up to stable isomorphism independent of the choice of V. This allows to talk about the invariant field of G without specifying V, denote it simply by F(G). In the paper under review some connections between invariant fields of reductive linear groups and division algebras are discussed.
The main result of the paper is the following Theorem 1. Suppose \(\phi\) : G\({}'\to G\) is a surjective homomorphism of linear connected reductive algebraic groups with central kernel cyclic of order n. Then \(F(G')\) is stably isomorphic to F(G)(A) for some central simple algebra \(A| F(G)\) of exponent n, where F(G)(A) is the function field of the Brauer- Severi variety defined by A. Furthermore some results are presented for some specific groups: \(SL_ n\), Spin groups of odd degree and \(SO_ n\). It should be noted the importance of the result about \(SL_ n\) and its images. Theorem 2. Let \(D=UD(F,n,2)\) be the generic division algebra over F of degree n in two variables, Z the center of D and A a division algebra in the class r[D]\(\in Br(Z)\). Set \(G_ r\) to be \(SL_ n(F)/C_ r\) where \(C_ r\) is the central cyclic subgroup of order r, let V be a \(G_ r\)-representation over F with an element with trivial stabilizer and \(F(G_ r)\) the invariant field of \(G_ r\) on F(V). Then \(F(G_ r)\) is stably isomorphic to Z(A), where Z(A) is the function field of the Brauer-Severy variety defined by A. finite dimensional representation; symmetric algebra; stable isomorphism; invariant fields; reductive linear groups; division algebras; function field; Brauer-Severi variety David J. Saltman, Invariant fields of linear groups and division algebras, Perspectives in ring theory (Antwerp, 1987) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 233, Kluwer Acad. Publ., Dordrecht, 1988, pp. 279 -- 297. Division rings and semisimple Artin rings, Other matrix groups over rings, Vector and tensor algebra, theory of invariants, Representation theory for linear algebraic groups, Brauer groups of schemes, Galois cohomology, Rational and unirational varieties Invariant fields of linear groups and division 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 \(S\) be a smooth complex algebraic surface of general type. If \(p_g(S)=q(S)\) (equivalently \(\chi(\mathcal O_S)=1\)), then \(p_g(S)\leq 4\). Surfaces with \(p_g=q=3\) or \(4\) are described by \textit{F. Catanese, C. Ciliberto} and \textit{M. Mendes Lopes} [Trans. Am. Math. Soc. 350, No. 1, 275--308 (1998; Zbl 0889.14019)], \textit{C. D. Hacon} and \textit{R. Pardini} [Trans. Am. Math. Soc. 354, No. 7, 2631--2638 (2002; Zbl 1009.14004)] and \textit{G. P. Pirola} [Manuscr. Math. 108, No. 2, 163--170 (2002; Zbl 0997.14009)], but surfaces with \(p_g=q\leq 2\) are still not completely classified.
Frequently quotients of surfaces by the action of a finite group have been used to produce new examples of algebraic surfaces. A smooth projective surface \(S\) is said to be \textit{isogenous to a product} if there exist two smooth curves \(C\), \(F\) and a finite group \(G\) acting freely on \(C\times F\) so that \(S=(C\times F)/G\).
The classification of surfaces with \(p_g=q=0\) which are isogenous to a product was obtained by \textit{I. C. Bauer, F. Catanese} and \textit{F. Grunewald} [Pure Appl. Math. Q. 4, No. 2, 547--586 (2008; Zbl 1151.14027)]. In this paper the classification for the case \(p_g=q=1\) is given. This is a sequel to the paper [\textit{F. Polizzi}, Commun. Algebra 36, No. 6, 2023--2053 (2008; Zbl 1147.14017)]. Some computations are done using the computer algebra program \texttt{GAP4}. surfaces of general type; isotrivial fibrations; actions of finite groups G. Carnovale and F. Polizzi, The classification of surfaces with \(p_g=q=1\) isogenous to a product of curves, Adv. Geom. 9 (2009), no. 2, 233--256. Surfaces of general type, Group actions on varieties or schemes (quotients), Computational aspects in algebraic geometry, Generators, relations, and presentations of groups The classification of surfaces with \(p_g=q=1\) isogenous to a product 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 [For the entire collection see Zbl 0722.00006.]
The Hartshorne-Rao module of a curve \(C\subset\mathbb{P}^ 3_ k\) is defined as \(M_ C=\sum_{n\in\mathbb{Z}}H^ 1({\mathcal I}_ C(n)))\), where \({\mathcal I}_ C\) denotes the ideal sheaf of \(C\). Let \(C_ i\), \(i=1,2\), be two curves. Is there a construction of a curve \(C\) (in terms of \(C_ i)\) such that \(M_ C\) is isomorphic to some shift of \(M_{C_ 1}\oplus M_{C_ 2}\)? This is solved by \textit{P. Schwartau} in his unpublished thesis ``Liaison addition and monomial ideals'' [Ph. D. thesis, Brandeis University 1982), see also \textit{J. Stückrad} and \textit{W. Vogel} [``Buchsbaum rings and applications. An interaction between algebra, geometry, and topology'', VEB Deutscher Verlag der Wissenschaft (Berlin 1986; Zbl 0606.13017); published simultaneous by Springer Verlag)] for a reproduction of his arguments.
In this paper the authors generalize Schwartau's result in several senses: (1) There is a liaison addition in any codimension in \(\mathbb{P}^ n_ k\), \(n\geq 3\), of schemes of mixed codimensions. --- (2) There is an addition of any number of schemes. --- (3) There is no need that the schemes are attached via a complete intersection (as in Schwartau's result).
The basic idea is the construction of a scheme \(Z\) consisting set- theoretically of the union of certain given schemes such that the cohomology of \(Z\) is related to those of the given schemes in terms of a long exact sequence. This far reaching generalization yields, in the case the underlying scheme is arithmetically Cohen-Macaulay, that the cohomology of \(Z\) is the shift of the direct sum of the involved schemes. There are a number of applications of this clever technique to the construction of arithmetically Buchsbaum schemes, the double linkage, Hilbert functions, etc. decomposition of Hartshorne-Rao module; arithmetical Cohen-Macaulay scheme; liaison addition; arithmetically Buchsbaum schemes; double linkage; Hilbert functions Linkage, Linkage, complete intersections and determinantal ideals A generalized liaison addition | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 construct and study an action of the affine braid group associated with a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a ``categorical version'' of Kazhdan-Lusztig-Ginzburg's construction of the affine Hecke algebra, and is used in particular by the first author and I. Mirković in the course of the proof of Lusztig's conjectures on equivariant \(K\)-theory of Springer fibers. braid groups; reductive algebraic groups; Lie algebras; Springer resolutions; affine Hecke algebras; dg-schemes; Fourier-Mukai transform Berukavnikov, R; Riche, S, Affine braid group actions on derived categories of Springer resolutions, Ann. Sci. l'Éc. Norm. Supèr. Quatr. Sér. 4, 45, 535-599, (2012) Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Braid groups; Artin groups, Modular Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Derived categories, triangulated categories Affine braid group actions on derived categories of Springer resolutions. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Das Buch ist aus einer Reihe von Vorträgen entstanden, die M. J. Taylor als Gastprofessor an der Universität Bordeaux gehalten hat.
Die klassische Theorie der komplexen Multiplikation liefert elliptische Funktionen, deren singuläre Werte abelsche Erweiterungen imaginär- quadratischer Zahlkörper erzeugen (Kroneckers Jugendtraum).
Das Hauptthema des Buches ist das Studium elliptischer Funktionen, deren singuläre Werte den Ring der ganzen algebraischen Zahlen von Klassenkörpern imaginär-quadratischer Zahlkörper erzeugen. Die richtige Wahl dazu sind die Funktionen aus \textit{R. Fueter}s in Vergessenheit geratenem Buch ''Vorlesungen über die singulären Moduln und die komplexe Multiplikation der elliptischen Funktionen'', Bde. I und II (Teubner, Leipzig-Berlin (1924; JFM 50.0101.01), (1927; JFM 53.0141.10).
Das zu referierende Buch ist sehr interessant, und da die Sprache der komplexen Funktionentheorie und der elliptischen Funktionen benutzt wird, besonders leicht lesbar. Zu kritisieren ist lediglich die unübersichtliche Art der Numerierung von Definitionen, Sätzen etc. und das Fehlen der Kapitelnummer auf jeder Seite, das ein Nachschlagen erschwert.
Nun zum Inhalt der einzelnen Kapitel: Das erste Kapitel enthält einige Resultate aus der Theorie der Kreiskörper, mit Hilfe derer später der Parallelismus zu den neuen Resultaten aufgezeigt wird. Das zweite und dritte Kapitel stellen eine kurze Zusammenfassung von Resultaten aus der Klassenkörpertheorie bzw. der Theorie der elliptischen Funktionen dar, die in den weiteren Kapiteln benötitgt werden.
Im vierten und fünften Kapitel werden die Eigenschaften der elliptischen Funktionen von Fueter bzw. der Ideale, die von Teilwerten dieser Funktionen erzeugt werden nach dem Muster von Fueters Buch studiert. Das sechste Kapitel enthält die Theorie der Abelschen Resolventen, das siebte und achte Kapitel die für das Weitere benötigten Ergebnisse aus der Theorie der Modulfunktionen und der klassischen komplexen Multiplikation.
In Kapitel neun wird untersucht, wie die Galoisgruppe von Klassenkörpern auf den singulären Werten verschiedener elliptischer Funktionen und Modulfunktionen operiert. Kapitel zehn beschreibt die Galoismodul-Struktur des Rings der ganzen algebraischen Zahlen einiger lokaler Erweiterungen mit Hilfe der Lubin-Tate-Theorie der formalen Gruppen.
Im elften Kapitel schließlich wird das Ziel des Buches erreicht, nämlich die Konstruktion des erzeugenden Elementes des Ringes der ganzen algebraischen Zahlen von gewissen algebraischen Erweiterungen imaginär-quadratischer Zahlkörper. Das Buch schließt mit einem Anhang über die Beziehungen zwischen den Funktionen von Fueter und elliptischen Einheiten. modular functions; automorphic functions; complex multiplication; abelian extensions; class field theory; elliptic functions; rings of algebraic integers; cyclotomic fields; abelian resolvents; Galois module structure; formal groups; Kroneckers Jugendtraum Cassou-Noguès, Ph.; Taylor, M. J., Elliptic Functions and Rings of Integers, Progr. Math., vol. 66, (1987), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc. Boston, MA Research exposition (monographs, survey articles) pertaining to number theory, Integral representations related to algebraic numbers; Galois module structure of rings of integers, Complex multiplication and moduli of abelian varieties, Modular and automorphic functions, Cyclotomic extensions, Class field theory, Elliptic functions and integrals, Formal groups, \(p\)-divisible groups Elliptic functions and rings or integers | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 briefly review the formal picture in which a Calabi-Yau \(n\)-fold is the complex analogue of an oriented real \(n\)-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of \textit{K. Behrend} and \textit{B. Fantechi} [Invent. Math. 128, 45--88 (1997; Zbl 0909.14006)] and \textit{Jun Li} and \textit{Gang Tian} [J. Am. Math. Soc. 11, 119--174 (1998; Zbl 0912.14004)] in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in \(\mathbb{P}^3\), and Donaldson- and Gromov-Witten-like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold \(X\), prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general \(K3\) fibration \(X\), enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the ``Mukai-dual'' 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of \(X\). Casson invariant; Calabi-Yau 3-fold; virtual moduli cycles; Hilbert schemes of curves; Gromov-Witten invariants; bundles on K3 fibrations R.\ P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds and bundles on K3-fibrations, J. Differential Geom. 54 (2000), 367-438. Calabi-Yau manifolds (algebro-geometric aspects), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Compact complex \(3\)-folds, Calabi-Yau theory (complex-analytic aspects), \(3\)-folds, \(K3\) surfaces and Enriques surfaces, Fibrations, degenerations in algebraic geometry A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on \(K3\) fibrations. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 special linear group \(\text{SL}_{n+1}(K)\) is the simply connected group and the projective linear group \(\text{PGL}_{n+1}(K)\) the adjoint group of Lie type \(A_n\). They are distinguished sections of the (reductive) general linear group \(\text{GL}_{n+1}(K)\) (admitting the same root system). We shall introduce corresponding connected algebraic groups resp. finite groups for each Lie type (to indecomposable root systems). These will be called universal (or general) groups of the given type. The universal groups involve the simply connected and adjoint groups and, in certain respects, appear to be better behaved (automorphisms, Schur multipliers, character tables).
For example, the general linear group \(\text{GL}_{n+1}(q)\), the unitary group \(\text{U}_{n+1}(q)\), the conformal symplectic group \(\text{CSp}_{2n}(q)\), the special Clifford group \(\text{G}^+_{2n+1}(q)\) to the nonsingular quadratic form of dimension \(2n+1\) and Witt index \(n\) (over the finite field \(K=\mathbb{F}_q\) with \(q\) elements) are the universal groups of type \(A_n\), \({^2A_n}\), \(C_n\), \(B_n\), respectively. For other types one obtains new ``classical groups''.
When \(K\) is algebraically closed, the universal groups are characterized in terms of the related simply connected and adjoint groups, as reductive groups with connected centres. In the finite case the groups are constructed via certain canonical Frobenius morphisms. Here, a corresponding group-theoretic description is more intricate (even for the general linear group). The underlying building presents a geometric link between related groups, the group of automorphisms generated by the root groups being understood as the nucleus of the family of groups of Lie type. special linear groups; projective linear groups; general linear groups; connected algebraic groups; root systems; universal groups; adjoint groups; unitary groups; conformal symplectic groups; special Clifford groups; reductive groups; Frobenius morphisms; root groups; groups of Lie type DOI: 10.1112/S0024610799008066 Linear algebraic groups over finite fields, Classical groups (algebro-geometric aspects) Simply connected, adjoint and universal groups of Lie type | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This survey article is an extended version of a talk by the author during the Ring Theory Conference held at Miskolc in July 1996. It contains an overview on module categories over finite-dimensional algebras which are tame.
Roughly speaking, an algebra \(A\) is tame if and only if all but finitely many \(A\)-modules of a given dimension can be parametrized by a finite number of one-parameter families. The representation theory of tame algebras has seen a dramatic development in the last 30 years. The author explains the basic ideas and methods, describes the main results and illustrates them by interesting examples.
The survey is divided into ten sections: the first one contains preliminaries on module categories and the second one the notions of tameness and wildness for algebras. Section three gives a short introduction to the method of quivers and their representations, including Galois coverings. Section four and five deal with results on the structure and shape of the Auslander-Reiten quiver and the component quiver of an algebra, respectively. In the next two sections, the author discusses recent progresses on affine varieties of modules and degenerations of algebras, which allow to study finite-dimensional modules using geometric methods, in particular methods from algebraic transformation groups and invariant theory. In section eight, methods arising from the theory of integral quadratic forms and their use in the characterization of the representation type for algebras are studied. The last two sections are devoted to two special classes of algebras, tame quasitilted algebras and tame simply connected algebras, for which very precise results have been proved. The paper closes with a bibliography containing 100 entries. survey; module categories over finite-dimensional algebras; representation theory of tame algebras; tameness; wildness; quivers; Galois coverings; Auslander-Reiten quivers; component quivers; affine varieties of modules; degenerations of algebras; finite-dimensional modules; integral quadratic forms; representation types; tame quasitilted algebras; tame simply connected algebras Representation type (finite, tame, wild, etc.) of associative algebras, Representations of quivers and partially ordered sets, Group actions on varieties or schemes (quotients), Quadratic and bilinear forms, inner products, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Torsion theories; radicals on module categories (associative algebraic aspects), Homological dimension (category-theoretic aspects), Module categories in associative algebras Tame module categories of finite dimensional algebras | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(\mathrm{GL}_n\) over a field \(k\) of degree \(d\) for \(d\leq n\), the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor. modular Springer theory; Schur algebras; Schur functors; Schur-Weyl duality; perverse sheaves; nilpotent cones; affine Grassmannians; categories of polynomial representations; general linear groups; representations of symmetric groups Mautner, C., A geometric Schur functor, Selecta Math. (N.S.), 20, 4, 961-977, (2014) Schur and \(q\)-Schur algebras, Representation theory for linear algebraic groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Coadjoint orbits; nilpotent varieties, Representations of finite symmetric groups A geometric Schur functor. | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities After the work of Seiberg and Witten, it has been seen that the dynamics of \(N=2\) Yang-Mills theory is governed by a Riemann surface \(\Sigma\). In particular, the integral of a special differential \(\lambda_{SW}\) over (a subset of) the periods of \(\Sigma\) gives the mass formula for BPS-saturated states. We show that, for each simple group \(G\), the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, \(G^{\lor}\), whose affine Dynkin diagram is the dual of that of \(G\). This curve is not unique, rather it depends on the choice of a representation \(\rho\) of \(G^{\lor }\); however, different choices of \(\rho\) lead to equivalent constructions. The Seiberg-Witten differential \(\lambda_{SW}\) is naturally expressed in Toda variables, and the \(N=2\) Yang-Mills pre-potential is the free energy of a topological field theory defined by the data \(\Sigma_{g,\rho}\) and \(\lambda_{SW}\). Riemann surface; Seiberg Witten differential; periodic Toda lattice; affine Dynkin diagram; topological field theory; Lie algebras E.J. Martinec and N.P. Warner, \textit{Integrable systems and supersymmetric gauge theory}, \textit{Nucl. Phys.}\textbf{B 459} (1996) 97 [hep-th/9509161] [INSPIRE]. Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Curves in algebraic geometry, Supersymmetric field theories in quantum mechanics Integrable systems and supersymmetric gauge 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 A ring homomorphism \(A\to A'\) is called pure, if the map \(M\to A'\otimes_ AM\) is injective for every A-module M. A morphism of affine schemes is pure, if so is the corresponding homomorphism of rings. In terms of a resolution \(Z\to X\) of singularities, one defines the notion of rational singularities.
Theorem. Let X'\(\to X\) be a pure morphism of affine schemes. Then if X' has rational singularities, then so does X.
This implies that quotient spaces of algebraic actions of algebraic reductive groups (over fields of zero characteristic) on affine scheme with rational singularities have rational singularities. This generalizes a few previous results. pure morphism of affine schemes; rational singularities; quotient spaces of algebraic actions of algebraic reductive groups Boutot, Jean-François, Singularités rationnelles et quotients par les groupes réductifs, Invent. Math., 88, 1, 65-68, (1987) Group actions on varieties or schemes (quotients), Singularities in algebraic geometry, Homogeneous spaces and generalizations, Schemes and morphisms Singularités rationnelles et quotients par les groupes réductifs. (Rational singularities and quotients by reductive groups) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities This paper completes the classification of algebras of finite type over a field which are Euclidean domains, begun by Armitage, Samuel, Leitzel, Madan, Queen, Lenstra, and the author. The main point proved here, left open by the previous work, is that Euclidean algebras of finite type over infinite ground fields are coordinate rings of affine open subschemes of the projective line \({\mathbb{P}}^ 1\). - A classification of Euclidean rings of S-integers in algebraic number fields was given, conditionally upon the generalised Riemann hypothesis, by Cooke, Weinberger, and Lenstra.
Several consequences of the classification are given; in particular, a question of Samuel is answered affirmatively: namely, that a principal ideal domain, which is an algebra of finite type over a field, is Euclidean if and only if the (so-called) minimal algorithm does not terminate at the second stage. - The proof of the main result uses Diophantine geometry. More precisely, it is based on Siegel's finiteness theorem for integral points on curves over number fields and an analogue proved in this paper for curves over fields of positive characteristic. Euclidean algorithm; generalised Jacobian varieties; algebras of finite type over a field; Euclidean domains; Diophantine geometry; integral points on curves Brown, M.L., Euclidean rings of affine curves, Math. Z., 208, 3, 467-488, (1991) 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 Given a simple, simply connected complex Lie group \(G\), the Verlinde formula is a combinatorial function \(V^ G: \mathbb{Z}\times \mathbb{Z}\to \mathbb{Z}\) associated with \(G\). The expressions \(V^ G (k,g)\) were first introduced by E. Verlinde in the context of conformal quantum field theory [cf. \textit{E. Verlinde}, Nucl. Physics B, Field Theory and Statistical Systems 300, No. 3, 360-376 (1988)]. Their significance in algebraic geometry stems from the (originally conjectural) fact that they are related to the Hilbert functions of moduli spaces of semi-stable vector bundles over compact Riemann surfaces of genus \(g\). The corresponding relation between those Hilbert functions of moduli spaces and the ``Verlinde numbers'' \(V^ G (k,g)\) used to be called the ``Verlinde conjecture'' for the respective moduli spaces, and its verification, mainly in the cases of \(G= \text{SU} (n)\) and \(G= \text{SL} (n)\), has been a central topic of research in the last five years.
More precisely, the Verlinde conjecture, in most general setting, can be roughly stated as follows. Let \(C\) be a compact Riemann surface of genus \(g\), and let \({\mathcal M}_ C^ G\) be the (factually existing) moduli space of principal G-bundles over \(C\). Then there is an ample line bundle \({\mathcal L}\) over \({\mathcal M}^ G_ C\), a so-called generalized theta bundle, such that \(\dim_ \mathbb{C} H^ 0 ({\mathcal M}_ C^ G, {\mathcal L}^{\otimes k})= V^ G (k_ h, g)\) for any \(k\in \mathbb{Z}\), where \(h\) denotes the dual Coxeter number of the group \(G\).
In this form, and for \(G= \text{SL} (n)\), the Verlinde conjecture has recently been verified by \textit{G. Faltings} [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]; \textit{A. Beauville} and \textit{Y. Laszlo} [Commun. Math. Phys. 164, No. 2, 385-419 (1994; Zbl 0815.14015)]; \textit{S. Kumar}, \textit{M. S. Narasimhan} and \textit{A. Ramanathan} [Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)]; \textit{A. Bertram} and the author [Topology 32, No. 3, 599-609 (1993; Zbl 0798.14004)], and others in less general cases. -- In the present paper under review, the author discusses the origin and properties of the Verlinde formulas \(V^ G (k,g)\) and, in addition, their connection with the famous Witten conjectures in the intersection theory of moduli spaces of algebraic curves. After a brief overview of the structure of topological field theories, fusion algebras and Verlinde's formal calculus, the explicit behavior of \(V^ G (k,g)\) as a function of \(k\) is studied, again in the special case of \(\text{SL} (n)\). The main result is a residue formula for \(V^{\text{SL} (n)} (k,g)\) yielding the deep fact that these numbers are integer-valued polynomials in \(k\). The author then shows how this residue formula for \(V^{\text{SL} (n)} (k,g)\) can be related, via the Grothendieck- Hirzebruch-Riemann-Roch theorem, to Witten's conjectures on the intersection numbers of moduli spaces of curves [cf. \textit{E. Witten}, J. Geom. Phys. 9, No. 4, 303-368 (1992; Zbl 0768.53042)]. Assuming the validity of Witten's formulas, a quick proof of the Verlinde conjecture (as stated above) is given.
This very instructive approach puts the Verlinde formulas into a wider context, and provides some more evidence of Witten's conjectures from this now well-established complex of results. vector bundles; conformal quantum field theory; Verlinde formula; Hilbert functions of moduli spaces of semi-stable vector bundles; compact Riemann surface; generalized theta bundle; Witten conjecture; intersection theory of moduli spaces of algebraic curves; topological field theories; fusion algebras Szenes, A.: The combinatorics of the Verlinde formulas In: Vector Bundles in Algebraic Geometry, Hitchin, N.J., et al., (eds.), Cambridge University Press, 1995 Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Theta functions and abelian varieties, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The combinatorics of the Verlinde formulas | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 was published twice, one times in the book Zbl 0527.00017.]
This paper is a sequel to the author's paper ''Fifteen characterizations of rational double points and simple critical points'' [Enseign. Math. 25, 132-163 (1979; Zbl 0418.14020)]. The characterizations of that paper are for complex varieties and complex functions, and involve the Dynkin diagrams \(A_ k\), \(D_ k\) and \(E_ k\). It turns out that the missing Dynkin diagrams \(B_ k\), \(C_ k\), and \(F_ 4\) (but not \(G_ 2)\) correspond to real singularities and real functions, and that a smaller number of similar characterizations are true for these as well. The main theorem of this paper contains four such characterizations: Let \(f:({\mathbb{R}}^ 3,0)\rightsquigarrow(\mathbb{R},0)\) be the germ at the origin of a real analytic function. Then the following are equivalent: (1) The germ f is right-left equivalent to one of the germs given in a certain list. (2) The germ f is simple (in the sense of Arnold). (3) The complexified variety \(f^{-1}(0)\) has a rational singularity at the origin. (4) A resolution of the real variety \(f^{-1}(0)\) is given in a certain list. The proof of the theorem proceeds by direct computation, or by referring to the corresponding theorem in the complex case. germ of a real analytic function; singularities of real varieties; rational double points; simple critical points; Dynkin diagrams; real singularities A. Durfee, 14 characterizations of rational double points (to appear). Singularities in algebraic geometry, Real algebraic and real-analytic geometry, Germs of analytic sets, local parametrization, Singularities of differentiable mappings in differential topology, Local complex singularities Four characterizations of real rational double points | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author systematically develops an approach based on singularity theory to the investigation of various special functions admitting an integral representation and occurring in physics, integral geometry, partial differential equations, etc. His book is devoted to the study of four important classes of functions: the volume functions, the Newton-Coulomb potentials, the Green functions of hyperbolic equations, and the multidimensional hypergeometric functions of Gelfand-Aomoto types.
The book consists of the preface, introduction, eight chapters and bibliography including 189 references. In the preface main results described in the book are listed. Next is the introduction written in a didactic style and containing an explanation of key ideas and basic results with a series of nonformal comments and remarks.
In the first two chapters the classical Picard-Lefschetz theory and its relations with the singularity theory of functions are discussed at great length. In particular, the author covers the following topics: critical points and values, Milnor fibre, monodromy and variation operators, Gauss-Manin connexion, Picard-Lefschetz formula, Dynkin diagrams, \({\mathcal R}\)-classification of real smooth and complex isolated hypersurface singularities, stratifications, intersection homology theory, etc. These chapters are particularly apt for those who want to enter topological aspects of singularity theory.
The third and seventh chapters deal with properties of the volume function whose notion (in the planar case) goes back to Isaac Newton. In the third chapter the non-algebraicity is proved of the volume function defined by any convex compact hypersurface in the even-dimensional real space. In the odd-dimensional case there are many topological obstructions for such a function to be algebraic. This leads to the conjecture that the only example of algebraic volume function is defined by the ellipsoids in \({\mathbb R}^{2n-1}\) (the so-called Archimedes' example). In chapter VII the author studies the Newton-Coulomb potential of hyperbolic algebraic surfaces in \({\mathbb R}^n.\) Three centuries ago Newton found that the potential of a sphere is constant inside the sphere. One hundred twenty two years later, J. Ivory proved the analogous statement for ellipsoids. Recently, natural generalizations of these classical results to the case of arbitrary hyperbolic hypersurfaces in \({\mathbb R}^n\) have been obtained by \textit{V. I. Arnold} and \textit{V. A. Vassiliev} [Notices Am. Math. Soc. 36, No. 9, 1148--1154 (1989; Zbl 0693.01005)]. Against such a historical background the author examines the case of a smooth hyperbolic hypersurface in \({\mathbb R}^n\) and analyses the behavior of Newtonian potentials outside the hyperbolicity domain. As a result, he describes all cases when such potentials of general hyperbolic hypersurfaces are algebraic. Furthermore conditions under which the potential is algebraic outside the hypersurface are determined and investigated; an approach to the study of the odd-dimensional case is also discussed. The main idea of the author's observations is to use an integral representation of the potential function, easy considerations from the monodromy theory of complete intersections, and properties of homology groups with coefficients in a local system.
Chapters IV and V are devoted to the study of the lacuna problem for hyperbolic differential operators with constant coefficients. First the author recalls the classification of the singular points of wave fronts for hyperbolic operators with constant coefficients, the description of local lacunae close to nonsingular points of fronts (following A. M. Davydova and V. A. Borovikov) and to the singularities \(A_2\) and \(A_3\) [following \textit{L. Gårding}, Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, Suppl., 53--68 (1977; Zbl 0369.35062)]. Of course, these results are reformulated in terms of singularity theory. Then the local Petrovskii cycles of strictly hyperbolic operators are also expressed in such a manner and the equivalence of local regularity of fundamental solutions of hyperbolic PDEs and the topological Petrovskii-Atiyah-Bott-Gårding condition is proved. Finally, a combinatorial algorithm which enumerates local lacunae close to all simple (and many nonsimple) singular points of wave fronts is described in detail.
Chapter VI is devoted to the investigation of homology of local systems, twisted monodromy theory and regularization problem of improper integration cycles. It contains, among other things, a description of twisted vanishing homology of complete intersection singularities and extensions of results from the second chapter that are mainly based on stratified Picard-Lefschetz theory with twisted coefficients.
The final chapter is concerned with the theory of multidimensional hypergeometric functions in the sense of Gelfand and Aomoto; the ramification, singularities, resonance and integral representations of such functions are also investigated. In particular, the author proves a well-known theorem on the analytic continuation of the multidimensional hypergeometric functions and discusses related problems; under certain condition he also obtains a detail description of the case of real plane arrangements [\textit{V. A. Vassiliev}, \textit{I. M. Gelfand} and \textit{A. V. Zelevinskij}, Funkts. Anal. Prilozh. 21, No. 1, 23--38 (1987; Zbl 0614.33008)], and so on.
It should be remarked that the book under review can be considered as an expanded and revised version of [\textit{V. A. Vassiliev}, Ramified integrals, singularities and lacunas. Mathematics and its Application. 315 Kluwer (1995; Zbl 0935.32026)]; a significant part of new materials is based on recent results and ideas of the author and his collaborators. The book contains many clear examples, comments, remarks, open questions and problems illustrated by lot of tables, pictures, and diagrams as well as important applications with instructive references and historical background. With no doubt this book is comprehensible, interesting and useful for graduate students; the variety of topics covered makes it also highly valuable for researchers, lecturers, and practicians working in either of the above mentioned fields of mathematics and its applications. Picard-Lefschetz theory; monodromy theory; isolated singularities; Dynkin diagrams; Gauss-Manin connexion; intersection homology theory; volume functions; Newton-Coulomb potentials; Green functions; hyperbolic equations; lacuna problem; non-integrability of ovals; twisted vanishing homology; ramification of potentials; homology of complements of plane arrangements; Grassmannians; multidimensional hypergeometric functions and integrals Vassiliev, V.A.: Applied Picard-Lefschetz Theory. American Mathematical Society, Providence (2002a) Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry, Integral representations, integral operators, integral equations methods in higher dimensions, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Continuation and prolongation of solutions to PDEs, Other hypergeometric functions and integrals in several variables, Shocks and singularities for hyperbolic equations Applied Picard-Lefschetz 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 This survey is based on the lecture given by the author at the International Conference on Representations of Algebras in Bielefeld in 2012. \textit{J. F. Carlson} gave a talk on this subject at ICRA XII in Toruń in 2007 with the survey [in: Trends in representation theory of algebras and related topics. Proceedings of the 12th international conference on representations of algebras and workshop (ICRA XII), Toruń, Poland, August 15--24, 2007. Zürich: European Mathematical Society (EMS). 167--200 (2008; Zbl 1210.20013)] published in the same series. In the current article we try to pick up where Carlson left off although some overlaps to set the stage were unavoidable. We also focus on the general case of a finite group scheme as opposed to a finite group highlighted in [loc. cit.]. cohomology of finite group schemes; \(\pi\)-points; modules of constant Jordan type; coherent sheaves Modular representations and characters, Group schemes, Cohomology of groups Representations and cohomology of finite group 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 This survey article deals with some classes of algebras arising in the representation theory of finite dimensional algebras which are related to exceptional curves. Roughly speaking, an exceptional curve is a possibly noncommutative curve which is noetherian, smooth, projective and admits an exceptional sequence of coherent sheaves. The concept of exceptional curves presented in this article generalizes the notion of weighted projective lines which was investigated by \textit{W. Geigle} and the author [in: Singularities, representations of algebras and vector bundles, Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)]\ in order to study module categories for canonical algebras (see [\textit{C. M. Ringel}, Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)]) from a geometrical point of view. Whereas exceptional commutative spaces are quite rare, in fact the only exceptional commutative curve over an algebraically closed field is the projective line, there is a rich supply of noncommutative exceptional curves.
An exceptional curve \(\mathbb{X}\) is called homogeneous if for each point of \(\mathbb{X}\) there is only one simple sheaf which is concentrated in that point. It is shown that the geometry of homogeneous curves can be controlled completely by the representation theory of a finite dimensional tame bimodule algebra. In contrast to the commutative case it may happen that there exist more than one simple sheaf concentrated in a single point. The author describes how these curves arise from the homogeneous curves applying a process of insertion of weights and explains how this concept is related to that of vector bundles with parabolic structures in the sense of \textit{C. S. Seshadri} [Fibrés vectoriels sur les courbes algébriques, Astérisque 96 (1982; Zbl 0517.14008)].
Furthermore the author discusses many applications of the concept of exceptional curves. He shows that it applies besides to finite dimensional tame hereditary and canonical algebras also to coordinate algebras of surface singularities, preprojective algebras of tame hereditary algebras, algebras of automorphic forms and two-dimensional factorial algebras. For more details concerning these topics we refer to the papers of \textit{W. Geigle} and \textit{H. Lenzing} [J. Algebra 144, No. 2, 273-343 (1991; Zbl 0748.18007)], \textit{D. Baer, W. Geigle} and \textit{H. Lenzing} [Commun. Algebra 15, 425-457 (1987; Zbl 0612.16015)], \textit{H. Lenzing} [in: Finite dimensional algebras and related topics, 191-212 (1994; see the following review Zbl 0895.16004)]\ and \textit{D. Kussin} [Graded factorial algebras of dimension two, Bull. Lond. Math. Soc. (to appear)]. survey; finite dimensional algebras; exceptional curves; noncommutative curves; exceptional sequences of coherent sheaves; weighted projective lines; module categories for canonical algebras; homogeneous curves; finite dimensional tame bimodule algebras; vector bundles with parabolic structures; coordinate algebras; surface singularities; tame hereditary algebras H. Lenzing, Representations of finite dimensional algebras and singularity theory, \textit{Trends in ring theory} (Miskolc, Hungary, 1996), \textit{Canadian Math. Soc. Conf. Proc.,}\textbf{22} (1998), Am. Math. Soc., Providence, RI (1998), 71-97. Representations of quivers and partially ordered sets, Singularities of curves, local rings, Representation type (finite, tame, wild, etc.) of associative algebras, Elliptic curves, Vector bundles on curves and their moduli Representations of finite dimensional algebras and singularity 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 This is a beautiful report on a beautiful subject. Sklyanin algebras are graded noncommutative algebras having strong regularity properties [\textit{M. Artin, J. Tate} and \textit{M. van den Bergh}, Invent. Math. 106, 335-388 (1991; Zbl 0763.14001); \textit{M. Artin} and \textit{W. Schelter}, Adv. Math. 66, 171-216 (1987; Zbl 0633.16001)]. Initially, they arose from Baxter's solution of the Yang-Baxter equation [\textit{E. K. Sklyanin}, Funkts. Anal. Prilozh. 16, 27-34 (1982; Zbl 0513.58028)]. A geometrical construction and generalizations were provided by \textit{A. Odeskij} and \textit{B. Feigin} [Funkts. Anal. Prilozh. 23, 45-54 (1989; Zbl 0687.17001)] and more recently by \textit{J. Tate} and \textit{M. van den Bergh}, Homological properties of Sklyanin algebras (preprint 1993)]. Let us fix a Sklyanin algebra \(A = A(E,\tau)\) of dimension 4; it is constructed from an elliptic curve \(E\), a line bundle of degree 4 over \(E\) and a point \(\tau \in E\) not of order 4. (Sklyanin algebras of dimension 3 are defined in a similar way, taking instead a line bundle of degree 3.) The main part of the survey under review deals with the classification of all the irreducible finite dimensional \(A\)-modules. Work in this direction was done, besides the already mentioned authors, by Levasseur, Smith, Stafford, Staniszkis. The main steps are: (1) The problem is reduced to the classification of all the irreducible objects in the category \(\text{Proj }A\) (this is the category of finitely generated graded modules localized at the subcategory of finite dimensional ones). Indeed, a simple finite dimensional \(A\)-module is always the quotient of an irreducible from \(\text{Proj }A\). One should also decide which of those irreducibles from \(\text{Proj }A\) have a finite dimensional quotient. (2) The first approximation to the irreducibles of \(\text{Proj }A\) is to classify the point modules. These are cyclic modules having the same Hilbert series as a polynomial ring in one variable, and for a commutative graded algebra, these are in bijective correspondence with the points of the associated projective variety. In the present case, the point modules are in bijective correspondence with the points of \(E\) plus 4 more points. (3) The irreducible modules which are not point ones are called fat. There is also the notion of line modules: cyclic modules having the same Hilbert series as a polynomial ring in two variables. In this case, they are in correspondence with the lines secant to \(E\), and any fat module is a quotient of a line module. If \(\tau\) is not of finite order, the classification can now be worked out (see the following review Zbl 0809.16052). If \(\tau\) is of finite order, the classification requires more work and was done by the author [The four dimensional Sklyanin algebra at points of finite order (Preprint 1992)]. The difference between the two cases has its roots at the center of the Sklyanin algebra. Indeed, in the infinite case, the center is generated by 2 elements first found by Sklyanin, whereas in the second case \(A\) is a finite module over the center. Sklyanin algebras; graded noncommutative algebras; regularity; Yang- Baxter equation; elliptic curve; line bundle; survey; irreducible finite dimensional \(A\)-modules; category of finitely generated graded modules; point modules; cyclic modules; Hilbert series; projective variety; irreducible modules Smith, S. P., The four-dimensional Sklyanin algebras, \(K\)-Theory. Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I (Antwerp, 1992), 8, 1, 65-80, (1994) Graded rings and modules (associative rings and algebras), Quantum groups (quantized enveloping algebras) and related deformations, Elliptic curves, Homological dimension in associative algebras, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Noetherian rings and modules (associative rings and algebras), Finite rings and finite-dimensional associative algebras The four-dimensional Sklyanin 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 constructs real forms of one-parameter algebraic families of complex affine algebraic groups. For the notion of an algebraic family, see [``Algebraic families of Harish-Chandra pairs'', Preprint, \url{arXiv:1610.03435}; ``Contractions of representations and algebraic families of Harish-Chandra modules'', Preprint, \url{arXiv:1703.04028}] by \textit{J. Bernstein}, \textit{N. Higson} and \textit{E. Subag}. \par Let $\sigma_1$ and $\sigma_2$ be two commuting antiholomorphic involutions of a complex affine algebraic group $G$. Its main result states that there exist an algebraic family $\boldsymbol{G}$ of affine algebraic groups and an antiholomorphic involution $\boldsymbol{\sigma}$ of the family $\boldsymbol{G}$ that interpolates between the real forms $G^{\sigma_1}$ and $G^{\sigma_2}$. More precisely, if $[\alpha: \beta] \in \mathbb{RP}^1$ then \[ \boldsymbol{G}^{\boldsymbol{\sigma}}|_{[\alpha:\beta]} \cong \begin{cases} G^{\sigma_1}, & \alpha\beta >0,\\ (G^{\sigma_1}\cap G^{\sigma_2}) \ltimes (\mathfrak{g}^{\sigma_1} \cap \mathfrak{g}^{-\sigma_2}), & \alpha \beta =0,\\ G^{\sigma_2}, & \alpha \beta <0. \end{cases} \] algebraic families of complex algebraic groups; algebraic families of Lie algebras; commuting involutions; real structure; symmetric pairs; Lie groups Linear algebraic groups over the reals, the complexes, the quaternions, General properties and structure of real Lie groups, Structure of families (Picard-Lefschetz, monodromy, etc.) Algebraic families of groups and commuting involutions | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(A={\mathbb{C}}\{X_ 1,...,X_ r\}/I\), where I is an \({\mathfrak m}\)-primary ideal, be a finite local analytic \({\mathbb{C}}\)-algebra. The main results of this thesis are lower bounds for the \({\mathbb{C}}\)-dimension of \(Der_{{\mathbb{C}}}(A)\), the module of derivations of A, (theorems 4.3 and 4.9) and of \(D_{{\mathbb{C}}}(A)\), the universally finite module of differentials (theorem 5.7 and 5.8). E.g., theorem 4.9 states: Let \(n=\dim_{{\mathbb{C}}}(A)\), \(\nu =ord(I)\) and \(\alpha =\min \{k| \quad {\mathfrak m}^ k\subset I\}\) then \(\dim_{{\mathbb{C}}}Der_{{\mathbb{C}}}(A)=r(n- 1)-\dim_{{\mathbb{C}}}Hom(I/{\mathfrak m}^{\alpha},{\mathfrak m}^{\quad \nu}/I).\) The author arrives at these results by considering those algebras as fibers of finite morphisms of complex analytic spaces. Very useful is an explicit construction of the Hilbert scheme \(Hilb^ n{\mathbb{P}}^ r\), given in section 3, and the concept of an embedded deformation (section 2). This interesting paper concludes with explicit examples and a computer program for computing the Hilbert-Samuel function of an ideal generated by homogeneous polynomials in four variables having the same total degree. finite analytic algebras; modules of differentials; module of; derivations; Hilbert scheme; embedded deformation; computing the Hilbert- Samuel function Modules of differentials, Analytic algebras and generalizations, preparation theorems, Deformations and infinitesimal methods in commutative ring theory, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to commutative algebra, Morphisms of commutative rings, Formal methods and deformations in algebraic geometry Über Deformationen und Derivationen endlicher \({\mathbb{C}}\)-Algebren. (On deformations and derivations of finite \({\mathbb{C}}\)-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 \textit{F. J. Grunewald, D. Segal} and \textit{G. C. Smith} [Invent. Math. 93, No. 1, 185-223 (1988; Zbl 0651.20040)] introduced the notion of a zeta function for an infinite group \(G\) encoding (normal) subgroups of finite index. The zeta functions counting subgroups of finite index in infinite nilpotent groups depend upon the behaviour of some associated system of algebraic varieties on reduction mod \(p\). Further, \textit{M. du Sautoy} [Isr. J. Math. 126, 269-288 (2001; Zbl 0998.20033) and J. Reine Angew. Math. 549, 1-21 (2002; Zbl 1001.20032)] constructed a group whose local zeta function was determined by the number of points on the elliptic curve \(E\colon Y^2=X^3-X\).
In this work the author generalises du Sautoy's construction to define a class of groups whose local zeta functions are dependent upon the number of points on the reduction of a given elliptic curve with a rational point. He also constructs a class of groups that behave the same way in relation to any curve of genus 2 with a rational point. The author ends with a discussion of problems arising from this work. nilpotent groups; zeta functions; elliptic curves; rational points; numbers of subgroups; subgroups of finite index Nilpotent groups, Other Dirichlet series and zeta functions, Elliptic curves, Associated Lie structures for groups, Rational points, Subgroup theorems; subgroup growth Associating curves of low genus to infinite nilpotent groups via the zeta 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 Given a simple algebraic group \(L\) over a characteristic zero algebraically closed field and let \(P\) be a parabolic subgroup with aura, i.e., with Abelian unipotent radical. Let \(l=\text{Lie }L\), and let \(G=l(0)\). Here the representation of \(G\) on \(l(1)\) is examined. The author gives a unified construction of a \(G\)-equivalent resolution of singularities for both the closure of an orbit of \(G\) in \(l(1)\) as well as for the closure of the conormal bundle of the orbit.
It is shown that \(L\cdot l(1)\cap(l(1)\oplus l(-1))=\bigcup_i{\mathfrak E}_i\), where \({\mathfrak E}_i\) is the closure of the orbit \({\mathcal O}_i\). Also, \(\bigcup_i{\mathfrak E}_i=\{(x,y)\mid x\in l(1),\;y\in l(-1),\;[x,y]=0\}\). The \(G\)-orbit structure of this variety is related to the double coset space \(G\backslash L/P\).
If \(U\) is the maximal unipotent subgroup of \(G\), then \(k[l(1)]\) is a free \(k[l(1)]^U\)-module, which here is equivalent to the quotient map \(\pi_{l(1)}\colon l(1)\to l(1)/ /U\) being equidimensional. To prove this, the paper provides a sufficient condition for the quotient map \(\pi_X\colon X\to X/ /U\) to be flat, where \(X\) is an affine \(G\)-variety. As a result, the irreducible representations of simple algebraic groups with aura are classified. parabolic subgroups; simple algebraic groups; Abelian unipotent radicals; resolutions of singularities; orbit structure; irreducible representations M. Brion, \textit{Invariants et covariants des groupes algébriques réductifs}, in: \textit{Théorie des Invariants et Géometrie des Variétés Quotients}, Travaux en Cours, t. 61, Paris, Hermann, 2000, pp. 83-168. Representation theory for linear algebraic groups, Group actions on varieties or schemes (quotients), Actions of groups on commutative rings; invariant theory Parabolic subgroups with Abelian unipotent radical as a testing site for invariant theory | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author in [``Nonabelian Jacobian of smooth projective varieties'', Science China, Vol. 56, No. 1, 1--42 (2013)] is a survey of the work done in [J. Differ. Geom. 74, No. 3, 425--505 (2006; Zbl 1106.14030)] as well as in [{I. Reider}, ``Nonabelian Jacobian of smooth projective surfaces and representation theory'', \url{arXiv:1103.4749}]. Central in this work is the construction of a nonabelian Jacobian \(J(X;L,d)\) of a smooth projective surface \(X\), a scheme over the Hilbert scheme \(X^{[d]}\) of subschemes of length \(d\) in \(X\), with a morphism to the stack of torsion free sheaves of rank 2 on \(X\) with determinant \(\mathcal{O}_X(L)\) and second Chern class \(d\). Among the various constructions covered in this survey, the existence of a sheaf of reductive Lie algebras on \(J(X;L,d)\) takes center stage. It originates from a well-chosen filtration on \(X^{[d]}\). This sheaf, which is the object of the second part of the paper, paves the way for the use of representation theoretic methods in the study of projective surfaces. There is some interesting work covered in the survey, though what makes it alluring are the possible applications. The seasoned algebraic geometer will no doubt appreciate the expansive coverage done in this work; worthy of further investigation are the connections with quantum gravity, homological mirror symmetry, the geometric Langlands program as well as quiver representations/Gromov-Witten invariants. Jacobian; Hilbert scheme; vector bundle; sheaf of reductive Lie algebras; Fano toric varieties; period maps; stratifications; Hodge-like structures; relative Higgs structures; perverse sheaves; Langlands program Reid, I, Nonabelian Jacobian of smooth projective surfaces -- a survey, Sci China Math, 56, 1-42, (2013) Parametrization (Chow and Hilbert schemes), Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Representations of orders, lattices, algebras over commutative rings, Variation of Hodge structures (algebro-geometric aspects), Geometric Langlands program (algebro-geometric aspects), Jacobians, Prym varieties Nonabelian Jacobian of smooth projective surfaces -- a survey | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\mathbb{C}\) be the field of complex numbers and let \(GL(n,\mathbb{C})\) denote the general linear group of order \(n\) over \(\mathbb{C}\). Let \(n\) be the sum of \(m\) positive integers \(p_ 1,\dots,p_ m\) and consider the block diagonal subgroup \(GL(p_ 1,\mathbb{C})\times\dots\times GL(p_ m,\mathbb{C})\). The adjoint representation of the Lie group \(GL(n,\mathbb{C})\) on its Lie algebra gives rise to the coadjoint representation of \(GL(n,\mathbb{C})\) on the symmetric algebra of all polynomial functions on \({\mathfrak Gl}(n,\mathbb{C})\). The polynomials that are fixed by the restriction of the coadjoint representation to the block diagonal subgroup form a subalgebra called the algebra of invariants. A finite set of generators of this algebra is explicitly determined and the connection with the generalized Casimir invariant differential operators is established. general linear group; block diagonal subgroup; adjoint representation; Lie group; Lie algebra; coadjoint representation; symmetric algebra; polynomial functions; algebra of invariants; generators; generalized Casimir invariant differential operators DOI: 10.1016/0021-8693(92)90184-N Representations of Lie and linear algebraic groups over real fields: analytic methods, Linear algebraic groups over the reals, the complexes, the quaternions, Finite-dimensional groups and algebras motivated by physics and their representations, Group actions on varieties or schemes (quotients), Applications of Lie groups to the sciences; explicit representations, Vector and tensor algebra, theory of invariants Invariant theory of the block diagonal subgroups of \(GL(n,\mathbb{C})\) and generalized Casimir operators | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The theory of theta functions is rather like the planet Venus -- it has two aspects, like the Evening and Morning Star. One of these is the role that theta functions play in the function theory and in particular the general theory of uniformization, the other is a large number of beautiful formulae which these relate functions to one another and which often have remarkable number-theoretic, representation-theoretic or combinatorial interpretations. Although both aspects are both beautiful and of great significance they are, like the Evening and Morning Star rarely to be seen at the same time -- here one should note that it does happen that Venus passes to the North of the Sun so that in Northern latitudes it passes from being the Evening Star; this phenomenon will be visible in nothern Norway in May, 2002. In this beautiful book the authors unite these two aspects. The basis of their treatment is function-theoretic and they describe the theory of the theta functions with characteristics and theta constants much more carefully than the standard texts do; whereas this theory is ``well-known'' it is often very difficult to find statements one needs and the authors point out that one of their motivations was just this problem. Having built up the function-theoretic basis in the first two chapters they then study many special cases of the function theory of modular varieties using theta constants to provide enough functions and forms. In the fourth chapter the emphasis changes to identities between theta constants. In the final three chapters it is the applications to combinatorial and arithmetical questions that comes into the centre of the considerations. There are many identities and congruences given here that are new, at least to the reviewer. In proving them the authors have made use of modern computer-based methods to both find and prove identities. This is not only heuristically and technically useful, it also gives one confidence in the truth of the identities, for many developments since the time of Ramanujan need more than human patience to verify. It is a real pleasure to read this book. In the introduction to his ``A brief introduction to theta functions''; Holt, 1961, \textit{R. Bellman} writes, ``The theory of elliptic functions is the fairyland of mathematics. The mathematician who once gazes upon this enchanting and wondrous domain crowed with the most beautiful relations and concepts is forever captivated.'' With this book the modern mathematician has a contemporary path into this world. theta functions; theta constants; modular varieties; partition functions; Ramanujan congruences; modular forms of \({1\over 2}\)-integral weight H. M. Farkas, I. Kra, Theta constants, Riemann surfaces and the modular group. Graduate Studies in Mathematics 37. American Mathematical Society, Providence, RI, (2001). Zbl0982.30001 MR1850752 Research exposition (monographs, survey articles) pertaining to functions of a complex variable, Research exposition (monographs, survey articles) pertaining to number theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Differentials on Riemann surfaces, Dedekind eta function, Dedekind sums, Hecke-Petersson operators, differential operators (one variable), Fourier coefficients of automorphic forms, Elementary theory of partitions, Analytic theory of partitions, Partitions; congruences and congruential restrictions, Theta functions and curves; Schottky problem, Special algebraic curves and curves of low genus, Riemann surfaces; Weierstrass points; gap sequences, Fuchsian groups and their generalizations (group-theoretic aspects) Theta constants, Riemann surfaces and the modular group. An introduction with applications to uniformization theorems, partition identities and combinatorial number 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 author characterizes the possible Hilbert functions of 0-dimensional subschemes of \(\mathbb{P}^3\) lying on an irreducible curve \(C\) which is contained in a smooth quadric surface \(Q\subseteq\mathbb{P}^3\) solely using the type \((a,b)\) of the curve. More precisely, he shows that in case \(1\leq a\leq b\), there is such a 0-dimensional scheme \(X\), if and only if its Hilbert function \(H_X\) satisfies
(1) \(\Delta H_X(i)=2i+1\) for \(i=0,\dots,b-1\),
(2) \(\Delta H_X(i)=a+b\) for \(i=b,\dots,s-1\) with some \(s\geq b\),
(3) \(\Delta H_X(s-1)>\Delta H_X(s)>\Delta H_X(s+1)+1>\cdots>\Delta H_X(s+e-1)+e-1>0\) for some \(0\leq e\leq a\), and
(4) \(\Delta H_X(i)=0\) for \(i\geq s+e\).
His proof uses an ``ad hoc'' construction and is partly based on prior work by \textit{G. Raciti} [cf. Commun. Algebra 18, No. 9, 3041-3053 (1990; Zbl 0721.14002) and in: Curves Semin. Queen's, Vol. VI, Queen's Pap. Pure Appl. Math. 83, Exposé J (1989; Zbl 0714.14035)]. curves on a surface; type of curve; Hilbert functions; 0-dimensional subschemes Zappalà G.,0-dimensional subschemes of curves lying on a smooth quadric surface, Le Mathematiche,52 (1997), 115--127. Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Plane and space curves \(0\)-dimensional subschemes of curves lying on a smooth quadric surface | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(B\) be an indefinite quaternion algebra over \(\mathbb Q\), and \(G\) be the `standard' split unitary group of 2 variables over \(B\). When \(B\) is split, \(G\) is the usual symplectic group of 4 variables. For any integer \(N >0\), the principal congruence group \(\Gamma(N)\) acts on the Siegel upper half space, and let \(Y(N)\) be the corresponding nonsingular moldular 3-fold (after adding cusps and resolving the singularities).
In this paper, the author proves that \(Y(N)\) is of general type when \(N d(B)\) is sufficiently large and \(d(B) >1\), where \(d(B)\) is the discriminant of \(B\). An explicit bound is also given. The case \(d(B) =1\) was proved by \textit{T. Yamazaki} using similar method [Am. J. Math. 98, 39-53 (1976; Zbl 0345.10014)]. The main idea is to use \textit{F. Arakawa}'s explicit formula [J. Math. Soc. Japan 33, 125-145 (1981; Zbl 0458.10023)] on the space of cusp forms for \(\Gamma (N)\) and the defect technique of \textit{F. Knöller} [Manuscr. Math. 37, 135-161 (1982; Zbl 0486.14009)] and \textit{S. Tsuyumine}'s [Invent. Math. 80, 269-281 (1985; Zbl 0576.14036)] on studying cusps. modular 3-folds of general type; quaternion modular forms; defects; quaternion unitary groups; quaternion algebra; Siegel modular groups Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Surfaces of general type, \(3\)-folds, 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 Modular 3-folds obtained from quaternion unitary groups of degree 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 In a series of papers the authors and \textit{Ed Cline}, known as CPS, developed a theory of abstract Kazhdan-Lusztig theories for a highest weight category \(\mathcal C\). It is well-known that \(\mathcal C\) realizes as the category of \(S\)-modules for a finite dimensional quasi-hereditary algebra \(S\). If \(S\) is positively graded, CPS also developed a theory on graded Kazhdan-Lusztig theories for the category \(\mathcal C\) of finitely generated graded \(S\)-modules. The property of having a graded Kazhdan-Lusztig theory is closely related to the Koszul property of \(S\). More precisely, \(\mathcal C\) has a graded Kazhdan-Lusztig theory if and only if \(S\) is a Koszul algebra and \(\mathcal C\) has an (ungraded) Kazhdan-Lusztig theory [see CPS, Proc. Lond. Math. Soc., III. Ser. 68, No. 2, 294-316 (1994; Zbl 0819.20045)].
This paper considers the relation of the Koszul property of \(S\) with some automorphisms of \(S\). One of the main results in the paper states essentially that, given an algebra \(S\) and an automorphism \(\sigma\), if the Ext-algebra associated to \(S\) can be given its natural graded structure by means of the eigenvalues of the induced action of \(\sigma\), then necessarily \(S\) is (graded and is) a Koszul algebra. This result, applied to quasi-hereditary algebras \(S\), shows how the existence of certain automorphisms on \(S\) can lead not only to a graded structure on \(S\), but also, when the highest weight category \({\mathcal C}=\text{mod-}S\) has an ungraded Kazhdan-Lusztig theory, to a graded Kazhdan-Lusztig theory on the associated graded module category. In the paper, the above result is, in particular, applied to the category of \(\ell\)-adic perverse sheaves on the flag variety \(G/B\), where \(G\) is a semisimple algebraic group, and \(B\) is one of its Borel subgroups. This category is shown to be a highest weight category, and, in positive characteristic, the Frobenius morphism on \(G/B\) naturally induces an automorphism on the associated quasi-hereditary algebra, satisfying the required conditions. Therefore, the associated graded module category \(\mathcal C\) has a graded Kazhdan-Lusztig theory. In this way, the authors obtain a natural proof that the algebra associated to the category of \(\ell\)-adic perverse sheaves on \(G/B\) is Koszul. (This result has been proved by \textit{A. Beilinson, V. Ginzburg, W. Soergel} in a different way. See their paper [in J. Am. Math. Soc. 9, No. 2, 473-527 (1996)].) Moreover, since the category of \(\ell\)-adic perverse sheaves on \(G/B\) is known to be equivalent to the principal block \(\mathcal O_{\text{triv}}\) of the category \(\mathcal O\) of the associated complex Lie algebra, it is obtained, as a corollary, that the algebra associated to \(\mathcal O_{\text{triv}}\) is Koszul. (This result was first announced by \textit{A. Beilinson, V. Ginzburg} with an outline of a proof [see their preprint Mixed categories, Ext-duality, and representations (results and conjectures)]. In an earlier version of the above-mentioned preprint, Beilinson-Ginzburg-Soergel gave a proof entirely different from the proof presented in this paper). finitely generated graded modules; abstract Kazhdan-Lusztig theories; highest weight category; finite dimensional quasi-hereditary algebras; graded Kazhdan-Lusztig theories; Koszul property; automorphisms; category of \(\ell\)-adic perverse sheaves; flag varieties; semisimple algebraic groups; Borel subgroups; principal blocks; category \(\mathcal O\) Parshall B., Quart. J. Math. Oxford 2 pp 345-- (1995) Representation theory for linear algebraic groups, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Graded rings and modules (associative rings and algebras), Homological dimension in associative algebras, Group actions on varieties or schemes (quotients), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Koszul algebras and the Frobenius automorphism | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension two | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities Let \(\Lambda\) be a finite dimensional algebra over an algebraically closed field. The authors prove that, for any semisimple object \(T\in\Lambda\text{-mod}\), the class of those \(\Lambda\)-modules with fixed dimension vector \(d\) and top \(T\) which do not permit any proper top-stable degenerations possesses a fine moduli space \(\mathfrak{ModuliMax}_d^T\) that is a projective variety. The authors show that any projective variety arises as \(\mathfrak{ModuliMax}_d^T\) for suitable \(\Lambda\), \(T\) and \(d\) (Example 5.4).
The following classification theorem is proved. Theorem B. For any semisimple \(T\in\Lambda\text{-mod}\), the modules of dimension vector \(d\) which are degeneration-maximal among those with top \(T\) have a fine moduli space \(\mathfrak{ModuliMax}_d^T\), that classifies them up to isomorphism. The variety \(\mathfrak{ModuliMax}_d^T\) is projective. Moreover, given any module \(M\) whose top \(M/JM\) is contained in \(T\), the closed sub-variety of \(\mathfrak{ModuliMax}_d^T\) consisting of the points that correspond to degenerations of \(M\) is a fine moduli space for the maximal top-\(T\) degenerations of \(M\).
In Theorem A a structural characterization of modules with no proper top-stable degenerations is given.
For part I see \textit{B. Huisgen-Zimmermann} [Proc. Lond. Math. Soc. (3) 96, No. 1, 163-198 (2008; Zbl 1207.16010)]. finite-dimensional algebras; finite-dimensional representations; top-stable degenerations; fine moduli spaces; projective varieties; degenerations of modules; representations of quivers 10.1016/j.aim.2014.02.008 Representations of associative Artinian rings, Fibrations, degenerations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Representations of quivers and partially ordered sets, Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) Top-stable degenerations of finite dimensional representations. 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 It is shown that the homology of the spin moduli spaces \({\mathcal M}_ g[\epsilon]\) of Riemann surfaces of genus g with spin structure of Arf invariant \(\epsilon\in {\mathbb{Z}}/2{\mathbb{Z}}\) (resp. of the corresponding spin mapping class groups) is stable, i.e. independent of g and \(\epsilon\) for sufficiently large g. As the author notes, the interest in these moduli spaces comes from fermionic string theory. For a second paper the computation of the first (integer coefficients) and second (rational coefficients) homology, and thus of the Picard group, of the spin moduli spaces is announced. All of this generalizes results and methods (constructing simplicial complexes from configuration of simple closed curves on a surface on which the mapping class groups act, then applying spectral sequence arguments) of two of the author's previous papers in which he obtained analogous results for the ordinary mapping class groups resp. moduli spaces. homology of the spin moduli spaces of Riemann surfaces with spin structure; Arf invariant; spin mapping class groups; fermionic string theory; Picard group; configuration of simple closed curves on a surface Harer J.L. (1990) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure. Math. Ann. 287(2): 323--334 Topology of Euclidean 2-space, 2-manifolds, General low-dimensional topology, Teichmüller theory for Riemann surfaces, Homology of classifying spaces and characteristic classes in algebraic topology, Differential topological aspects of diffeomorphisms, Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) Stability of the homology of the moduli spaces of Riemann surfaces with spin structure | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 generalize a result of \textit{J. Harris} [Math. Ann. 249, 191-204 (1980; Zbl 0449.14006)] on the Hilbert functions of a zero dimensional subscheme of \(P^ 2\) which is linked to an other one in a complete intersection to the case of locally Cohen-Macaulay subschemes of \(P^ n\) of pure codimension \(d\geq 2.\)
As an application they compute the speciality and the Hilbert function of curves in \(P^ 3\), obtaining a bound for the least integer from which the speciality vanishes, improving the bound of \textit{L. Gruson} and \textit{C. Peskine} [see Algebr. Geom. Proc., Tromsø Symp. 1977, Lect. Notes Math. 687, 31-59 (1978; Zbl 0412.14011)]. liaison; linked schemes; Hilbert functions; complete intersection; locally Cohen-Macaulay subschemes of \(P^ n\); speciality Complete intersections, Étale and other Grothendieck topologies and (co)homologies, Projective techniques in algebraic geometry, Parametrization (Chow and Hilbert schemes) On the cohomology groups of linked schemes | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The article studies relations between ADE-Lie theory and ADE-singularities of surfaces.
Let \(X\) denote a compact complex surface with a rational double point and \(\pi :Y \to X\) the minimal resolution of. If \(C_1 , \dots , C_n\) in \(Y\) are the irreducible components of the exceptional locus, then the dual graph of the exceptional divisor \(\sum_{i=1}^n C_i\) is a Dynkin diagram of one of the types A, D, E.
For the integer homology of \(Y\), there is a natural decomposition \(H^2(Y, {\mathbb{Z}} )= H^2(X, {\mathbb{Z}} ) \oplus \Lambda \), where \(\Lambda = \{ \sum _i a_i[C_i] | a_i \in {\mathbb{Z}} \} \). The subset \(\Phi := \{ \alpha \in \Lambda | \alpha^2 =-2 \}\) is an ADE root system of a simple Lie-algebra \(\mathbf g \). Its associated Lie algebra bundle over \(Y\) is defined as
\[
{\mathcal E}_0^{\mathbf g}:= {\mathcal O_Y}^n \oplus \{ \bigoplus_{\alpha \in \Phi}{\mathcal O_Y} (\alpha ) \} .
\]
This bundle does not descend to the original surface \(X\). The authors show, if \(p_g(X)=0\) then \( {\mathcal E}_0^{\mathbf g}\) has a deformation to a bundle which can descend to \(X\).
Their result generalizes the work of Friedman-Morgan for \(E_n\)-bundles over del Pezzo surfaces [\textit{R. Friedman} and \textit{J. W. Morgan}, Contemp. Math. 312, 101--115 (2002; Zbl 1080.14533)].
Furthermore, the authors describe the minuscule representation bundles of these Lie algebra bundles in terms of configurations of (reducible) \((-1)\)-curves in \(Y\). ADE bundle; ADE singularity; singularities of surfaces; simple Lie algebra; Lie algebra bundle; minimal resolution of an ADE singularity Chen, YX; Leung, NC, ADE bundles over surfaces with ADE singularities, Int. Math. Res. Not., 15, 4049-4084, (2014) Singularities of surfaces or higher-dimensional varieties, Singularities in algebraic geometry, Special surfaces, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Structure theory for Lie algebras and superalgebras \(ADE\) bundles over surfaces with \(ADE\) 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 We study the classification problem for left-symmetric algebras with commutation Lie algebra \(\mathfrak{gl}(n)\) in characteristic \(0\). The problem is equivalent to the classification of étale affine representations of \(\mathfrak{gl}(n)\). Algebraic invariant theory is used to characterize those modules for the algebraic group \(\text{SL}(n)\) which belong to affine étale representations of \(\mathfrak{gl}(n)\). From the classification of these modules we obtain the solution of the classification problem for \(\mathfrak{gl}(n)\). As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal. left symmetric algebras; Lie-admissible algebras; semisimple algebraic groups; algebra of invariants; representations Baues, O., Left-symmetric algebras for \(g l(n)\), \textit{Transactions of the American Mathematical Society}, 351, 7, 2979-2996, (1999) Nonassociative algebras satisfying other identities, Lie algebras of linear algebraic groups, Group actions on varieties or schemes (quotients), Lie-admissible algebras, Representation theory for linear algebraic groups Left-symmetric algebras for \(\mathfrak{gl}(n)\). | 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 Let \(G\) be a finite subgroup of \(\text{SL}(n, \mathbb C)\), \(S_G\) the covariant algebra of \(G\) and \(\langle S_G\rangle_i\) the subspace of \(S_G\) of homogeneous degree \(i\). For each irreducible representation \(\rho\) of \(G\) let \(\langle \rho, (S_G)_i\rangle _G\) be the multiplicity of \(\rho\) in \((S_G)_i.\) The Molian series \(P_{S_{G, \rho}} (t)\) of \(S_G\) for \(\rho\) is defined by
\[
P_{S_{G, \rho}}(t)=\sum\langle \rho, (S_G)_i\rangle_G t^i.
\]
Explicit formulas for the Molian series are given in the case that \(G\) is one of the exceptional finite subgroups of \(\text{SL}(3, \mathbb C)\). Let \(\text{Hilb}^G(\mathbb C^n)\) be the universal subscheme of the Hilbert scheme \(\text{Hilb}^{| G| } (\mathbb C^n)\) parameterizing all smoothable scheme theoretic \(G\)-orbits of length \(| G| \). \(\text{Hilb}^G(\mathbb C^3)\) is studied, especially the fiber \(\pi^{-1}(0)\) of the Hilbert-Chow morphism \(\pi: \text{Hilb}^G(\mathbb C^3) \to \mathbb C^3/G\) in case that \(G\) is a finite subgroup of SO(3). An SO(3)-version of the McKay correspondence similar to the SU(2) case is given. Hilbert scheme; invariant theory; McKay quiver; McKay correspondence Gomi, Y; Nakamura, I; Shinoda, K, Coinvariant algebras of finite subgroups of SL\_{}\{3\}\(\mathbb{C}\), Can. J. Math., 56, 495-528, (2004) Group actions on varieties or schemes (quotients), Linear boundary value problems for ordinary differential equations, Parametrization (Chow and Hilbert schemes), Singularities of surfaces or higher-dimensional varieties Coinvariant algebras of finite subgroups of \(\text{SL} (3,\mathbb{C})\) | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 \(\theta\) be an involutive automorphism of \(G\) with corresponding linear involution \(d\theta :\mathfrak{g}\rightarrow\mathfrak{g}\) where \(\mathfrak{g}=\)Lie\(\left( G\right) .\) Then \(\mathfrak{g=k\oplus p}\) where \(\mathfrak{k}=\left\{ x\in\mathfrak{g\,}|\,d\theta\left( x\right) =x\right\} \) and \(\mathfrak{p} =\left\{ x\in\mathfrak{g}\,|\,d\theta\left( x\right) =-x\right\} .\) If we let \(G^{\theta}=\left\{ g\in G\,|\,\theta\left( g\right) =g\right\} ,\) then \(G^{\theta}\) acts on \(\mathfrak{p},\) and while this action is understood when \(k\) has characteristic zero, it is not as well understood in the positive characteristic case. The characteristic zero case, as developed in [\textit{B. Kostant} and \textit{S. Rallis}, Am. J. Math. 93, 753--809 (1971; Zbl 0224.22013)] which use compactness properties and \(\mathfrak{sl}\left( 2\right) \)-triples which do not work in positive characteristic.
Here, the author considers this action in the case where char\(\;k=p\) for a good prime \(p\), that is, when \(G\) is a reductive algebraic group with a root system \(\Phi\) such that when the longest element of each irreducible component of \(\Phi\) is expressed as a linear combination of a basis each coefficient is less than \(p\). Much of the paper proceeds as in the work cited above, however some adjustments must be made to the characteristic zero theory. Suppose furthermore that the derived subgroup is simply connected and that there exists a symmetric \(G\)-invariant non-degenerate bilinear form \(\mathfrak{g\times g}\rightarrow k\). It is proved that this bilinear form can be chosen to be \(\theta\)-equivariant as well. The notion of \(\mathfrak{sl} \left( 2\right) \)-triples is replaced by associated cocharacters, which allows us to compute the number of irreducible components of the variety \(\mathcal{N}\) of nilpotent elements of \(\mathfrak{p}\). This variety has a dense open orbit, which is also true for each fibre of the map \(\mathfrak{p\rightarrow p}//G^{\theta}.\) The corresponding statement for \(G\), a conjecture of Richardson, is shown to be false. Lie algebras; Ring of invariants; Geometric invariant theory P. Levy, Involutions of reductive Lie algebras in positive characteristic, Adv. Math. 210 (2007), no. 2, 505--559. Modular Lie (super)algebras, Other algebraic groups (geometric aspects) Involutions of reductive Lie algebras in positive characteristic | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities In this paper, we study properties of the Hilbert schemes of ideals of finite algebras over an algebraically closed field. We prove a duality theorem for the Hilbert schemes of a finite Gorenstein algebra. We also study some properties of finite algebras obtained from informations on their Hilbert schemes. We give examples of finite algebras \(A\) such that the sequences \(\{\chi(\mathrm{Hilb}^r(A))\}_r\) are unimodal. They are examples of a generalization of a combinatorial conjecture by Stanton. finite algebras; Hilbert schemes Structure of finite commutative rings, Parametrization (Chow and Hilbert schemes) On the Hilbert schemes of finite algebras over an algebraically closed field | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 proves existence theorems for higher order singularities of a finite morphism to \({\mathbb{P}}^ m\) and deduces a result on simple connectivity of varieties admitting a finite morphism of bounded singularity. - The singularities are obtained by successive degeneration of double points. Our main tool is R. Schwarzenberger's notion of generalized secant sheaves and the connectedness theorem by W. Fulton and the author. morphism to projective space; higher order singularities of a finite morphism; simple connectivity of varieties; successive degeneration of double points Singularities in algebraic geometry, Rational and birational maps, Ramification problems in algebraic geometry, Topological properties in algebraic geometry Higher order singularities of morphisms to projective space | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow 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 unitary group takes the form \(G=\{g\in GL(3\mathbb{C})| ^ t\bar g R g=R\}\)
\[
\text{where\quad} R = \begin{pmatrix} S & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix},
\]
\(R\) is Hermitian and \(-iR\) has signature (2,1). The entries of \(R\) are in an imaginary quadratic number field \(K\). There are some remarks about a generalization to the case of signature \((p+1,1)\), \(p\geq 1\). G acts on the symmetric domain
\[
\mathcal D = \{w,z\in \mathbb{C} | \text{Im}z > -\frac12\bar wSw\}.
\]
If \(\Gamma =G\cap SL(3,\mathcal O_ k)\) with \(\mathcal O_ k\) the ring of integers in \(K\), then a \(\Gamma\)-automorphic form \(f(w,z)\) has an expansion \(f = \sum_{r}g_ r(w)e(rz)\), \(e(x)=e^{2\pi ix}\), where \(g_ r(w)\) are theta functions. Given another automorphic function \(g(w,z)\) with expansion \(\sum_{r}h_ r(w)e(rz)\), then the main result is the analytic continuation of the series \(\sum_{r}<g_ r,h_ r> r^{-s}\); this continuation is realized by an integral of Rankin type
\[
\int_{P_\Gamma \setminus \mathcal D} f\bar g\left(\text{Im}z+ \frac i2 \bar wSw\right)^ s dw d\bar w dz d\bar z.
\]
Dirichlet series; Rankin convolution of automorphic forms on unitary groups; Jacobi forms; automorphic forms; Eisenstein series; theta functions; analytic continuation; integral of Rankin type Theta series; Weil representation; theta correspondences, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Automorphic functions in symmetric domains, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Dirichlet series and automorphic forms on unitary groups | 0 |
The article consists of two parts. In the first part, the authors discuss the relations between simple singularities, the finite subgroups of \(\text{SL}(2,\mathbb{C})\), simple Lie groups and algebras, quivers of finite type, modular invariant partition functions in two dimensions, and pairs of von Neumann algebras of type \(\text{II}_1\). In particular, it is discussed that any simple singularity is a quotient singularity by a finite subgroup of \(\text{SL}(2,\mathbb{C})\), and the resolution of singularity has the corresponding Dynkin diagram. McKay showed how to recover the Dynkin diagram by using representation theory. \textit{Gonzalez-Sprinberg} and \textit{J.-L. Verdier} gave an explicit bijection between the set of exceptional curves in the resolution of a simple singularity and the representations of the corresponding finite subgroup \(G\).
In the second part of the paper the authors study this McKay correspondence by using Hilbert scheme techniques. To any point of the exceptional set they introduce in a natural way a \(G\)-module. The equivalence class of this module is constant along every exceptional divisor. simple singularities; McKay correspondence; Hilbert schemes; simple Lie groups; simple Lie algebras; quivers of finite type; modular invariant partition functions; von Neumann algebras; Dynkin diagram Y. Ito, I. Nakamura, \textit{Hilbert schemes and simple singularities}, in: \textit{New Trends in Algebraic Geometry} (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 151-233. Singularities in algebraic geometry, Parametrization (Chow and Hilbert schemes), Global theory and resolution of singularities (algebro-geometric aspects), Representations of quivers and partially ordered sets Hilbert schemes and simple singularities The author give an explicit geometric description to some of H. Nakajima's quiver varieties. More precisely, let \(X = {\mathbb{C}}^2\), \(\Gamma \subset \text{SL}({\mathbb{C}}^2)\) be a finite subgroup, and \(X_\Gamma\) be a minimal resolution of \(X/\Gamma\). The main result states that \(X^{\Gamma [n]}\) (the \(\Gamma\)-equivariant Hilbert scheme of \(X\)) and \(X_\Gamma^{[n]}\) (the Hilbert scheme of \(X_\Gamma\)) are quiver varieties for the affine Dynkin graph corresponding to \(\Gamma\) via the McKay correspondence with the same dimension vectors but different parameters.
In section two, basic concepts such as the definition of quivers, quiver varieties, representation of quivers and the construction of Crawley-Boevey were reviewed. In section three, the author reproduced in a short form a geometric version of the McKay correspondence based on investigation of \(X_\Gamma\), and proved a generalization of certain result of \textit{M. Kapranov} and \textit{E. Vasserot} [Math. Ann. 316, No. 3, 565--576 (2000; Zbl 0997.14001)]. The main result mentioned above was verified in section four. In particular, it follows that the varieties \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) are diffeomorphic. In section five, \(({\mathbb{C}}^* \times {\mathbb{C}}^*)\)-actions on \(X^{\Gamma [n]}\) and \(X_\Gamma^{[n]}\) for cyclic \(\Gamma \cong {\mathbb{Z}}/d {\mathbb{Z}}\) were considered. The author proved the combinatorial identity \(UCY(n, d) = CY(n, d)\) where \(UCY\) and \(CY\) denote the number of uniformly colored diagrams and the number of collections of diagrams respectively. quiver varieties; Hilbert schemes; McKay correspondence; moduli space Kuznetsov, A.: Quiver varieties and Hilbert schemes. Moscow Math. J. \textbf{7}, 673-697 (2007). arXiv:math.AG/0111092 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Parametrization (Chow and Hilbert schemes), Representations of quivers and partially ordered sets, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Quiver varieties and Hilbert schemes. | 0 |
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