Algebraic description of -modules associated to matrices

2014

We give an answer to the abstract Capelli problem: Let (G, V) be a multiplicity-free finitedimensional representation of a connected reductive complex Lie Group G and G ′ be its derived subgroup. Assume that the categorical quotient V //G is one-dimensional , i.e., there exists a polynomial f generating the algebra of G ′-invariant polynomials on V (C[V ] G ′ = C[f ]) and such that f ∈ C[V ] G. We prove that the category of regular holonomic DV-modules invariant under the action of G is equivalent to the category of graded modules of finite type over a suitable algebra A. This has been conjectured by T. Levasseur [27, Conjecture 5.17, p. 508] (after we had already proved it in some cases: [34], [35], [36], [37]).

Journal of Algebra, 2017

We give an answer to the abstract Capelli problem: Let (G, V) be a multiplicity-free finitedimensional representation of a connected reductive complex Lie Group G and G ′ be its derived subgroup. Assume that the categorical quotient V //G is one-dimensional , i.e., there exists a polynomial f generating the algebra of G ′-invariant polynomials on V (C[V ] G ′ = C[f ]) and such that f ∈ C[V ] G. We prove that the category of regular holonomic DV-modules invariant under the action of G is equivalent to the category of graded modules of finite type over a suitable algebra A. This has been conjectured by T. Levasseur [27, Conjecture 5.17, p. 508] (after we had already proved it in some cases: [34], [35], [36], [37]).

arXiv (Cornell University), 2014

We give an answer to the abstract Capelli problem: Let (G, V) be a multiplicity-free finitedimensional representation of a connected reductive complex Lie Group G and G ′ be its derived subgroup. Assume that the categorical quotient V //G is one-dimensional , i.e., there exists a polynomial f generating the algebra of G ′-invariant polynomials on V (C[V ] G ′ = C[f ]) and such that f ∈ C[V ] G. We prove that the category of regular holonomic DV-modules invariant under the action of G is equivalent to the category of graded modules of finite type over a suitable algebra A. This has been conjectured by T. Levasseur [27, Conjecture 5.17, p. 508] (after we had already proved it in some cases: [34], [35], [36], [37]).

Selecta Mathematica, New Series

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of "smallest" such modules are irreducibleĝ-modules and all irreducible gintegrableĝ-modules at the critical level arise in this way. X ⊗H X , where H X is the algebra of differential polynomials on H 1 (X, Ω 1 X → Ω 2,cl X ). Apart from serving as a prototype, algebras of twisted differential operators are directly linked to algebras of twisted chiral differential operators via the notion of the Zhu algebra [Zhu], and this is another topic of the present paper. Zhu attached to each graded vertex algebra V an associative algebra, Zhu(V ). We show that the sheaf associated to the presheaf X ⊃ U → Zhu(D ch,tw X (U )) is precisely D tw X . Zhu(V ) controls representation theory of V , the subject to which we now turn.

2021

A theorem of Goodwillie [11] shows that the periodic cyclic homology has homotopy invariance in characteristic zero. Getzler [9] constructs, under a suitable condition, a flat connection on the relative periodic cyclic homology of an associative algebra over a base commutative ring of characteristic zero. The connection can be thought of as a noncommutative analogue of Gauss-Manin connection. Morita invariant property is of fundamental importance in noncommutative algebraic geometry. Unfortunately, to the knowledge of the author, Morita invariance of the original connection remains open. Nowadays, in view of several motivations including homological mirror symmetry, the significance of Morita invariance of the associated connection becomes clear. Equivariant contexts (that is, categories with group actions) naturally arise from interesting situations: for example matrix factorizations [22], schemes with group actions, etc. However, explicit formulas often destroy symmetries, and it ...

We give a classification of regular holonomic D-modules on skew-symmetric matrices attached to the linear action of the general linear group GLN .

Mathematische Annalen, 2014

We give an algebraic classification of regular holonomic D-modules on (C 2n) 2 related to the action of the group Sp(2n, C) × GL(2, C) product of the symplectic linear transformations group by the general linear group.

Selecta Mathematica, 2011

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of "smallest" such modules are irreducibleĝ-modules and all irreducible gintegrableĝ-modules at the critical level arise in this way. X ⊗H X , where H X is the algebra of differential polynomials on H 1 (X, Ω 1 X → Ω 2,cl X ). Apart from serving as a prototype, algebras of twisted differential operators are directly linked to algebras of twisted chiral differential operators via the notion of the Zhu algebra [Zhu], and this is another topic of the present paper. Zhu attached to each graded vertex algebra V an associative algebra, Zhu(V ). We show that the sheaf associated to the presheaf X ⊃ U → Zhu(D ch,tw X (U )) is precisely D tw X . Zhu(V ) controls representation theory of V , the subject to which we now turn.

Transformation Groups, 2010

Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in [CP2] and were motivated by representations of quantum affine algebras. In [FL] a more general case was considered by replacing the polynomial ring with the coordinate ring of an algebraic variety and partial results analogous to those in were obtained. In this paper, we show that there is a natural definition of the local and global Weyl modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing results in [CP2] and [FL]. We also analyze the fundamental Weyl modules and show that unlike the case of the affine Lie algebras, the Weyl functors need not be left exact.

2008

We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the N-Maurer-Cartan equation.