Vera Serganova

Kazhdan-Lusztig polynomials and character formula for the Lie superalgebragI(m/n)

We find the character formula for irreducible finite-dimensionalgl(m/n)-modules. Also multiplicities of the composition factors in a Kac module are calculated.

Generic Irreducible Representations of Finite-Dimensional Lie Superalgebras

A theory of highest weight modules over an arbitrary finite-dimensional Lie superalgebra is constructed. A necessary and sufficient condition for the finite-dimensionality of such modules is proved. Generic finite-dimensional irreducible representations are defined and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic finite-dimensional irreducible module over any finite-dimensional Lie superalgebra. The conjecture is proved for several classes of Lie superalgebras, in particular for all solvable ones, for all simple ones, and for certain semi-simple ones.

Cohomology of generalized supergrassmannians and character formulae for basic classical Lie superalgebras

In this paper, we use geometrical methods adapted from the Borel-Weil-Bott theory to compute the character of every finite dimensional simple module over a basic classical Lie superalgebra.

On generalizations of root systems

We define a generalization of a root system as a set of vectors in a vector space with some symmetry property. The main difference with the usual root systems is the existence of isotropic roots. We classify irreducible generalized root systems. As follows from our classification all such root systems are root systems of contragredient Lie superalgebras which were classified by V.Kac in 1977.

Kac–Moody Superalgebras and Integrability

The first part of this paper is a review and systematization of known results on (infinite-dimensional) contragredient Lie superalgebras and their representations. In the second part, we obtain character formulae for integrable highest weight representations of sl (1|n) and osp (2|2n); these formulae were conjectured by Kac–Wakimoto.

Characters of Finite-Dimensional Irreducible q(n) -Modules

The solution of the Kac character problem for the‘queer’ series of Lie superalgebras q(n) is announced. An explicit algorithm which computes the character of an arbitrary finite-dimensional irreducible q(n)-module is presented. As an illustration, the ‘correction terms’ to the generic character formula of Penkov (Monatsh. Math.118 (1994), 419) are written down for all finite-dimensional irreducible representations of q(n), for n≤4, with nongeneric character.

On the Superdimension of an Irreducible Representation of a Basic Classical Lie Superalgebra

In this paper we prove the Kac-Wakimoto conjecture that a simple module over a basic classical Lie superalgebra has non-zero superdimension if and only if it has maximal degree of atypicality. The proof is based on the results of [Duflo and Serganova, On associated variety for Lie superalgebras, math/0507198] and [Gruson and Serganova, Proceedings of the London Mathematical Society, doi:10.1112/plms/pdq014].We also prove the conjecture in [Duflo and Serganova, On associated variety for Lie superalgebras, math/0507198] about the associated variety of a simple module and the generalized Kac-Wakimoto conjecture in [Geer, Kujawa and Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, arXiv:1001.0985v1] for the general linear Lie superalgebra.

Classification of Finite-Growth General Kac–Moody Superalgebras

A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree. In this article we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case.

On representations of the Lie superalgebra p(n)

Abstract We introduce a new way to study representations of the Lie superalgebra p(n). Since the center of the universal enveloping algebra U acts trivially on all irreducible representations, we suggest to study the quotient algebra U by the radical of U. We show that U has a large center which separates typical finite-dimensional irreducible representations. We give a description of U factored by a generic central character. Using this description we obtain character formulae of generic (infinite-dimensional) irreducible representations. We also describe some geometric properties of the supervariety Spec Gr U in the coadjoint representation.

A reduction method for atypical representations of classical Lie superalgebras

Abstract We introduce a reduction method for studying representations of classical Lie superalgebras with atypical central character. We show that the atypical quotient of universal enveloping algebra has a non-trivial Jacobson radical. The factor by this radical has a new center, which is calculated for sl(1| n ) and psl(2|2). Using this center we obtain new character formulae, generalization of Borel–Weil–Bott and Beilinson–Bernstein localization theorems.