Maria Gorelik

Shapovalov determinants of Q-type Lie superalgebras

We define an analogue of Shapovalov forms for Q-type Lie superalgebras and factorize the corresponding Shapovalov determinants which are responsible for simplicity of highest weight modules. We apply the factorization to obtain a description of the centres of Q-type Lie superalgebras.

The Kac Construction of the Centre of U(g) for Lie Superalgebras

Abstract In 1984, Victor Kac [8] suggested an approach to a description of central elements of a completion of for any Kac-Moody Lie algebra . The method is based on a recursive procedure. Each step is reduced to a system of linear equations over a certain subalgebra of meromorphic functions on the Cartan subalgebra. The determinant of the system coincides with the Shapovalov determinant for . We prove that the Kac approach can also be applied to finite dimensional Lie superalgebras with Cartan matrix A (as claimed in [8]) and reproduce for them Sergeev’s description of the centers of [14]. In order to prove this, one needs to show that the recursive procedure stops after a finite number of steps. The original paper [8] does not indicate how to check this fact. Here we give a detailed presentation of the Kac approach and apply it to finite dimensional Lie superalgebras . In particular, we deduce the Kac formulas for the Shapovalov determinants and verify the finiteness of the recursive procedure.

Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

The annihilation theorem for the completely reducible Lie superalgebras

Abstract.A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1,2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not generated by its intersection with the centre of universal enveloping algebra. More generally, Musson produced in [Mu1] a family of such “singular” Verma modules for osp(1,2l) cases. In this article we give a necessary and sufficient condition on the highest weight of a osp(1,2l)-Verma module for its annihilator to be generated by its intersection with the centre. This answers a question of Musson. The classical proof of the Duflo theorem is based on a deep result of Kostant which uses some delicate algebraic geometry reasonings. Unfortunately these arguments can not be reproduced in the quantum and super cases. This obstruction forced Joseph and Letzter, in their work on the quantum case (see [JL]), to find an alternativeapproach to the Duflo theorem. Following their ideas, we compute the factorization of the Parthasarathy–Ranga-Rao–Varadarajan (PRV) determinants. Comparing it with the factorization of Shapovalov determinants we find, unlike to the classical and quantum cases, that the PRV determinant contains some extrafactors. The set of zeroes of these extrafactors is precisely the set of highest weights of Verma modules whose annihilators are not generated by their intersection with the centre. We also find an analogue of Hesselink formula (see [He]) giving the multiplicity of every simple finite dimensional module in the graded component of the harmonic space in the symmetric algebra.

Denominator identities for finite-dimensional Lie superalgebras and Howe duality for compact dual pairs

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups, and, as an application of these formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras.

Advances in lie superalgebras

This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory of generalized Drinfeld-Sokolov hierarchies.Using a super-affine version of Kostant’s cubic Dirac operator, we prove a very strange formula for quadratic finite-dimensional Lie superalgebras with a reductive even subalgebra.

The prime spectrum of a quantum analogue of the ring of regular functions on the open Bruhat cell translate

Abstract The prime spectrum of a quantum analogue R o w of the ring of regular functions on the w -translate of the open Bruhat cell is studied in the spirit of [3]. We define a stratification of the spectrum into components indexed by pairs ( y 1 , y 2 ) of elements of the Weyl group satisfying y 1 ≤ w ≤ y 2 . Each component admits a unique minimal ideal which is explicitly described. We show the inclusion relation of closures to be that induced by Bruhat order.

The annihilation theorem for the Lie superalgebra osp(1,2ℓ)

Abstract We give necessary and sufficient conditions for the annihilator of a Verma module over a Lie superalgebra osp(1,2l) to be generated by its intersection with the centre of the universal enveloping superalgebra.

Integrable Modules Over Affine Lie Superalgebras \({\mathfrak{sl}(1|n)^{(1)}}\)

We describe the category of integrable