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Maria Gorelik

Publications and source records attributed to Maria Gorelik.

35 records · Page 2Linked to original sources
arXiv2017

Integrable modules over affine Lie superalgebras sl(1|n)^

We describe the category of integrable sl(1|n)^ -modules with the positive central charge and show that the irreducible modules provide the full set of irreducible representations for the corresponding simple vertex algebra.

math.RT↗
arXiv2015

Generalized Reflection Root Systems

We study a combinatorial object, which we call a GRRS (generalized reflection root system); the classical root systems and GRSs introduced by V. Serganova are examples of finite GRRSs. A GRRS is finite if it contains a finite number of vectors and is called affine if it is infinite and has a finite minimal quotient. We prove that an irreducible GRRS containing an isotropic root is either finite or affine; we describe all finite and affine GRRSs and classify them in most of the cases.

math.RT↗
arXiv2014

Characters of (relatively) integrable modules over affine Lie superlagebras

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules $L$ over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras $\mathfrak{g}$. The problems consists of two parts. First, it is the reduction of the problem to the $\overline{\mathfrak{g}}$-module $F(L)$, where $\overline{\mathfrak{g}}$ is the associated to $L$ integral Lie superalgebra and $F(L)$ is an integrable irreducible highest weight $\overline{\mathfrak{g}}$-module. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional $\mathfrak{g}$-modules when $\mathfrak{g}$ is basic, and all maximally atypical non-critical integrable $\mathfrak{g}$-modules when $\mathfrak{g}$ is affine with non-zero dual Coxeter number.

math.RT↗
arXiv2013

Bounded highest weight modules of q(n)

A classification of the simple highest weight bounded modules of the queer Lie superalgebras q(n) is obtained. To achieve this classification we introduce a new combinatorial tool - the star action. Our result leads in particular to a classification of all simple weight q(n)-modules.

math.RT↗
arXiv2012

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. As an application of our 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.

math.RT↗
arXiv2010

Weyl denominator identity for the affine Lie superalgebra gl(2|2)^

We prove the Weyl denominator identity for the affine Lie superalgebra gl(2|2)^ conjectured by V. Kac and M. Wakimoto. As it was pointed out in their paper, this gives another proof of Jacobi identity for the number of presentations of a given integer as a sum of eight squares.

math.RT↗
arXiv2007

On a generic Verma module at the critical level over affine Lie superalgebras

We describe the structure of a Verma module with a generic highest weight at the critical level over a symmetrizable affine Lie superalgebra not of the type A(2k,2l)^{(4)}. We obtain the character formula for a simple module with a generic highest weight at the critical level conjectured by V. G. Kac and D. A. Kazhdan.

math.RT↗
arXiv2005

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.

math.RT↗
arXiv2000

Strongly typical representations of the basic classical Lie superalgebras

The category of representations with a strongly typical central character of a basic classical Lie superalgebra is proven to be equivalent to the category of representations of its even part corresponding to an appropriate central character. For a Lie superalgebra $osp(1,2l)$ the category of representations with a "generic" weakly atypical central character is described.

math.RT↗
arXiv1999

On the Ghost Centre of Lie Superalgebras

We define a notion of ghost centre of a Lie superalgebra g=g_0+g_1 which is a sum of invariants with respect to the usual adjoint action (centre) and invariants with respect to a twisted adjoint action (``anticentre''). We calculate the anticentre in the case when the top external degree of g_1 is a trivial g_0-module. We describe the Harish-Chandra projection of the ghost centre for basic classical Lie superalgebras and show that for these cases the ghost centre coincides with the centralizer of the even part of the enveloping algebra. The ghost centre of a Lie superalgebra plays a role of the usual centre of a Lie algebra in some problems of representation theory. For instance, for osp(1,2l) the annihilator of a Verma module is generated by the intersection with the ghost centre.

math.RT↗
arXiv1999

The Annihilation theorem for the completely reducible Lie superalgebras

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.

math.RT↗
arXiv1997

The prime spectrum of a quantum Bruhat cell translate

The prime spectra of two families of algebras, $S^w$ and $\check{S}^w$, $w\in W,$ indexed by the Weyl group $W$ of a semisimple finitely dimensional are studied. The algebras $S^w$ have been introduced by A.~Joseph; they are $q$-analogues of the algebras of regular functions on $w$-translates of the open Bruhat cell of a semisimple Lie group $G$ corresponding to the Lie algebra $\fg$. We define a stratification of the spectra into components indexed by pairs $(y_1,y_2)$ of elements of the Weyl group satisfying $y_1\leq w\leq y_2$. 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.

q-alg↗