A character formula for singly atypical modules of the lie superalgebra sl(m/n) (original) (raw)

Modular invariant representations of infinite-dimensional Lie algebras and superalgebras

Proceedings of the National Academy of Sciences, 1988

In this paper, we launch a program to describe and classify modular invariant representations of infinite-dimensional Lie algebras and superalgebras. We prove a character formula for a large class of highest weight representations L(A) of a Kac-Moody algebra g with a symmetrizable Cartan matrix, generalizing the Weyl-Kac character formula [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70]. In the case of an affine g, this class includes modular invariant representations of arbitrary rational level m = t/u, where t E Z and u E N are relatively prime and m + g 2 g/u (g is the dual Coxeter number). We write the characters of these representations in terms of theta functions and calculate their asymptotics, generalizing the results of Kac and Peterson [

Modules over affine Lie superalgebras

Arxiv preprint hep-th/9304122, 1993

Modules over affine Lie superalgebras G are studied, in particular, for G = OSP (1, 2). It is shown that on studying Verma modules, much of the results in Kac-Moody algebra can be generalized to the super case. Of most importance are the generalized Kac-Kazhdan formula and the Malikov-Feigin-Fuchs construction, which give the weights and the explicit form of the singular vectors in the Verma module over affine Kac-Moody superalgebras. We have also considered the decomposition of the admissible representation of OSP (1, 2) into that of SL(2)⊗Virasoro algebra, through which we get the modular transformations on the torus and the fusion rules. Different boundary conditions on the torus correspond to the different modings of the current superalgebra and characters or super-characters, which might be relevant to the Hamiltonian reduction resulting in Neveu-Schwarz or Ramond superconformal algebras. Finally, the Felder BRST complex, which consists of Wakimoto modules by the free field realization, is constructed.

Finite-Dimensional Half-Integer Weight Modules over Queer Lie Superalgebras

Communications in Mathematical Physics, 2016

We give a new interpretation of representation theory of the finitedimensional half-integer weight modules over the queer Lie superalgebra q(n). It is given in terms of Brundan's work of finite-dimensional integer weight q(n)-modules by means of Lusztig's canonical basis. Using this viewpoint we compute the characters of the finite-dimensional half-integer weight irreducible modules. For a large class of irreducible modules whose highest weights are of special types (i.e., totally connected or totally disconnected) we derive closed-form character formulas that are reminiscent of Kac-Wakimoto character formula for classical Lie superalgebras. 2010 Mathematics Subject Classification. 17B67.

Some new simple modular lie superalgebras

2007

Two new simple modular Lie superalgebras will be obtained in characteristics 3 and 5, which share the property that their even parts are orthogonal Lie algebras and the odd parts their spin modules. The characteristic 5 case will be shown to be related, by means of a construction of Tits, to the exceptional ten dimensional Jordan superalgebra of Kac Date: December 29, 2005.

Characters of (relatively) integrable modules over affine Lie superlagebras

2014

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules LLL over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras mathfrakg\mathfrak{g}mathfrakg. The problems consists of two parts. First, it is the reduction of the problem to the overlinemathfrakg\overline{\mathfrak{g}}overlinemathfrakg-module F(L)F(L)F(L), where overlinemathfrakg\overline{\mathfrak{g}}overlinemathfrakg is the associated to LLL integral Lie superalgebra and F(L)F(L)F(L) is an integrable irreducible highest weight overlinemathfrakg\overline{\mathfrak{g}}overlinemathfrakg-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 mathfrakg\mathfrak{g}mathfrakg-modules when mathfrakg\mathfrak{g}mathfrakg is basic, and all maximally atypical non-critical integrable mathfrakg\mathfrak{g}mathfrakg-modules when mathfrakg\mathfrak{g}mathfrakg is affine with non-zero dual Coxeter number.

Models of some simple modular Lie superalgebras

Pacific Journal of Mathematics, 2009

Models of the exceptional simple modular Lie superalgebras in characteristic p ≥ 3, that have appeared in the classification due to Bouarroudj, Grozman and Leites [BGLb] of the Lie superalgebras with indecomposable symmetrizable Cartan matrices, are provided. The models relate these exceptional Lie superalgebras to some low dimensional nonassociative algebraic systems.

A class of quadratic deformations of Lie superalgebras

2010

We study certain Z_2-graded, finite-dimensional polynomial algebras of degree 2 which are a special class of deformations of Lie superalgebras, which we call quadratic Lie superalgebras. Starting from the formal definition, we discuss the generalised Jacobi relations in the context of the Koszul property, and give a proof of the PBW basis theorem. We give several concrete examples of quadratic Lie superalgebras for low dimensional cases, and discuss aspects of their structure constants for the `type I' class. We derive the equivalent of the Kac module construction for typical and atypical modules, and a related direct construction of irreducible modules due to Gould. We investigate in detail one specific case, the quadratic generalisation gl_2(n/1) of the Lie superalgebra sl(n/1). We formulate the general atypicality conditions at level 1, and present an analysis of zero-and one-step atypical modules for a certain family of Kac modules.

ON CHARACTERS AND DIMENSION FORMULAS FOR REPRESENTATIONS OF THE LIE SUPERALGEBRA gl(M|N)

Lie Theory and Its Applications in Physics V, 2004

We derive a new expression for the supersymmetric Schur polynomials s λ (x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m|n) and gives rise to a determinantal formula for s λ (x/y). In the second part, we use this determinantal formula to derive new expressions for the dimension and superdimension of covariant representations V λ of the Lie superalgebra gl(m|n). In particular, we derive the t-dimension formula, giving a specialization of the character corresponding to the Z-grading of V λ. For a special choice of λ, the new t-dimension formula gives rise to a Hankel determinant identity.

Primitive vectors of Kac-modules of the Lie superalgebras sl(m/n)

Journal of Mathematical Physics, 2000

To study finite-dimensional modules of the Lie superalgebras, Kac introduced the Kac-modules V(Λ) and divided them into typical or atypical modules according as they are simple or not. For Λ being atypical, Hughes et al have an algorithm to determine all the composition factors of a Kac-module; they conjectured that there exists a bijection between the composition factors of a Kac-module and the so-called permissible codes. The aim of this paper is to prove this conjecture. We gives a partial proof here, i.e., to any unlinked code, by constructing explicitly the primitive vector, we prove that there corresponds a composition factor of the Kac-module. We will give a full proof of the conjecture in another paper.