Ngau Lam

Super duality and irreducible characters of ortho-symplectic Lie superalgebras

We formulate and establish a super duality which connects parabolic categories O for the ortho-symplectic Lie superalgebras and classical Lie algebras of BCD types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category O, which includes all finite dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.

Irreducible Characters of General Linear Superalgebra and Super Duality

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. Furthermore, we prove that certain parabolic BGG categories over the general linear algebra and over the general linear superalgebra are equivalent. We also verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category of the general linear superalgebra.

Character formula for infinite-dimensional unitarizable modules of the general linear superalgebra

The Fock space of m+p bosonic and n+q fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra glm+p|n+q. Its tensor powers decompose into direct sums of infinite-dimensional irreducible highest-weight glm+p|n+q-modules. We obtain an explicit decomposition of any tensor power of this Fock space into irreducibles, and develop a character formula for the irreducible glm+p|n+q-modules arising in this way.

The brundan–kazhdan–lusztig conjecture for general linear lie superalgebras

In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundans conjecture and its variants associated to all Borel subalgebras in full generality.

A BGG-Type Resolution for Tensor Modules over General Linear Superalgebra

We construct a Bernstein–Gelfand–Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector.

Infinite-Dimensional Lie Superalgebras and Hook Schur Functions

Abstract: Making use of a Howe duality involving the infinite-dimensional Lie superalgebra and the finite-dimensional group GLl of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of in terms of hook Schur functions. We use the reduction procedure of to to derive a character formula for a certain class of level 1 highest weight irreducible representations of, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra . These modules turn out to form the complete set of integrable -modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible -modules may be written as a sum of products of hook Schur functions.

Quasi-finite modules for Lie superalgebras of infinite rank

We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras gl∞ | ∞, & and D, and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.

Finite conformal modules over N=2,3,4 superconformal algebras

In this paper we continue the study of representation theory of formal distribution Lie superalgebras initiated by Cheng and Kac [Asian J. Math. 1, 181–193 (1997); 2, 153–156 (1998) (erratum)]. We study finite Verma-type conformal modules over the N=2, N=3 and the two N=4 superconformal algebras and also find explicitly all singular vectors in these modules. From our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the N=2, N=3 and the two N=4 superconformal algebras.

An inversion formula for some Fock spaces

A symmetric bilinear form on a certain subspace Tˆb of a completion of the Fock space Tb is defined. The canonical and dual canonical bases of Tˆb are dual with respect to the bilinear form. As a consequence, the inversion formula connecting the coefficients of the canonical basis and that of the dual canonical basis of Tˆb expanded in terms of the standard monomial basis of Tb is obtained. Combining with the Brundans algorithm for computing the elements in the canonical basis of Tˆbst, we have an algorithm computing the elements in the canonical basis of Tˆb for arbitrary b.

Super Duality and Homology of Unitarizable Modules of Lie Algebras

The u-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types gl(n), sp(2n) and so(2n) are obtained. As a consequence, we recover the Enrights formulas for three Hermitian symmetric pairs of classical types (SU(p; q); SU(p) X SU(q)), (Sp(2n);U(n)) and (SO*(2n);U(n)).