Irreducible Characters of Kac-Moody Lie superalgebras

TL;DR

This paper extends super duality to Kac-Moody Lie superalgebras, establishing character formulas for irreducible modules using Kazhdan-Lusztig polynomials and analyzing integrable modules' structure.

Contribution

It introduces a super duality formalism for Kac-Moody Lie superalgebras, providing character formulas and a semisimple tensor category of integrable modules.

Findings

Characters expressed via Kazhdan-Lusztig polynomials

Equivalence of categories between superalgebras and algebras

Semisimple tensor subcategory of integrable modules

Abstract

Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a large family of irreducible highest weight modules over a symmetrizable Kac-Moody Lie superalgebra are then given in terms of Kazhdan-Lusztig polynomials for the first time. We formulate a notion of integrable modules over a symmetrizable Kac-Moody Lie superalgebra via super duality, and show that these integrable modules form a semisimple tensor subcategory, whose Littlewood-Richardson tensor product multiplicities coincide with those in the Kac-Moody algebra setting.