We classify good Z-gradings of basic Lie superalgebras over an algebraically closed field F of characteristic zero. Good Z-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the construction of super W -algebras. We also describe the centralizer of a nilpotent even element and of an sl2-triple in gl(m|n) and osp(m|2n).

A contragredient Lie superalgebra is a superalgebra defined by a Cartan matrix. A contragredient Lie superalgebra has finite-growth if the dimensions of the graded components (in the natural grading) depend polynomially on the degree. In this paper we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case.

Abstract. Character formulas for Lie superalgebras have been shown to have important applications to number theory and combinatorics. We prove the Kac-Wakimoto character formula for the general linear Lie superalgebra gl (m|n). This formula specializes to the well-known Kac-Weyl character formula when the modules are typical and to the Weyl denominator identity when the module is trivial. We also prove a determinantal character formula for KW-modules using the Kac-Wakimoto character formula.

We define regular Kac-Moody superalgebras and classify them using integrable modules. We give conditions for irreducible highest weight modules of regular Kac-Moody superalgebras to be integrable. This paper is a major part of the proof for the classification of finite-growth contragredient Lie superalgebras.

We show that the Duflo-Serganova functor on the category of finite-dimensional modules over a finitedimensional contragredient Lie superalgebra induces a ring homomorphism on a natural quotient of the Grothendieck ring, which is isomorphic to the ring of supercharacters. We realize this homomorphism as a certain evaluation of functions related to the supersymmetry property. We use this realization to describe the kernel and image of the homomorphism induced by the Duflo-Serganova functor.