Crystal Hoyt

Classification of Finite-Growth General Kac–Moody Superalgebras

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 article we classify finite-growth contragredient Lie superalgebras. Previously, such a classification was known only for the symmetrizable case.

Simplicity of vacuum modules over affine Lie superalgebras

We prove an explicit condition on the level

Integrable sl(∞)-modules and category O for gl(m|n): INTEGRABLE sl(∞)-MODULES AND CATEGORY O FOR gl(m|n)

k

Good gradings of basic Lie superalgebras

for the irreducibility of a vacuum module

Regular Kac–Moody superalgebras and integrable highest weight modules

V^{k}

Kac -Moody superalgebras of finite growth

over a (non-twisted) affine Lie superalgebra, which was conjectured by M. Gorelik and V.G. Kac. An immediate consequence of this work is the simplicity conditions for the corresponding minimal W-algebras obtained via quantum reduction, in all cases except when the level

Integrable

k

sl(\infty)

is a non-negative integer.

-modules and Category

We introduce and study new categories Tg,k of integrable g = sl(∞)-modules which depend on the choice of a certain reductive in g subalgebra k ⊂ g. The simple objects of Tg,k are tensor modules as in the previously studied category Tg [DPS]; however, the choice of k provides for more flexibility of nonsimple modules in Tg,k compared to Tg. We then choose k to have two infinite-dimensional diagonal blocks, and show that a certain injective object Km|n in Tg,k realizes a categorical sl(∞)-action on the category O Z m|n, the integral category O of the Lie superalgebra gl(m|n). We show that the socle of Km|n is generated by the projective modules in O m|n, and compute the socle filtration of Km|n explicitly. We conjecture that the socle filtration of Km|n reflects a “degree of atypicality filtration” on the category O m|n. We also conjecture that a natural tensor filtration on Km|n arises via the Duflo–Serganova functor sending the category O m|n to O Z m−1|n−1. We prove a weaker version of this latter conjecture for the direct summand of Km|n corresponding to finite-dimensional gl(m|n)-modules. Mathematics subject classification (2010): Primary 17B65, 17B10, 17B55.