Integrable $sl(\infty)$-modules and Category $\mathcal O$ for $\mathfrak{gl}(m|n)$

TL;DR

This paper introduces new categories of integrable sl(∞)-modules related to Lie superalgebras, constructs an injective object that realizes a categorical sl(∞)-action on category O, and explores its socle structure and related conjectures.

Contribution

It defines new categories T(g,k) for integrable sl(∞)-modules, constructs an injective object K(m|n) realizing a categorical sl(∞)-action on category O, and analyzes its socle filtration and conjectural relations.

Findings

The injective object K(m|n) realizes a categorical sl(∞)-action on O(m|n).

The socle of K(m|n) is generated by projective modules in O(m|n).

Explicit socle filtration of K(m|n) is computed.

Abstract

We introduce and study new categories T(g,k)of integrable sl(\infty)-modules which depend on the choice of a certain reductive subalgebra k in g=sl(\infty). The simple objects of these categories are tensor modules as in the previously studied category, however, the choice of k provides more flexibility of nonsimple modules. We then choose k to have two infinite-dimensional diagonal blocks, and show that a certain injective object K(m|n) in T(g,k) realizes a categorical sl(\infty)-action on the integral category O(m|n) of the Lie superalgebra gl(m|n). We show that the socle of K(m|n) is generated by the projective modules in O(m|n), and compute the socle filtration of K(m|n) explicitly. We conjecture that the socle filtration of K(m|n) reflects a "degree of atypicality filtration" on the category O(m|n). We also conjecture that a natural tensor filtration on K(m|n) arises via the Duflo--Serganova functor sending the category O(m|n) to O(m-1|n-1). We prove this latter conjecture for a direct summand of K(m|n) corresponding to the finite-dimensional gl(m|n)-modules.