Tensor representations of $\mathfrak q(\infty)$

TL;DR

This paper develops a new categorical framework for modules over the infinite-dimensional queer superalgebra, revealing tensor product structures, socle filtrations, and Koszul self-duality.

Contribution

It introduces a symmetric monoidal category for modules over $q(ty)$, providing new insights into tensor products and duality properties of these modules.

Findings

Tensor products of natural and conatural modules are injective.

Explicit socle filtrations and tensor product formulas are derived.

The category exhibits Koszul self-duality.

Abstract

We introduce a symmetric monoidal category of modules over the direct limit queer superalgebra $\q (\infty)$. The category can be defined in two equivalent ways with the aid of the large annihilator condition. Tensor products of copies of the natural and the conatural representations are injective objects in this category. We obtain the socle filtrations and formulas for the tensor products of the indecoposable injectives. In addition, it is proven that the category is Koszul self-dual.