It takes two spectral sequences

TL;DR

This paper develops spectral sequences for the representation theory of bgl(1|1), leading to new functors and branching rules for modules over Lie superalgebras, advancing understanding of superdimension zero modules.

Contribution

It introduces two spectral sequences that classify superdimension zero modules and constructs explicit semisimplification functors for bgl(1|1) representations.

Findings

Spectral sequences annihilate superdimension zero modules.

Constructed explicit semisimplification functors.

Proved branching rules for modules over Kac-Moody and queer Lie superalgebras.

Abstract

We study the representation theory of the Lie superalgebra $\mathfrak{gl}(1|1)$, constructing two spectral sequences which eventually annihilate precisely the superdimension zero indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on $\mathrm{Rep} (\mathfrak{gl}(1|1))$. These two spectral sequences are related by contragredient duality, and from their limits we construct explicit semisimplification functors, which we explicitly prove are isomorphic up to a twist. We use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to $\mathfrak{gl}(1|1)$-subalgebras.