On supergroups and their semisimplified representation categories

TL;DR

This paper studies the structure of supergroup representation categories, identifies the largest tensor ideal, and describes the resulting semisimple category, including an explicit case for $GL(m|1)$.

Contribution

It characterizes the semisimplification of supergroup representation categories and explicitly determines the reductive supergroup for $GL(m|1)$.

Findings

Identification of the largest proper tensor ideal in supergroup categories

Description of the semisimple quotient category and its properties

Explicit determination of the reductive supergroup for $GL(m|1)$

Abstract

The representation category $\mathcal{A} = Rep(G,\epsilon)$ of a supergroup scheme $G$ has a largest proper tensor ideal, the ideal $\mathcal{N}$ of negligible morphisms. If we divide $\mathcal{A}$ by $\mathcal{N}$ we get the semisimple representation category of a pro-reductive supergroup scheme $G^{red}$. We list some of its properties and determine $G^{red}$ in the case $GL(m|1)$.