# Semisimple algebraic tensor categories

## TL;DR

This paper characterizes semisimple algebraic tensor categories over an algebraically closed field of characteristic zero as categories of finite-dimensional twisted super representations of affine reductive supergroups, detailing their structural properties.

## Contribution

It provides a classification of such tensor categories in terms of supergroups and describes the conditions under which these supergroups are reductive, linking algebraic and superalgebraic structures.

## Key findings

- Semisimple algebraic tensor categories correspond to representations of affine reductive supergroups.
- Reductivity of the supergroup is characterized by its connected component and Lie superalgebra structure.
- The Lie superalgebra divided by its center decomposes into classical Lie algebras and orthosymplectic types.

## Abstract

A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a supergroup is reductive if and only if its connected component is reductive. The connected component is reductive if and only if the Lie superalgebra divided by its center is a product of simple Lie algebras of classical type and Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.