Nilpotence, radicaux et structures mono\"{\i}dales

TL;DR

This paper studies the structure of Wedderburn $K$-linear categories with locally nilpotent radicals, proving existence and uniqueness of sections, and applies these results to Tannakian categories and affine group schemes.

Contribution

It establishes new existence and uniqueness theorems for sections in Wedderburn categories, including monoidal cases, and generalizes the Jacobson-Morozov theorem for affine group schemes.

Findings

Existence of sections in Wedderburn categories.

Uniqueness of these sections under certain conditions.

Application to pro-reductive envelopes of affine group schemes.

Abstract

For $K$ a field, a Wedderburn $K$-linear category is a $K$-linear category $\sA$ whose radical $\sR$ is locally nilpotent and such that $\bar \sA:=\sA/\sR$ is semi-simple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection $\sA\to \bar\sA$, in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when $\sA$ has a monoidal structure for which $\sR$ is a monoidal ideal. The latter applies notably to Tannakian categories over a field of characteristic zero, and we get a generalisation of the Jacobson-Morozov theorem: the existence of a pro-reductive envelope $\Pred(G)$ associated to any affine group scheme $G$ over $K$. Other applications are given in this paper as well as in a forthcoming one on motives.