Nilpotence, radicaux et structures monoïdales

Details Nilpotence, radicaux et structures monoïdales Nilpotence, radicaux et structures monoïdales

2002-03-26

arxiv.org/abs/math/0203273v3

Rendiconti del Seminario Matematico dell'Universit\~A? di Padova 108 (2002), 107--291

Mathematics

Category Theory

145 pages. Version a paraitre aux Rendiconti del Seminario Matematico dell'Universita' di Padova

Scientific paper

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.

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