Nilpotence, radicaux et structures monoïdales
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- March 2002
- DOI:
- 10.48550/arXiv.math/0203273
- arXiv:
- arXiv:math/0203273
- Bibcode:
- 2002math......3273A
- Keywords:
-
- Mathematics - Category Theory;
- Mathematics - Commutative Algebra;
- Mathematics - Representation Theory;
- 16N;
- 16D;
- 18D10;
- 18E;
- 14L;
- 16G60;
- 13E10;
- 17C
- E-Print:
- 145 pages. Version a paraitre aux Rendiconti del Seminario Matematico dell'Universita' di Padova