Yves André and Bruno Kahn: Nilpotence, radicaux et structures monoidales

andre@math.jussieu.fr, kahn@math.jussieu.fr

Submission: 2001, Jun 21, revised: 2001, Oct 29, and 2002, Apr 24

For $K$ a field, a \emph{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 \emph{pro-reductive envelope} $\Pred(G)$ associated to any affine group scheme $G$ over $K$ ($\Pred(\bG_a)=SL_2$, and $\Pred(G)$ is infinite-dimensional for any bigger unipotent group). Other applications are given in this paper as well as in a forthcoming one on motives.

In this revised manuscript, we also provide an abstract version of recent results of Kimura on finite-dimensional Chow motives, and relate his theory to the problem of algebraicity of the sum of even (or odd) Künneth projectors.

2000 Mathematics Subject Classification: 16N, 16D, 18D10, 18E, 14L, 16G60, 13E10, 17C

Keywords and Phrases: nilpotence, radical, monoidal structures, Tannakian categories, motives, proreductive envelopes of affine group schemes

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