Nilpotence, radicaux et structures mono\"ıdales
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| Format: | Preprint |
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2002
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| _version_ | 1866908502357704704 |
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| author | André, Yves Kahn, Bruno O'Sullivan, Peter |
| author_facet | André, Yves Kahn, Bruno O'Sullivan, Peter |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_math_0203273 |
| institution | arXiv |
| publishDate | 2002 |
| record_format | arxiv |
| spellingShingle | Nilpotence, radicaux et structures mono\"ıdales André, Yves Kahn, Bruno O'Sullivan, Peter Category Theory Commutative Algebra Representation Theory 16N, 16D, 18D10, 18E, 14L, 16G60, 13E10, 17C 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. |
| title | Nilpotence, radicaux et structures mono\"ıdales |
| topic | Category Theory Commutative Algebra Representation Theory 16N, 16D, 18D10, 18E, 14L, 16G60, 13E10, 17C |
| url | https://arxiv.org/abs/math/0203273 |