## Nilpotence, radicals and monoidal structures. With an appendix by Peter O’Sullivan. (Nilpotence, radicaux et structures monoïdales.)(French)Zbl 1165.18300

Rend. Semin. Mat. Univ. Padova 108, 107-291 (2002); erratum ibid. 113, 125-128 (2005).
Authors’ abstract: For $$K$$ a field, a Wedderburn $$K$$-linear category is a $$K$$-linear category $$\mathcal A$$ whose radical $$\mathcal R$$ is locally nilpotent and such that $$\overline{A}:={\mathcal A}/{\mathcal R}$$ is semi-simple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection $$A\to\overline{A}$$, in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when $$A$$ has a monoidal structure for which $$R$$ 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 $$^p$$Red$$(G)$$ associated to any affine group scheme $$G$$ over $$K(^p\text{Red} (G_a)=\text{SL}_2$$, and $$^p$$Red$$(G)$$ is infinite-dimensional for any bigger unipotent group). Other applications are given in this paper as well as in the authors’ note [C. R., Math., Acad. Sci. Paris 334, No. 11, 989–994 (2002; Zbl 1052.14021)] on motives.

### MSC:

 18D10 Monoidal, symmetric monoidal and braided categories (MSC2010) 14F42 Motivic cohomology; motivic homotopy theory 14L15 Group schemes 16N99 Radicals and radical properties of associative rings 18E40 Torsion theories, radicals

Zbl 1052.14021
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