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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

Citations:

Zbl 1052.14021
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References:

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