Virtually indecomposable tensor categories

DOI10.4310/MRL.2011.V18.N5.A13zbMATH Open1256.18006arXiv1101.3054OpenAlexW2963981021MaRDI QIDQ2428828

Publication date: 21 April 2012

Published in: Mathematical Research Letters (Search for Journal in Brave)

Abstract: Let k be any field. J-P. Serre proved that the spectrum of the Grothendieck ring of the k-representation category of a group is connected, and that the same holds in characteristic zero for the representation category of a Lie algebra over k. We say that a tensor category C over k is virtually indecomposable if its Grothendieck ring contains no nontrivial central idempotents. We prove that the following tensor categories are virtually indecomposable: Tensor categories with the Chevalley property; representation categories of affine group schemes; representation categories of formal groups; representation categories of affine supergroup schemes (in characteristic

e 2); representation categories of formal supergroups (in characteristic

e 2); symmetric tensor categories of exponential growth in characteristic zero. In particular, we obtain an alternative proof to Serre's Theorem, deduce that the representation category of any Lie algebra over k is virtually indecomposable also in positive characteristic (this answers a question of Serre), and (using a theorem of Deligne in the super case, and a theorem of Deligne-Milne in the even case) deduce that any (super)Tannakian category is virtually indecomposable (this answers another question of Serre).

Full work available at URL: https://arxiv.org/abs/1101.3054

zbMATH Keywords

tensor category Grothendieck ring J.-P. Serre

Mathematics Subject Classification ID

Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) (17B10) Representation theory of groups (20C99) Formal groups, (p)-divisible groups (14L05) Grothendieck groups (category-theoretic aspects) (18F30) Representation theory of associative rings and algebras (16G99) Group schemes (14L15) Groupoids, semigroupoids, semigroups, groups (viewed as categories) (18B40) Hopf algebras and their applications (16T05)

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