Virtually indecomposable tensor categories

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 \ne 2); representation categories of formal supergroups (in characteristic \ne 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).

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