Virtually indecomposable tensor categories

TL;DR

This paper proves that various classes of tensor categories, including those related to algebraic groups and Lie algebras, are virtually indecomposable, extending Serre's classical results and answering open questions in the field.

Contribution

It establishes the virtual indecomposability of broad classes of tensor categories, including positive characteristic cases, and provides alternative proofs of Serre's theorem.

Findings

Tensor categories with the Chevalley property are virtually indecomposable.

Representation categories of affine and formal groups are virtually indecomposable.

Super Tannakian categories are also virtually indecomposable.

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 \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).