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 {\em characteristic zero} for the representation category of a Lie algebra over $k$ \cite{se}. We say that a tensor category $\C$ over $k$ is \emph{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 {\em positive characteristic} (this answers a question of Serre \cite{se}), and (using a theorem of Deligne \cite{d} in the super case, and a theorem of Deligne–Milne \cite{dm} in the even case) deduce that any (super)Tannakian category is virtually indecomposable (this answers another question of Serre \cite{se}).

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Published 28 October 2011