  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).
