Virtually indecomposable tensor categories (Q2428828)

The author calls a tensor category virtually indecomposable, if its Grothendieck ring (considered over \(\mathbb Z\)) has no nontrivial central idempotents, and proves that a number of tensor categories, such as tensor categories with the Chevalley property, representation categories of affine (super)group schemes and of formal (super)groups, and symmetric tensor categories of exponential growth in characteristic zero, are virtually indecomposable. Among other, this answers affirmatively a question of Serre about connectedness of the spectrum of the Grothendieck ring of a Tannakian category.