Monoidal abelian envelopes and a conjecture of Benson--Etingof

Abstract

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply connected algebraic group is the abelian envelope of the category of tilting modules. Benson and Etingof conjectured that a certain limit of finite symmetric tensor categories is tensor equivalent to the finite dimensional representations of $S{L}_2$ in characteristic $2$. We use our results on the abelian envelopes to prove this conjecture and its variants for any prime $p$.

ver 3: Generalized the results to arbitrary primes p (ver 1,2 dealt with the case p=2), ver 2: minor fix in the definition of abelian envelope

References in corpus (2)

• Monoidal abelian envelopes
• Tannakian categories in positive characteristic