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 in characteristic . We use our results on the abelian envelopes to prove this conjecture and its variants for any prime .
Kevin Coulembier. Inna Entova-Aizenbud. Thorsten Heidersdorf. "Monoidal abelian envelopes and a conjecture of Benson and Etingof." Algebra Number Theory 16 (9) 2099 - 2117, 2022. https://doi.org/10.2140/ant.2022.16.2099