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Monoidal abelian envelopes and a conjecture of Benson and Etingof

Kevin Coulembier, Inna Entova-Aizenbud and Thorsten Heidersdorf

Vol. 16 (2022), No. 9, 2099–2117
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 SL 2 in characteristic 2. We use our results on the abelian envelopes to prove this conjecture and its variants for any prime p.

Keywords
tensor category, tilting modules, abelian envelope
Mathematical Subject Classification 2010
Primary: 18D10
Secondary: 14L15, 16D90
Milestones
Received: 11 December 2019
Revised: 15 July 2021
Accepted: 17 August 2021
Published: 19 December 2022
Authors
Kevin Coulembier
School of Mathematics and Statistics
University of Sydney
Australia
Inna Entova-Aizenbud
Department of Mathematics
Ben Gurion University of the Negev
Be’er-Sheva
Israel
Thorsten Heidersdorf
Mathematisches Institut
Universität Bonn
Germany