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Simple algebras and exact module categories

Kevin Coulembier, Mateusz Stroiński, Tony Zorman

Abstract

We verify a conjecture of Etingof and Ostrik, stating that an algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras. Towards that end, we introduce an analogue of the Jacobson radical of an algebra object, similar to the Jacobson radical of a finite-dimensional algebra. We give applications of our main results in the context of incompressible finite symmetric tensor categories.

Simple algebras and exact module categories

Abstract

We verify a conjecture of Etingof and Ostrik, stating that an algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras. Towards that end, we introduce an analogue of the Jacobson radical of an algebra object, similar to the Jacobson radical of a finite-dimensional algebra. We give applications of our main results in the context of incompressible finite symmetric tensor categories.
Paper Structure (23 sections, 48 theorems, 65 equations, 3 figures)

This paper contains 23 sections, 48 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite tensor categories
  4. Mixed subfunctors
  5. Algebra ideals and mixed subfunctors, first approach
  6. The set-up
  7. The pseudo-Hopf algebra
  8. Action maps
  9. The bijection
  10. Algebra ideals and mixed subfunctors, second approach
  11. Reduction of presheaves to projective objects
  12. Mixed subfunctors and representable presheaf ideals
  13. Mixed subfunctors and Cp-stable ideals
  14. From Cp-stable ideals to mixed subfunctors
  15. From mixed subfunctors to Cp-stable ideals
  16. ...and 8 more sections

Key Result

Theorem 1

Let $A$ be an algebra object in a finite tensor category $\mathcal{C}$. The following are equivalent:

Figures (3)

  • Figure 4.1: Verification of condition \ref{['mixed2']}.
  • Figure 4.2: Verification of condition \ref{['Day2']}.
  • Figure 5.1: Verification of the first part of condition \ref{['mixed3']}.

Theorems & Definitions (106)

  • Theorem : \ref{['thm:mainresult']}
  • Definition 1
  • Definition 2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • ...and 96 more