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Finite tensor categories

Pavel Etingof, Viktor Ostrik

TL;DR

The paper develops a comprehensive framework for finite tensor categories beyond semisimplicity, introducing exact module categories and duality concepts to extend classical Hopf-algebra freeness results to the categorical setting. It establishes foundational properties, such as preservation of projectives by surjective quasi-tensor functors and a categorical version of Frobenius-Perron theory, and develops Morita theory and Drinfeld center techniques to relate a category to its dual. The work then applies this machinery to classify indecomposable exact module categories over key examples, including representations of finite groups in positive characteristic, finite supergroups, and Taft algebras, illustrating the reach of the theory in representation theory and related areas. Overall, the results provide a robust nonsemisimple analogue of fusion-category structure theory with broad implications for quantum groups, logarithmic CFT, and group-theoretic tensor categories.

Abstract

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.

Finite tensor categories

TL;DR

The paper develops a comprehensive framework for finite tensor categories beyond semisimplicity, introducing exact module categories and duality concepts to extend classical Hopf-algebra freeness results to the categorical setting. It establishes foundational properties, such as preservation of projectives by surjective quasi-tensor functors and a categorical version of Frobenius-Perron theory, and develops Morita theory and Drinfeld center techniques to relate a category to its dual. The work then applies this machinery to classify indecomposable exact module categories over key examples, including representations of finite groups in positive characteristic, finite supergroups, and Taft algebras, illustrating the reach of the theory in representation theory and related areas. Overall, the results provide a robust nonsemisimple analogue of fusion-category structure theory with broad implications for quantum groups, logarithmic CFT, and group-theoretic tensor categories.

Abstract

We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular, we generalize to the categorical setting the Hopf and quasi-Hopf algebra freeness theorems due to Nichols-Zoeller and Schauenburg, respectively. We also give categorical versions of the theory of distinguished group-like elements in a finite dimensional Hopf algebra, of Lorenz's result on degeneracy of the Cartan matrix, and of the absence of primitive elements in a finite dimensional Hopf algebra in zero characteristic. We also develop the theory of module categories and dual categories for not necessarily semisimple finite tensor categories; the crucial new notion here is that of an exact module category. Finally, we classify indecomposable exact module categories over the simplest finite tensor categories, such as representations of a finite group in positive characteristic, representations of a finite supergroup, and representations of the Taft Hopf algebra.

Paper Structure

This paper contains 25 sections, 58 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. General properties of finite tensor categories
  3. Definitions and notation
  4. Projectivity of tensor products and duals
  5. Surjective quasi-tensor functors
  6. Frobenius-Perron dimensions
  7. Projectivity defect
  8. Proof of Theorem \ref{['projsurj']}
  9. Categorical freeness
  10. The distinguished character
  11. Dimensions of projective objects and degeneracy of the Cartan matrix
  12. Absence of primitive elements
  13. The Ext algebra of a finite tensor category
  14. Surjective and injective functors
  15. Exact module categories
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $P$ be a projective object in ${\mathcal{C}}$, and $X$ any object of ${\mathcal{C}}$. Then $P\otimes X$ is projective.

Theorems & Definitions (130)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Theorem 2.5
  • Proposition 2.6
  • proof
  • ...and 120 more