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Lectures on Symmetric Tensor Categories

Pavel Etingof, Arun S. Kannan

Abstract

This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of the current state of the theory of symmetric tensor categories and discuss their applications, including ones unavailable in the literature.

Lectures on Symmetric Tensor Categories

Abstract

This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of the current state of the theory of symmetric tensor categories and discuss their applications, including ones unavailable in the literature.

Paper Structure

This paper contains 39 sections, 26 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Lecture 1: Symmetric tensor categories and Deligne's theorem
  3. Symmetric tensor categories
  4. Additive structure
  5. Monoidal structure
  6. Symmetric structure
  7. Rigid categories
  8. Symmetric tensor categories
  9. Monoidal functors and equivalences
  10. Symmetric pseudotensor categories
  11. Representations of Affine Group Schemes and Proalgebraic Completions
  12. The Category of Supervector Spaces
  13. Trace and dimension
  14. Affine Supergroup Schemes
  15. Moderate Growth of Symmetric Tensor Categories and Deligne Categories
  16. ...and 24 more sections

Key Result

Proposition 2.10

${\rm Rep}_\mathbb K(G)$ is a symmetric tensor category over $\mathbb K$.

Theorems & Definitions (76)

  • Remark 2.1
  • Definition 2.2: Monoidal Structure
  • Remark 2.3
  • Definition 2.4: Braided Monoidal Category
  • Definition 2.5: Symmetric Monoidal Category
  • Definition 2.6: Rigidity
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9: Symmetric Tensor Category
  • Proposition 2.10
  • ...and 66 more