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The Drinfeld center of an oligomorphic tensor category

Pavel Etingof, Andrew Snowden

Abstract

Recently, Harman and the second author introduced a new construction of pre-Tannakian tensor categories based on oligomorphic groups. We develop tools for analyzing the Drinfeld centers of these categories, and compute the center explicitly in a number of cases. In particular, we find several finitely tensor-generated pre-Tannakian categories (including the Delannoy category) that are identified with their own center via the canonical functor; prior to this work, we knew no such examples besides the category of vector spaces.

The Drinfeld center of an oligomorphic tensor category

Abstract

Recently, Harman and the second author introduced a new construction of pre-Tannakian tensor categories based on oligomorphic groups. We develop tools for analyzing the Drinfeld centers of these categories, and compute the center explicitly in a number of cases. In particular, we find several finitely tensor-generated pre-Tannakian categories (including the Delannoy category) that are identified with their own center via the canonical functor; prior to this work, we knew no such examples besides the category of vector spaces.

Paper Structure

This paper contains 56 sections, 69 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Descriptions of some centers
  3. Overview of method
  4. Tensor category terminology
  5. Notation
  6. Background on Drinfeld centers
  7. The fundamental group
  8. Drinfeld center
  9. Finite groups
  10. Oligomorphic groups and tensor categories
  11. Oligomorphic groups
  12. Measures and integration
  13. Permutation modules
  14. The completed group algebra
  15. Fraïssé limits
  16. ...and 41 more sections

Key Result

Theorem 1.1

Let $\mathcal{T}$ be one of the following pre-Tannakian categories: Then the natural functor $\mathcal{T} \to \mathcal{Z}(\mathcal{T})$ is an equivalence. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A tree $A$ (left) and its double $B$ (right). There is an embedding $A \to B$ by mapping each leaf to the leaf with the same label.

Theorems & Definitions (169)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 3.1
  • Definition 3.2
  • Example 3.3
  • ...and 159 more