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The representation theory of Brauer categories I: triangular categories

Steven V Sam, Andrew Snowden

Abstract

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

The representation theory of Brauer categories I: triangular categories

Abstract

This is the first in a series of papers in which we study representations of the Brauer category and its allies. We define a general notion of triangular category that abstracts key properties of the triangular decomposition of a semisimple complex Lie algebra, and develop a highest weight theory for them. We show that the Brauer category, the partition category, and a number of related diagram categories admit this structure.

Paper Structure

This paper contains 70 sections, 66 theorems, 60 equations, 2 figures.

Table of Contents

  1. Overview
  2. Main results
  3. Other diagram categories
  4. Motivation
  5. Relation to previous work
  6. Diagram algebras
  7. Deligne categories
  8. Twisted commutative algebras
  9. Open problems
  10. Introduction
  11. Triangular categories
  12. The Brauer category
  13. Relation to previous work
  14. Outline
  15. Representations of categories
  16. ...and 55 more sections

Key Result

Theorem 2.1

The Brauer category $\mathfrak{G}$ is naturally a triangular category.

Figures (2)

  • Figure 1: Concepts connected to the Brauer category.
  • Figure 2: Summary of triangular categories.

Theorems & Definitions (146)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 136 more