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Diagram categories of Brauer type

Sigiswald Barbier

Abstract

This paper introduces monoidal (super)categories resembling the Brauer category. For all categories, we can construct bases of the hom-spaces using Brauer diagrams. These categories include the Brauer category, its deformation the BWM-category, the periplectic Brauer category, and its deformation the periplectic $q$-Brauer category but also some new exotic categories. We show that the BWM-category is the unique deformation of the Brauer category in this framework, while the periplectic Brauer category has two deformations, which are each other monoidal opposite.

Diagram categories of Brauer type

Abstract

This paper introduces monoidal (super)categories resembling the Brauer category. For all categories, we can construct bases of the hom-spaces using Brauer diagrams. These categories include the Brauer category, its deformation the BWM-category, the periplectic Brauer category, and its deformation the periplectic -Brauer category but also some new exotic categories. We show that the BWM-category is the unique deformation of the Brauer category in this framework, while the periplectic Brauer category has two deformations, which are each other monoidal opposite.

Paper Structure

This paper contains 16 sections, 6 theorems, 52 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Diagram algebras and categories
  3. The diagram categories of Brauer type
  4. Structure of the paper
  5. Brauer diagrams
  6. Monoidal supercategories
  7. Definition of a monoidal supercategory
  8. Example: the marked Brauer category
  9. Motivating the definition of categories of Brauer type
  10. The defining relations of the category
  11. Establishing the equations
  12. Categories of Brauer type
  13. The case f=f'=0.
  14. The subcase S non-zero and l'=l.
  15. The subcase S non-zero and l' not =l.
  16. ...and 1 more sections

Key Result

Proposition \oldthetheorem

Every $(m,n)$-Brauer diagram has a unique decomposition into fundamental diagrams of the form $UXA$ where $U \in I(r,n)$, $X \in H(r)$ and $A \in J(m,r)$.

Figures (4)

  • Figure 1.1: An example of a $(8,6)$-Brauer diagram with one cup, two caps, and four propagating lines.
  • Figure 1.2: Some examples of relations to simplify diagrams.
  • Figure 2.1: Two different decompositions into fundamental diagrams of the same Brauer diagram: $s_1^2 a_1^2 a_1^4s_2^6=a_1^2a_2^4s_3^6s_5^6$.
  • Figure 2.2: Example of the diagram $I_1^{7,7}I_2^{5,4} I_0^{3,2} I_1^{1,1}$ in $I(1,9)$.

Theorems & Definitions (18)

  • Definition \oldthetheorem: Brauer diagram
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem: Strict monoidal supercategory
  • Definition \oldthetheorem: Monoidal superfunctors
  • Definition \oldthetheorem: Monoidal opposite
  • Definition \oldthetheorem
  • ...and 8 more