Generalized diagram categories and monoids, and their representations

TL;DR

This paper introduces a broad family of diagram categories and monoids derived from two-dimensional cobordisms, providing a unified framework for their representation theory and extending classical examples like Temperley--Lieb and Brauer categories.

Contribution

It generalizes classical diagram categories by incorporating twists, creating a new family, and develops a unified approach to their representation theory.

Findings

Introduces a new family of diagram categories and monoids.

Provides a unified framework for their representation theory.

Extends classical diagram categories with additional twists.

Abstract

Classical diagram categories and monoids, including the Temperley--Lieb, Brauer, and partition cases, arise as special instances of the category of two dimensional cobordisms and admit additional twists that produce a large new family of diagram categories and monoids. In this paper we introduce this family and develop a unified approach to their representation theory.