On geometrically defined extensions of the Temperley-Lieb category in the Brauer category

TL;DR

This paper introduces a hierarchy of subcategories within the partition category based on left-height, connecting Temperley-Lieb and Brauer categories, and studies their algebraic representation theory.

Contribution

It defines an infinite chain of subcategories parameterized by left-height, extending classical categories and analyzing their associated algebra representations.

Findings

Established the chain from Temperley-Lieb to Brauer categories

Analyzed the algebraic structure of End sets for each subcategory

Provided insights into the representation theory of these algebras

Abstract

We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=\infty$) category. The End sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.