Temperley-Lieb and Birman-Murakami-Wenzl like relations from multiplicity free semi-simple tensor system

TL;DR

This paper investigates conditions under which projection operators in multiplicity free semi-simple tensor categories satisfy Temperley-Lieb and Birman-Murakami-Wenzl relations, linking category data to algebra representations.

Contribution

It establishes sufficient conditions for representing the Birman-Murakami-Wenzl algebra using braided multiplicity free semi-simple tensor categories.

Findings

Projection operators satisfy Temperley-Lieb relations under specific conditions.

Sufficient conditions are provided for representing the Birman-Murakami-Wenzl algebra.

Results utilize the data of the categories, with connections to diagrammatic proofs.

Abstract

In this article we consider conditions under which projection operators in multiplicity free semi-simple tensor categories satisfy Temperley-Lieb like relations. This is then used as a stepping stone to prove sufficient conditions for obtaining a representation of the Birman-Murakami-Wenzl algebra from a braided multiplicity free semi-simple tensor category. The results are found by utalising the data of the categories. There is considerable overlap with the results found in arXiv:1607.08908, where proofs are shown by manipulating diagrams.