A Temperley-Lieb analogue for the BMW algebra

TL;DR

This paper introduces a new quotient of the BMW algebra analogous to the Temperley-Lieb algebra, exploring its properties and connections to quantum sl(2) representations, and discusses general results on cellular algebra radicals.

Contribution

It defines and studies a Temperley-Lieb analogue for the BMW algebra, extending the understanding of algebraic structures related to quantum groups.

Findings

Established properties of the new quotient algebra.

Proved results on the radical of cellular algebras.

Connected the algebra to quantum sl(2) representations.

Abstract

The Temperley-Lieb algebra may be thought of as a quotient of the Hecke algebra of type A, acting on tensor space as the commutant of the usual action of quantum sl(2) on the n-th tensor power of the 2-dimensional irreducible module. We define and study a quotient of the Birman-Wenzl-Murakami algebra, which plays an analogous role for the 3-dimensional representation of quantum sl(2). In the course of the discussion we prove some general results about the radical of a cellular algebra, which may be of independent interest.