Simple Modules for Temperley-Lieb Algebras and related Algebras

TL;DR

This paper presents an algorithm to compute the dimensions of simple modules for Temperley-Lieb and related algebras over any field, leveraging tilting theory of quantum groups, with applications to Jones quotient and BMW algebras.

Contribution

It introduces a new, practical algorithm for determining simple module dimensions using quantum group tilting theory, applicable in various characteristics.

Findings

Algorithm efficiently computes simple module dimensions in characteristic zero.

Method extends to positive characteristic with more complexity.

Results provide a complete classification for Jones quotient algebras.

Abstract

Let $k$ be an arbitrary field and let $q \in k\setminus\{0\}$. In this paper we use the known tilting theory for the quantum group $U_q(sl_2)$ to obtain the dimensions of simple modules for the Temperley-Lieb algebras $TL_n(q+q^{-1})$ and related algebras over $k$. Our main result is an algorithm which calculates the dimensions of simple modules for these algebras. We take advantage of the fact that $TL_n(q+q^{-1})$ is isomorphic to the endomorphism ring of the $n$'th tensor power of the natural $2$-dimensional module for the quantum group for $sl_2$. This algorithm is easy when the characteristic is $0$ and more involved in positive characteristic. We point out that our results for the Temperley-Lieb algebras contain a complete description of the simple modules for the Jones quotient algebras. Moreover, we illustrate how the same results lead to corresponding information about simple modules for the BMW-algebras and other algebras closely related with endomorphism algebras of families of tilting modules for $U_q(sl_2)$.