Temperley–Lieb algebras at roots of unity, a fusion category and the Jones quotient

Abstract

When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $\TL_n(q)$ is non-semisimple for almost all $n$. In this work, using cellular methods, we give explicit generating functions for the dimensions of all the simple $\TL_n(q)$-modules. Jones showed that if the order $|q^2|=\ell$ there is a canonical symmetric bilinear form on $\TL_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_\ell\in\TL_{\ell-1}(q)\subseteq\TL_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. Although the algebras $Q_n(\ell):=\TL_n(q)/R_n(q)$, which we refer to as the Jones algebras (or quotients), are not the largest semisimple quotients of the $\TL_n(q)$, our results include dimension formulae for all the simple $Q_n(\ell)$-modules. This work could therefore be thought of as generalising that of Jones {\it et al.} on the algebras $Q_n$. We also treat a fusion category $\cC_{\rm red}$ introduced by Reshitikhin, Turaev and Andersen, whose objects are the quantum $\fsl_2$-tilting modules with non-zero quantum dimension, and which has an associative truncated tensor product (the fusion product). We show $Q_n(\ell)$ is the endomorphism algebra of a certain module in $\cC_{\rm red}$ and use this fact to recover a dimension formula for $Q_n(\ell)$. We also show how a fusion rule for $K(Q_\infty):=\bigoplus_{n\geq 1}K_0(Q_n(\ell))$ is determined from the structure of $\cC_{\rm red}$.