Contents Online

# Mathematical Research Letters

## Volume 26 (2019)

### Number 1

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

Pages: 121 – 158

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n1.a8

#### Authors

#### Abstract

When the parameter $q$ is a root of unity, the Temperley–Lieb algebra $\mathrm{TL}_n$ 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 $\mathrm{TL}_n (q)$-modules. Jones showed that if the order $\lvert q^2 \rvert = \ell$ there is a canonical symmetric bilinear form on $\mathrm{TL}_n (q)$, whose radical $R_n (q)$ is generated by a certain idempotent $E_{\ell-1} \in \mathrm{TL}_{\ell-1} (q) \subseteq \mathrm{TL}_n (q)$, 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 $Q_n (\ell) := \mathrm{TL}_n (q) / R_n (q)$, which we refer to as the Jones algebras (or quotients), are not the largest semisimple quotients of the $\mathrm{TL}_n (q)$, 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 *et al.* on the algebras $Q_n (\ell)$. We also treat a fusion category $\mathcal{C}_{\mathrm{red}}$ introduced by Reshitikhin, Turaev and Andersen, whose simple objects are the quantum $\mathfrak{sl}_2$-tilting modules with non-zero quantum dimension, and which has an associative truncated tensor product referred to below as the fusion product. We show $Q_n (\ell)$ is the endomorphism algebra of a certain module in Cred and use this fact to recover a dimension formula for $Q_n (\ell)$. We also show how to construct a “stable limit” $K (Q_{\infty})$ of the corresponding fusion category of the $Q_n (\ell)$, whose structure is determined by the fusion rule of $\mathcal{C}_{\mathrm{red}}$, and observe a connection with a fusion category of affine $\mathfrak{sl}_2$.

Received 13 September 2017

Accepted 30 January 2018

Published 7 June 2019