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



K. Iohara (Laboratoire de recherche en mathématiques, Institut Camille-Jordan, Université Claude Bernard Lyon, Villeurbanne, France)

G. I. Lehrer (School of Mathematics and Statistics, University of Sydney, NSW, Australia)

R. B. Zhang (School of Mathematics and Statistics, University of Sydney, NSW, Australia)


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