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

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 et al. on the algebras $Q_n(\ell)$. We also treat a fusion category $\mathcal{C}_{\rm red}$ introduced by Reshitikhin, Turaev and Andersen, whose objects are the quantum $\mathfrak{sl}_2$-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 $\mathcal{C}_{\rm red}$ 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 $\mathcal{C}_{\rm red}$, and observe a connection with a fusion category of affine $\mathfrak{sl}_2$ and the Virosoro algebra.