The Jones quotients of the Temperley-Lieb algebras

Abstract

When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $T{L}_n(q)$ is non-semisimple for almost all $n$. Jones showed that there is a canonical symmetric bilinear form on $T{L}_n(q)$, whose radical $R_n(q)$ is generated by a certain idempotent $E_{\ell }\in T{L}_{\ell -1}(q)\subseteq T{L}_n(q)$, which is now referred to as the Jones-Wenzl idempotent, for which an explicit formula was subsequently given by Graham and Lehrer. In this work, we study the quotients $Q_n(\ell ):=T{L}_n(q)/R_n(q)$, where $|q^2|=\ell$, which are precisely the algebras generated by Jones' projections. We give the dimensions of their simple modules, as well as $\dim (Q_n(\ell ))$; en route we give generating functions and recursions for the dimensions of cell modules and associated combinatorics. When the order $|q^2|=4$, we obtain an isomorphism of $Q_n(\ell )$ with the even part of the Clifford algebra, well known to physicists through the Ising model. When $|q^2|=5$, we obtain a sequence of algebras whose dimensions are the odd-indexed Fibonacci numbers. The general case is described explicitly.