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- W2725508022 abstract "When the parameter $q$ is a root of unity, the Temperley-Lieb algebra $TL_n(q)$ is non-semisimplefor 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 bya certain idempotent $E_ellinTL_{ell-1}(q)subseteqTL_n(q)$, 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 thealgebras $Q_n$. We also treat a fusion category $cC_{rm red}$ introduced by Reshitikhin, Turaev and Andersen, whose objectsare the quantum $fsl_2$-tilting modules with non-zero quantum dimension, and which has an associativetruncated tensor product (the fusion product). We show $Q_n(ell)$ is the endomorphism algebraof 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 $K(Q_infty):=bigoplus_{ngeq 1}K_0(Q_n(ell))$ is determined from the structure of $cC_{rm red}$." @default.
- W2725508022 created "2017-07-14" @default.
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- W2725508022 date "2017-03-01" @default.
- W2725508022 modified "2023-09-23" @default.
- W2725508022 title "Temperley-Lieb At Roots of Unity, A Fusion category and the Jones Quotient" @default.
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