From tensor category to Temperley-Lieb algebra representation

Abstract

We construct a representation of the Temperley-Lieb algebra from a multiplicity-free semisimple monoidal Abelian category ${\cal C}$, with two simple objects $\lambda$ and $\nu$ such that $\lambda\otimes\nu$ is simple and Hom$_{\cal C}(\lambda\otimes \lambda, \nu)$ is not empty. A self-contained manual to tensor categories is also provided as well as a summary of the best known example of the construction: Schur-Weyl duality for $U_q(sl_2))$.