From tensor category to Temperley-Lieb algebra representation

We construct a representation of the Temperley-Lieb algebra from a multiplicity-free semisimple monoidal Abelian category ${\cal C}$, with two simple objects $λ$ and $ν$ such that $λ\otimesν$ is simple and Hom$_{\cal C}(λ\otimes λ, ν)$ 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 $U_q(sl_2))$.

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