From tensor category to Temperley-Lieb algebra representation

TL;DR

This paper constructs a representation of the Temperley-Lieb algebra using a specific type of monoidal category, providing a manual and connecting it to known examples like Schur-Weyl duality for quantum groups.

Contribution

It introduces a new method to derive Temperley-Lieb algebra representations from tensor categories, expanding the understanding of their algebraic structures.

Findings

Constructed a Temperley-Lieb algebra representation from a semisimple monoidal category.

Provided a manual for tensor categories and summarized a key example involving Schur-Weyl duality.

Connected the construction to known algebraic dualities in quantum group theory.

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 $U_q(sl_2))$.