Markov elements in affine Temperley-Lieb algebras

Abstract

We define a tower of affine Temperley-Lieb algebras of type $\widetilde{A_n}$ and we define Markov elements in those algebras. We prove that any trace over an affine Temperley-Lieb algebras of type $\widetilde{A_2}$ is uniquely defined by its values on the Markov elements.

We proved in arXiv:1311.7092 that any trace over an affine Temperley-Lieb algebras of type $\widetilde{A_n}$ is uniquely defined by its values on the Markov elements, for $n\ge 3$. We here provide the proof for $n=2$ for completeness