Markov elements in affine Temperley-Lieb algebras

TL;DR

This paper introduces a tower of affine Temperley-Lieb algebras of type A_{n} and characterizes traces on these algebras, showing they are uniquely determined by Markov elements in the A_{2} case.

Contribution

It defines Markov elements within affine Temperley-Lieb algebras and proves the uniqueness of traces based on these elements for type A_{2}.

Findings

Trace over affine Temperley-Lieb algebra of type A_{2} is uniquely determined by Markov elements.

Introduces a new framework for understanding traces in affine Temperley-Lieb algebras.

Establishes foundational properties of Markov elements in the affine setting.

Abstract

We define a tower of affine Temperley-Lieb algebras of type $\tilde{A_{n}}$ and we define Markov elements in those algebras. We prove that any trace over an affine Temperley-Lieb algebras of type $\tilde{A_{2}}$ is uniquely defined by its values on the Markov elements.