Commuting Families in Temperley-Lieb Algebras

TL;DR

This paper introduces analogs of Jucys-Murphy elements for the affine Temperley-Lieb algebra, explores their properties, and connects them to affine Hecke algebras and quantum groups, providing explicit eigenvalues and specializations.

Contribution

It defines and analyzes Jucys-Murphy elements in the affine Temperley-Lieb algebra, linking them to affine Hecke algebras and quantum groups, with explicit eigenvalues and basis expansions.

Findings

Explicit expansion of Jucys-Murphy elements in terms of planar Brauer diagrams

Eigenvalues of these elements on generic irreducible representations

Connections to affine Hecke algebra and quantum group Casimir element

Abstract

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements in the affine Hecke algebra of type A, which in turn come from the Casimir element of the quantum group $U_h\mathfrak{gl}_n$. We also give the explicit specializations of these results to the finite Temperley-Lieb algebra.