Commuting families in Hecke and Temperley-Lieb algebras

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.