The Temperley–Lieb algebra and its generalizations in the Potts andXXZ models

Abstract

We discuss generalizations of the Temperley–Lieb algebra in the Potts andXXZ models. These can be used to describe the addition of integrable boundary terms ofdifferent types.We use the Temperley–Lieb algebra and its one-boundary, two-boundary, and periodicextensions to classify different integrable boundary terms in the two-, three-, and four-statePotts models. The representations always lie at critical points where the algebras becomesnon-semisimple and possess indecomposable representations. In the one-boundary case weshow how to use representation theory to extract the Potts spectrum from anXXZ model with particular boundary terms and hence obtain the finite size scaling of the Potts models.In the two-boundary case we find that the Potts spectrum can be obtained by combining severalXXZ models with different boundary terms. As in the Temperley–Lieb case, there is a directcorrespondence between representations of the lattice algebra and those in the continuumconformal field theory.