Temperley-Lieb algebras and the long distance properties of statistical mechanical models

Abstract

The author writes down the regular representation of the Temperley-Lieb algebra T k (q) in the basis of reduced words on the k Temperley-Lieb generators. When k=2n-1, representations of these generators may be used to construct the transfer matrices for statistical mechanical models on an n-site wide lattice. He shows that the generically irreducible representation of T 2n-1 (q) responsible for the unique free energy in such models may be restricted to the regular representation of T n-1 (q). He gives equivalent forms for the regular representation, derived from lattice models, which manifest its indecomposable structure when k and q are such that T k (q) is semisimple. He hence generalises to obtain the structure of T k (q) when not semisimple. A transfer matrix built with generators in the regular representation gives the long distance properties of all possible such models. He shows how to find the semisimple quotient algebra which gives these long distance properties when T k (q) itself is not semisimple. He hence classifies the long distance properties of these statistical mechanical models.