ON AN ALGEBRAIC APPROACH TO HIGHER-DIMENSIONAL STATISTICAL-MECHANICS

We study representations of Temperley-Lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary lattices. We first discuss a new hyperfinite algebra, the Diagram algebra D(nunderbar)(Q), which is a quotient of the Temperley-Lieb algebra appropriate for Potts models in the mean field case, and in which the algebras appropriate for all transverse lattice shapes G appear as subalgebras. We give the complete structure of this subalgebra in the case A(n) (Potts model on a cylinder). The study of the Full Temperley Lieb algebra of graph G reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more (complex) parameters associated to topological effects such as links. We give a complete classification in the A(n) case where the only such effects are loops and twists.