On an algebraic approach to higher dimensional statistical mechanics

TL;DR

This paper explores algebraic structures related to statistical mechanics models, introducing a new hyperfinite algebra and classifying representations for specific lattice shapes, advancing understanding of algebraic methods in higher-dimensional systems.

Contribution

It introduces the Diagram algebra $D_{\underline{n}}(Q)$ for Potts models and provides a complete classification of its irreducible representations for the ${\hat A}_n$ case.

Findings

Introduction of the Diagram algebra $D_{\underline{n}}(Q)$ for Potts models

Complete structure of the algebra for ${\hat A}_n$ lattices

Classification of irreducible representations with topological parameters

Abstract

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 $D_{\underline{n}}(Q)$, 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 ${\hat A}_n$ (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 ${\hat A}_n$ case where the only such effects are loops and twists.