On the algebraic approach to cubic lattice Potts models

TL;DR

This paper explores the algebraic structures related to cubic lattice Potts models, revealing their complexity and proposing methods to derive simpler quotient algebras to better understand three-dimensional statistical physics.

Contribution

It analyzes the structure of generalized Temperley-Lieb algebras for cubic lattice graphs and suggests approaches to extract meaningful quotient algebras for 3D Potts models.

Findings

Diagram algebras are too large for direct 3D analogy to planar cases.

Proposes measures to obtain quotient algebras.

Highlights the complexity of algebraic structures in 3D models.

Abstract

We consider Diagram algebras, $\Dg(G)$ (generalized Temperley-Lieb algebras) defined for a large class of graphs $G$, including those of relevance for cubic lattice Potts models, and study their structure for generic $Q$. We find that these algebras are too large to play the precisely analogous role in three dimensions to that played by the Temperley-Lieb algebras for generic $Q$ in the planar case. We outline measures to extract the quotient algebra that would illuminate the physics of three dimensional Potts models.