Multi-Colour Braid-Monoid Algebras

TL;DR

This paper introduces multi-colour braid-monoid algebras, providing explicit matrix representations linked to exactly solvable lattice models, and shows their algebraic structure underpins the models' solvability.

Contribution

It defines multi-colour braid-monoid algebras and connects them to solvable lattice models, extending the algebraic framework for integrable systems.

Findings

Two-colour braid-monoid algebra describes Yang-Baxter algebra of critical dilute A-D-E models

Explicit matrix representations are provided for these algebras

Solvability of models is a consequence of algebraic structure

Abstract

We define multi-colour generalizations of braid-monoid algebras and present explicit matrix representations which are related to two-dimensional exactly solvable lattice models of statistical mechanics. In particular, we show that the two-colour braid-monoid algebra describes the Yang-Baxter algebra of the critical dilute A-D-E models which were recently introduced by Warnaar, Nienhuis, and Seaton as well as by Roche. These and other solvable models related to dense and dilute loop models are discussed in detail and it is shown that the solvability is a direct consequence of the algebraic structure. It is conjectured that the Yang-Baxterization of general multi-colour braid-monoid algebras will lead to the construction of further solvable lattice models.