The Bubble Algebra: Structure of a Two-Colour Temperley-Lieb Algebra

TL;DR

This paper introduces a new class of diagram algebras called Bubble Algebras, generalizing the Temperley-Lieb algebra to model two-dimensional statistical mechanics systems with multiple colours, and explores their representation theory and applications.

Contribution

It defines the Bubble Algebra, a multiparameter generalization of the Temperley-Lieb algebra, and analyzes its structure and representation theory, extending the framework for statistical mechanics models.

Findings

Established the structure of Bubble Algebras.

Determined the generic representation theory.

Showed applications to solving Yang-Baxter equations.

Abstract

We define new diagram algebras providing a sequence of multiparameter generalisations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional Statistical Mechanics. These algebras give a rigorous foundation to the various "multi-colour algebras" of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in Statistical Mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations.