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

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.

24 pages, LaTeX using IOP styles, many figures included; typo in formula (7) corrected

References in corpus (7)

• Generalised twisted partition functions
• The spin-1/2 XXZ Heisenberg chain, the quantum algebra U_q[sl(2)], and duality transformations for minimal models
• Hecke algebraic approach to the reflection equation for spin chains
• Spectrum of a duality-twisted Ising quantum chain
• Duality and conformal twisted boundaries in the Ising model
• Exactly soluble isotropic spin-1/2 ladder models
• Generalized blob algebras and alcove geometry

Cited by in corpus (8)

• Non compact conformal field theory and the a_2^{(2)} (Izergin-Korepin) model in regime III
• The principal indecomposable modules of the dilute Temperley-Lieb algebra
• $U_q({\mathfrak{sl}}_n)$ web models and ${\mathbb{Z}}_n$ spin interfaces
• Fusion rules for the Temperley-Lieb algebra and its dilute generalisation
• Power-law entanglement and Hilbert space fragmentation in non-reciprocal quantum circuits
• A coupled Temperley-Lieb algebra for the superintegrable chiral Potts chain
• Representation theory for dilute lattice models
• The planar parafermion algebra: The ${\mathbb{Z}}_N$ clock model and the coupled Temperley-Lieb algebra