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

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

2003-07-08

arxiv.org/abs/math-ph/0307017v3

J.Phys.A36:10551-10572,2003

Physics

Mathematical Physics

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

Scientific paper

10.1088/0305-4470/36/42/010

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.

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