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On an algebraic approach to higher dimensional statistical mechanics

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arxiv hep-th/9208061 v1 pith:VHKIRAMA submitted 1992-08-24 hep-th nlin.SIsolv-int

P. Martin , Herbert Saleur This is my paper · ORCID
classification hep-thnlin.SIsolv-int
keywords algebracasealgebrasappropriateassociatedcompleteeffectsgive
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abstract

We study representations of Temperley-Lieb algebras associated with the transfer matrix formulation of statistical mechanics on arbitrary lattices. We first discuss a new hyperfinite algebra, the Diagram algebra $D_{\underline{n}}(Q)$, which is a quotient of the Temperley-Lieb algebra appropriate for Potts models in the mean field case, and in which the algebras appropriate for all transverse lattice shapes $G$ appear as subalgebras. We give the complete structure of this subalgebra in the case ${\hat A}_n$ (Potts model on a cylinder). The study of the Full Temperley Lieb algebra of graph $G$ reveals a vast number of infinite sets of inequivalent irreducible representations characterized by one or more (complex) parameters associated to topological effects such as links. We give a complete classification in the ${\hat A}_n$ case where the only such effects are loops and twists.

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Cited by 3 Pith papers

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  1. Modular covariant torus partition functions of dense $A_1^{(1)}$ and dilute $A_2^{(2)}$ loop models

    math-ph 2025-01 conditional novelty 7.0 of 10

    For all coprime (p,p'), the dense A_1^(1) and dilute A_2^(2) loop models are conjectured to have identical torus conformal partition functions, supporting a common logarithmic universality class.

  2. Logarithmic correlation functions for critical dense polymers on the cylinder

    cond-mat.stat-mech 2019-07 unverdicted novelty 7.0 of 10

    Explicit finite-n lattice correlators for dense polymers on a cylinder are computed via Temperley-Lieb algebra and shown to match ratios of c=-2 CFT correlators involving boundary fields of dimensions -1/8 and 0, with...

  3. Fuss--Catalan algebras on generalized Dyck paths via non-crossing partitions

    math.CO 2025-07 conditional novelty 6.0 of 10

    Fuss-Catalan algebras and their one- and two-boundary versions are realized on increasing chains of non-crossing partitions, with a new r=2 reflection equation solution.

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