On an algebraic approach to higher dimensional statistical mechanics

Details On an algebraic approach to higher dimensional statistical mechanics On an algebraic approach to higher dimensional statistical mechanics

1992-08-24

arxiv.org/abs/hep-th/9208061v1

Commun.Math.Phys. 158 (1993) 155-190

Physics

High Energy Physics

High Energy Physics - Theory

41 pages, 13 figures (two not included)

Scientific paper

10.1007/BF02097236

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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