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International Journal of Modern Physics B

TEMPERLEY-LIEB OPERATORS AND CRITICAL A-D-E MODELS

Paul A. Pearce

Published Apr 1, 1990• 13 citations
Publisher

Abstract

A review is given of recent work connecting exactly solvable lattice models with Temperley-Lieb and Hecke algebras, Dynkin diagrams of simply-laced Lie algebras, conformal and modular invariance and Virasoro characters.
DOI
10.1142/s0217979290000371

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An equivalence between generalized restricted solid-on-solid models, associated with sets of graphs, and multi-colour loop models is established. As an application the authors consider solvable loop models and, in this way, obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A L ,A L ) and related to the C 2 (1) vertex model, they give an off-critical extension, which breaks the Z 2 symmetry of the Dynkin diagrams.
We investigate critical restricted solid-on-solid models associated to Dynkin diagrams of type AAA, DDD and EEE, with fixed, periodic and twisted periodic boundary conditions. These models are endowed with an action of the diagrams of the Temperley-Lieb category. For each model, we obtain the decomposition of the state space as a direct sum of irreducible modules over the Temperley-Lieb algebra TLN(β)\mathsf{TL}_N(β)TLN​(β) or its periodic incarnation EPTLN(β)\mathsf{\mathcal EPTL}_N(β)EPTLN​(β). This allows us to recover the known conformal partition functions for these models in the continuum scaling limit. For each irreducible factor arising in the decompositions, we define an associated local operator on the lattice, which behaves like a connectivity operator. Using knowledge from the Temperley-Lieb representation theory at roots of unity, we show that these operators satisfy certain linear difference relations, which are lattice counterparts of the singular-vector relations in conformal field theory.