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arXiv:hep-th/9304009 · International Journal of Modern Physics B

INTERTWINERS AND A-D-E LATTICE MODELS

PAUL A. PEARCE, YU-KUI ZHOU

Published Sep 30, 1993• 26 citations
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Abstract

Intertwiners between A-D-E lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwining relations in studying the spectra of the A-D-E models is emphasized. In particular, it is shown that the existence of an intertwiner implies that many eigenvalues of the A-D-E row transfer matrices are exactly in common for a finite system and, consequently, that the corresponding central charges and scaling dimensions can be identified.

DOI
10.1142/s0217979293003449

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Intertwiners between A-D-E lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwining relations in studying the spectra of the A-D-E models is emphasized. In particular, it is shown that the existence of an intertwiner implies that many eigenvalues of the A-D-E row transfer matrices are exactly in common for a finite system and, consequently, that the corresponding central charges and scaling dimensions can be identified.
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Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras [Formula: see text], [Formula: see text], AL, DL and E6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.
We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite-dimensional quantum group modules is clarified. We find a curious phenomenon where the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r+1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent paper.