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Physical Review Letters

Structure of Temperley-Lieb algebras and its application to 2D statistical models

Dan Lévy

Published Jan 29, 1990• 28 citations
Publisher

Abstract

A simple derivation of the ideals in the Temperley Lieb algebras is presented together with a convenient basis. Using this basis it is shown how to derive the eigenvalue equations for operators defined on the algebra and a relation to the Bethe-Ansatz equations is noticed. The irreducible content of the spin-(1/2 XXZ model is also discussed.

DOI
10.1103/physrevlett.64.499

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A simple derivation of the ideals in the Temperley Lieb algebras is presented together with a convenient basis. Using this basis it is shown how to derive the eigenvalue equations for operators defined on the algebra and a relation to the Bethe-Ansatz equations is noticed. The irreducible content of the spin-(1/2 XXZ model is also discussed.
Lattice algebras are defined and used to study the symmetries of 2D lattice models. New and interesting examples of such algebras are provided by the affine Hecke algebra, owing to the possibility of constructing braid generators out of its generators. I propose an Ansatz for the braid generators and derive some solutions. A particular finite-dimensional quotient is shown to be a natural generalization of the Temperley-Lieb-Jones algebra. It is used to give a unified picture of known and unknown symmetries of the spin-½ xxz model with boundary terms. The Ising model in an external magnetic field is also a representation of this quotient.
In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL(q) (2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyze the finite-dimensional representations of the quantum SL(q) (2) with q(p) = +/- 1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type I representation), their properties, and the complete set of the states related to a definite energy where the new states are obtained by an appropriate limiting process.
In this paper, an XXZ model of the Heisenberg spin chain with the symmetry of the quantum SL_q(2) enveloping algebra is discussed. In terms of the structure of the Temperley-Lieb algebra, in this model we analyse the finite-dimensional representations of the quantum SL_q(2) with q~p=±1 in some detail, including the condition of appearance of the reducible but indecomposable representations (type Ⅰ representation), their properties, and the completeset of the states related to a definite energy where the new states are obtained by an appropriare limit process.