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Journal of the Physical Society of Japan

Yang-Baxter Relations in Terms ofn-jSymbols ofsuq(2) Algebra

Masao Nomura

Published Aug 15, 1989• 18 citations
Publisher

Abstract

A trigonometric function parametrization of Yang-Baxter (Y-B) relations is described in terms of n - j symbols of su q (2), a deformation of su (2). The formalism starts with deduction of the Y-B relation for the IRF (interaction round a face) model of su q (2) from a general operator relation for the vertex model. The relation obtained for the IRF model is transformed into the Y-B relation for the vertex model using the property that the 3- j symbol is an asymptotic limit of the 6- j symbol. A class of very general solutions to Y-B equations is also presented. This solution of the IRF model with infinite rapidity parameters is linked to a symmetry of the 12- j symbol of the second kind.

DOI
10.1143/jpsj.58.2694

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A trigonometric function parametrization of Yang-Baxter (Y-B) relations is described in terms of n - j symbols of su q (2), a deformation of su (2). The formalism starts with deduction of the Y-B relation for the IRF (interaction round a face) model of su q (2) from a general operator relation for the vertex model. The relation obtained for the IRF model is transformed into the Y-B relation for the vertex model using the property that the 3- j symbol is an asymptotic limit of the 6- j symbol. A class of very general solutions to Y-B equations is also presented. This solution of the IRF model with infinite rapidity parameters is linked to a symmetry of the 12- j symbol of the second kind.
Relations are exploited among Clebsch–Gordan (CG) and Racah coefficients in the algebra suq(2), known as a deformation of su(2). These are used to show that the Yang–Baxter (YB) relation for the IRF (interaction round a face) model results from one of the symmetry relations for the 9-j symbol specific to suq(2), and that in an asymptotic limit this YB relation becomes the YB relation for the two-dimensional vertex model. The Racah coefficient, which has a particularly simple dependence on q, is efficiently used such that an asymptotic limit of the Racah coefficient is the CG coefficient and another limit gives the factorized S matrix of the vertex model.
The Yang-Baxter algebras (YBA) are introduced in a general framework stressing their power to exactly solve the lattice models associated to them. The algebraic Bethe Ansatz is developed as an eigenvector construction based on the YBA. The six-vertex model solution is given explicitly. The generalization of YB algebras to face language is considered. The algebraic BA for the SOS model of Andrews, Baxter and Forrester is described using these face YB algebras. It is explained how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. This approach permit to define and solve rigorously massive QFT as an appropiate continuum limit of gapless vertex models. The deep links between the YBA and Lie algebras are analyzed including the quantum groups that underly the trigonometric/hyperbolic YBA. Braid and quantum groups are derived from trigonometric/hyperbolic YBA in the limit of infinite spectral parameter. To conclude, some recent developments in the domain of integrable theories are summarized.
An analog of the Yang-Baxter Algebra (YBA) is defined in face language. The operators tαα′,ββ′(θ) introduced here enjoy all essential properties of the vertex language YBA. Using this face YBA an algebraic Bethe Ansatz (BA) is constructed for SOS models (unrestricted IRF models). The face dual of the six-vertex model and the critical ABF model are worked out explicitly. Eigenvectors and eigenvalues of the transfer matrix are found and the corresponding BA equations derived and compared with the six vertex BAE.
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.
Abstract Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group A , we construct an operator solution of the Yang–Baxter equation generalizing the solution of the Faddeev–Volkov model. Based on a specific choice of a subgroup and by using the Weil transformation, we also give a new non-operator interpretation of the Yang–Baxter relation. That allows us to construct a lattice QFT-model of IRF-type with gauge invariance under independent B -translations of local ‘spin’ variables.