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2-D cyclic representations of U-q(sl(2)), C-G coefficients, and a free fermion 8-vertex model

J Zhang, GC Yang, Yan Huang

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

We construct explicitly the two-dimensional cyclic representation of quantum algebra U-q(sl(2)) with q = i and z central extension, The representation is irreducible and we write down explicitly the Clebsch-Gordan coefficients and the intertwiner of tensor representations in different orders, This intertwiner turns out to be an 8-vertex R-matrix that satisfies the free-Fermion condition and therefore gives an integrable model.

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Quantum R matrix for G2and a solvable 175-vertex model

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Journal of Physics A Mathematical and General•Apr 21, 1990•35 citations
We construct explicitly the two-dimensional cyclic representation of quantum algebra U-q(sl(2)) with q = i and z central extension, The representation is irreducible and we write down explicitly the Clebsch-Gordan coefficients and the intertwiner of tensor representations in different orders, This intertwiner turns out to be an 8-vertex R-matrix that satisfies the free-Fermion condition and therefore gives an integrable model.
In this paper we continue the study of Q-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). Taking a special limit in this R-matrix we obtained new formulas for the Q-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct Q-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-)integer spins.
We identify the precise relationship between the five-parameter τ (2) -family in the N -state chiral Potts model and XXZ chains with U q ( sl 2 )-cyclic representation. By studying the Yang–Baxter relation of the six-vertex model, we discover a one-parameter family of L -operators in terms of the quantum group U q ( sl 2 ). When N is odd, the N -state τ (2) -model can be regarded as the XXZ chain of cyclic representations with . The symmetry algebra of the τ (2) -model is described by the quantum affine algebra via the canonical representation. In general, for an arbitrary N , we show that the XXZ chain with a U q ( sl 2 )-cyclic representation for q 2 N = 1 is equivalent to two copies of the same N -state τ (2) -model.
The quantum R matrix for Cartan's exceptional simple Lie algebra G 2 in its seven-dimensional (minimal) representation is presented. The R matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra U q (G 2 ). This defines a new solvable seven-state 175-vertex model on the planar square lattice whose Boltzmann weights satisfy the Yang-Baxter equation. For the equivalent face model, the local state probability is obtained in terms of a relative of a level-1 string function related to the affine Lie algebra pair B 3 (1) contains/implies G 2 (1) .