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Journal of Physics A Mathematical and General

On some quantum R matrices associated with representations of Uq(sl(2,C)) when q is a root of unity

M Couture

Published Feb 7, 1991• 17 citations
Publisher

Abstract

The author constructs quantum R matrices associated with certain non-cyclic irreducible representations of U q (sl(2,c)) at roots of unity which are parametrized by one continuous parameter. The author finds solutions for the two- and three-dimensional representations via Baxterization. The two-dimensional case corresponds to the free fermion model of Fan and Wu (1970).

DOI
10.1088/0305-4470/24/3/001

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The author constructs quantum R matrices associated with certain non-cyclic irreducible representations of U q (sl(2,c)) at roots of unity which are parametrized by one continuous parameter. The author finds solutions for the two- and three-dimensional representations via Baxterization. The two-dimensional case corresponds to the free fermion model of Fan and Wu (1970).
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Abstract We present a general formula for constructing R-matrices with non-additive spectral parameters associated with a type-I quantum superalgebra. The spectral parameters originate from two one-parameter families of inequivalent finite-dimensional irreducible representations of the quantum superalgebra upon which the R-matrix acts. Applying to the quantum superalgebra Uq(gl(2|1)), we obtain the explicit expression for the Uq(gl(2|1))− invariant R-matrix which is of non-difference form in spectral parameters. Using this R­ matrix we derive a new two-parameter integrable model of strongly correlated electrons with pure imaginary pair hopping terms.
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.
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