Universal R-matrix for Non-standard Quantum Sl(2, Ir)

A universal R-matrix for the non-standard (Jordanian) quantum deformation of sl(2, IR) is presented. A family of solutions of the quantum Yang– Baxter equation is obtained from some finite dimensional representations of this Lie bialgebra quantization of sl(2, IR). The quantum Yang–Baxter equation (YBE) R12R13R23 = R23R13R12. (1) was discovered to play a relevant role as the integrability condition for (1+1) quantum field theories [1] and also in connection with two dimensional models in lattice statistical physics [2], conformal field theory [3] and knot theory [4]. Nowadays it is well known that the investigation on the algebraic properties of this equation and the obtention of new solutions are closely related to the study of quantum groups and algebras [5]. In fact, let A be a Hopf algebra and let R be an invertible element in A ⊗ A such that σ ◦∆(X) = R∆(X)R, ∀X ∈ A (2) where σ is the flip operator σ(x ⊗ y) = (y ⊗ x). If we write R = ∑ i ai ⊗ bi, R12 ≡ ∑ i ai ⊗ bi ⊗ 1, R13 ≡ ∑ i ai ⊗ 1⊗ bi, R23 ≡ ∑ i 1⊗ ai ⊗ bi and the relations (∆⊗ id)R = R13R23, (id⊗∆)R = R13R12, (3) are fulfilled, (A,R) is called a quasitriangular Hopf algebra [6]. In that case, R is easily proven to be a solution of (1). Hereafter, if A is a Hopf algebra and R fulfills both (1) and (2), we shall say that R is a (quantum) universal R-matrix for A. Obviously, different representations for the algebra A will give rise to different explicit solutions of the quantum YBE. In particular, let A be a quantum deformation Uz(g) of the universal enveloping algebra of a Lie algebra g. Then, the quasicocommutativity property (2) is translated, in terms of the Hopf algebra dual to A, into the known FRT relations defining the quantum group Funz(G) [5]. It is also known that each Uz(g) defines a unique Lie bialgebra structure on g that can be used to characterize the quantum deformation. For all semisimple Lie algebras, all these Lie bialgebra structures are coboundaries generated by classical r-matrices. In sl(2, IR), two outstanding Lie bialgebra structures can be mentioned: the standard one, generated by the classical r-matrix r = λ J+ ∧ J− and the non-standard (triangular) Lie bialgebra given by the element r = χJ3 ∧ J+ (λ, χ ∈ IR). The quantization of the former is the well known Drinfel’d–Jimbo deformation, whose quantum universal R-matrix was given in [7]. For the latter, the corresponding non-standard quantum algebra is the so called Jordanian deformation of sl(2, IR) [8] (introduced for the first time in [9] in a quantum group setting; see [10] and references therein and [11], where the nonstandard deformation of so(2, 2) was constructed). To our knowledge, no quantum universal R-matrix is known for this case (the R given in [8] is neither a solution of (1) nor verifies (2), as it has already been pointed out in [12, 13]). The aim of this letter is to provide it. Let us consider the sl(2, IR) Lie algebra with the following commutation relations [J3, J+] = 2J+, [J3, J−] = −2J−, [J+, J−] = J3. (4)