A nonlinear deformed su(2) algebra with a two-colour quasitriangular Hopf structure

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some of them with a Hopf algebraic structure is addressed by studying in detail a specific example, referred to as Aq (1). This algebra is shown to possess two series of (N + 1)-dimensional unitary irreducible representations, where N = 0, 1, 2, . . .. To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed by proceeding in two steps. In the first one, a variant and extension of the deforming functional technique is introduced: variant because a map between two deformed algebras, suq(2) and Aq (1), is considered instead of a map between a Lie algebra and a deformed one, and extension because use is made of a two-valued functional, whose inverse is singular. As a result, the Hopf structure of suq(2) is carried over to Aq (1), thereby endowing the latter with a double Hopf structure. In the second step, the definition of the coproduct, counit, antipode, and R-matrix is extended so that the double Hopf algebra is enlarged into a new algebraic structure. The latter is referred to as a two-colour quasitriangular Hopf algebra because the corresponding R-matrix is a solution of the coloured Yang-Baxter equation, where the ‘colour’ parameters take two discrete values associated with the two series of finite-dimensional representations. Running title: Two-colour quasitriangular Hopf algebra PACS: 02.10.Tq, 03.65.Fd, 11.30.Na