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

Abstract

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J_0 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 ${\cal A}^+_q(1)$. 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, su_q(2) and ${\cal A}^+_q(1)$, 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 su_q(2) is carried over to ${\cal A}^+_q(1)$, 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.