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.