Nonlinear deformed su(2) algebras involving two deforming functions

The most common nonlinear deformations of the su(2) Lie algebra, introduced by Polychronakos and Roek, involve a single arbitrary function ofJ _o and include the quantum algebra su _q (2) as a special case. In the present contribution, less common nonlinear deformations of su(2), introduced by Delbecq and Quesne and involving two deforming functions ofJ _o, are reviewed. Such algebras include Witten's quadratic deformation of su(2) as a special case. Contrary to the former deformations, for which the spectrum ofJo is linear as for su(2), the latter give rise to exponential spectra, a property that has aroused much interest in connection with some physical problems. Another interesting algebra of this type, denoted byA _q ⁺ (1), has two series of (N+1)-dimensional unitary irreducible representations, whereN=0, 1, 2, ... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed. The resulting algebraic structure, referred to as a two-colour quasitriangular Hopf algebra, is described.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.Directeur de recherches FNRS.