On the bicrossproduct structures for the Uλ(isoω2...ωN (N)) family of algebras

Deformed algebras (usually called ‘quantum groups’) have received great attention since the original works of Drinfel’d, Jimbo and Faddeev, Reshetikhin and Takhtajan [1, 2, 3, 4] which gave a (unique) deformation procedure for simple Lie algebras. However, the deformation of non-simple Lie algebras has been characterized by the lack of a definite prescription and this explains why inhomogeneous algebras do not have a unique deformation. A possible approach to deforming non-simple algebras is by extending the contraction of Lie algebras to the framework of deformed Hopf algebras, an idea originally introduced by Celeghini et al. [5, 6]. As is well known, the standard İnönü-Wigner [7] contraction of (simple) Lie algebras leads to non-simple algebras which have a semidirect structure, where the ideal is the abelianized part of the original algebra. By introducing higher powers in the contraction parameter or, equivalently, by performing two (or more) successive contractions it is also possible to arrive to algebras with a central extension structure. This simple mechanism becomes difficult to implement for deformed algebras for which it is usually necessary to redefine the deformation parameter in terms of the contraction one to have a well-defined contraction limit [5, 6]. This is the case, for instance, of the κ-Poincaré algebra [8, 9] in which the deformation parameter κ appears as a redefinition of the original (adimensional) parameter q of soq(3, 2) in terms of the De Sitter radius R. A way to skip some of the problems of the standard contraction procedure for deformed algebras is to use the method of ‘graded’ contractions. This mechanism was put forward by Moody, Montigny and Patera [10, 11] for Lie algebras and has been applied recently to describe a large set of deformed Hopf algebras [12, 13]. The scheme provides