On Braided Poisson and Quantum Inhomogeneous Groups

It is well known [1, 2] that the Lorentz part of any quantum (or Poisson) Poincaré group is triangular. This is in fact a general feature, which excludes the standard q-deformation from the context of inhomogeneous quantum groups [3]. In order to make the standard q-deformation compatible with inhomogeneous groups one has to consider some generalization of the notion of quantum (Poisson) group, such as, for example, a braided quantum (Poisson) group. The notion of a braided Hopf algebra is due to S. Majid [4]. It is a natural generalization of the notion of a Hopf algebra when we replace the usual symmetric monoidal category of vector spaces by a braided one (the incorporation of *-structures is more controversial — we follow here the approach of [5]). A characteristic feature of this generalization is that the comultiplication is a morphism of algebras when the product algebra is considered with a crossed tensor product structure rather than the ordinary one. On the Poisson level, it means that instead of ordinary Poisson groups (G, π) (where π is such a Poisson structure on G that the group multiplication is a Poisson map from the usual product Poisson structure π⊕π onG×G to π onG), we consider triples (G, π, π1), where π is a Poisson structure on G and π1 is a bi-vector field on G×G of the cross-type (i.e. having zero both projections on G) such that