POISSON-LIE STRUCTURE ON THE TANGENT BUNDLE OF A POISSON-LIE GROUP, AND POISSON ACTION LIFTING

We show in this paper that the tangent bundle TG, of a Poisson-Lie group G has a Poisson-Lie group structure given by the canonical lifting of that of G. We determine the dual group of TG, its Lie bialgebra and its double Lie algebra. We also show that any Poisson action of G on a Poisson manifold P is lifted on a Poisson action of TG on the tanget bundle TP.