Poisson-Lie Structure on the Tangent Bundle of a Poisson-Lie Group and Poisson Action Lifting

Abstract

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 $TP$.