## Poisson-Lie structure on the tangent bundle of a Poisson-Lie group, and Poisson action lifting.(English)Zbl 1132.53315

Summary: 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 by a Poisson action of $$TG$$ on the tangent bundle $$TP$$.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 17B63 Poisson algebras