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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
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