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 $TP$.
Mohamed Boumaiza. Nadhem Zaalani. "Poisson-Lie Structure on the Tangent Bundle of a Poisson-Lie Group and Poisson Action Lifting." J. Geom. Symmetry Phys. 4 1 - 18, 2005. https://doi.org/10.7546/jgsp-4-2005-1-18