On the Riemannian Geometry of tangent Poisson-Lie group

Abstract: Let $(G,\Pi_{G},\tilde{g})$ be a Poisson-Lie group equipped with a left invariant pseudo-Riemannian metric $\tilde{g}$ and let $(TG,\Pi_{TG},\tilde{g}^{c})$ be the Sanchez de Alvarez tangent Poisson-Lie group of $G$ equipped with the left invariant pseudo-Riemannian metric $\tilde{g}^{c}$, complete lift of $\tilde{g}$. In this paper, we express respectively the Levi-Civita connection, curvature and metacurvature of $(TG,\Pi_{TG},\tilde{g}^{c})$ in terms of the Levi-Civita connection, curvature and metacurvature of the basis Poisson-Lie group $(G,\Pi_{G},\tilde{g})$ and we prove that the space of differential forms $\Omega^{*}(G)$ on $G$ is a differential graded Poisson algebra if, and only if, $\Omega^{*}(TG)$ is a differential graded Poisson algebra . Moreover, we prove that the triplet $(G,\Pi_{G},\tilde{g})$ is a pseudo-Riemannian Poisson-Lie group if, and only if, $(TG,\Pi_{TG},\tilde{g}^{c})$ is also a pseudo-Riemannian Poisson-Lie group and we give an example of 6-dimensional pseudo-Riemannian Sanchez de Avarez tangent Poisson-Lie group.