Geometry of Tangent Poisson-Lie Groups

Let G be a Poisson-Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. 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 show that G is a pseudo-Riemannian Poisson-Lie group if, and only if, the Sanchez de Alvarez tangent Poisson-Lie group TG is also a pseudo-Riemannian Poisson-Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson-Lie groups are given.