Reduced Riemannian Poisson manifolds and Riemannian Poisson-Lie groups

Abstract

Let (M,ΛM,〈,〉M) be a Poisson manifold equipped with a Riemannian metric 〈,〉M compatible with the Poisson structure ΛM and H a Lie group that acts on M properly, freely, by isometries and preserving the Poisson structure on M. There exists a unique Poisson structure ΛM/H and a unique Riemannian metric 〈,〉M/H on the reduced manifold M/H, generated by ΛM and 〈,〉M respectively. In this paper, we give necessary and sufficient conditions so that the compatibility conditions between the Poisson tensor ΛM and the metric 〈,〉M on M remain verified on the reduced Poisson manifold (M/H,ΛM/H,〈,〉M/H). When M=G is a Poisson-Lie group and H is a normal and closed coisotropic subgroup of G, we give interesting algebraic consequences associated with the compatibility between the Poisson tensor and the metric on G/H.