Reduced Riemannian Poisson manifolds and Riemannian Poisson-Lie groups

Let (M, Lambda(M), <,>(M)) be a Poisson manifold equipped with a Riemannian metric <,>(M) compatible with the Poisson structure Lambda(M) 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 Lambda(M/H) and a unique Riemannian metric <,>(M/H) on the reduced manifold M/H, generated by Lambda(M) and <,>(M) respectively. In this paper, we give necessary and sufficient conditions so that the compatibility conditions between the Poisson tensor Lambda(M) and the metric <,>(M) on M remain verified on the reduced Poisson manifold (M/H, Lambda(M/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. (C) 2019 Elsevier B.V. All rights reserved.