A Note on Poisson Symmetric Spaces

We introduce the notion of a Poisson symmetric space and the associated infinitesimal object, a symmetric Lie bialgebra. They generalize corresponding notions for Lie groups due to V. G. Drinfel’d. We use them to give some geometric insight to certain Poisson brackets that have appeared before in the literature. 1 Motivation Let us recall briefly the best-known examples of Poisson manifolds. The most basic example is that of a symplectic manifold (M,ω), where the Poisson bracket is defined by {f1, f2} =< ω,Xf1 ∧Xf2 >, f1, f2 ∈ C (M) (1) and where for each f ∈ C(M) we have introduced the vector field Xf satisfying Xf y ω = df [1]. Every Poisson manifold foliates into symplectic manifolds [16]. The second well known class of Poisson brackets are the linear Poisson brackets on a vector space, which arise as follows. Let g be a Lie algebra and for each ξ ∈ g view the differential dξf of f ∈ C (g) as an element of g. Then we have the linear Poisson bracket {f1, f2}(ξ) =< ξ, [dξf1, dξf2] >, ξ ∈ g , f1, f2 ∈ C (g). (2) Moreover, every linear Poisson bracket has this form [16]. The next class one might consider is the class of quadratic Poisson brackets on a Lie group G. They can be described as follows. Let g = Lie(G) be the Lie algebra of G and assume that g has an ad-invariant inner product ( , ). Then to each skew-symmetric solution A: g → g of the classical Yang-Baxter equation