We generalize the notion of weight for Gelfand-Fuks cohomology theory of symplectic vector spaces to the homogeneous Poisson vector spaces, and try some combinatorial approach to Poisson cohomology groups.
We introduce Poisson double algebroids, and the equivalent concept of double Lie bialgebroid, which arise as second-order infinitesimal counterparts of Poisson double groupoids. We develop their underlying Lie theory, showing how these objects are re
lated by differentiation and integration. We use these results to revisit Lie 2-bialgebras by means of Poisson double structures.
For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $omega_xi^s$, where $xi in mathfrak{t}^*_+$ is in the positive Weyl chamber and $s in mathbb{R}$. The symplectic form $omega_xi^0$ is identified wit
h the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $xi$. The cohomology class of $omega_xi^s$ is independent of $s$ for a fixed value of $xi$. In this paper, we show that as $sto -infty$, the symplectic volume of $omega_xi^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K/T cong G/B$. This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on $Lie(K)^*$ [4, Conjecture 1.1].
The purpose of this paper is to study covariant Poisson structures on the complex Grassmannian obtained as quotients by coisotropic subgroups of the standard Poisson--Lie SU(n). Properties of Poisson quotients allow to describe Poisson embeddings generalizing those obtained in math.SG/9802082.
We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels
of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].
We apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $ngeq 2$, and prove the simplicity of their automorphism groups.