UNIVERSAL COCYCLES AND THE GRAPH COMPLEX ACTION ON HOMOGENEOUS POISSON BRACKETS BY DIFFEOMORPHISMS

摘要

The graph complex acts on the spaces of Poisson bi-vectors P by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e., P = Lv(P) w.r.t. the Lie derivative along some vector field V, but not quadratic (the coefficients of P are not degree-two homogeneous polynomials), and whenever its velocity bi-vector P＝Q(P), also homogeneous w.r.t. V by Lv(Q) = nQ whenever Q(P) = Or(γ)(P~(⊗~n)) is obtained using the orientation morphism Or from a graph cocycle γ on n vertices and 2n-2 edges, then the 1-vector Χ= Or(γ)(V⊗P~(⊗~(n-1))) is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors P satisfying the above assumptions, on all finite-dimensional affine manifolds M. Still, if the bi-vector Q (≠) 0 is exact in the respective Poisson cohomology, so there exists a vector field У such that Q(P) = [У,P], then the universal cocycle X does not belong to the coset of У mod ker [P,·]. We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the R-matrices for the Lie algebra gl(2).