Universal Cocycles and the Graph Complex Action on Homogeneous Poisson Brackets by Diffeomorphisms

R. Buring*, A. V. Kiselev

*Corresponding author for this work

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Abstract

Abstract: 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(3) w.r.t. the Lie derivative along some vector field (Formula presented.)., but not quadratic (the coefficients of P are not degree-two homogeneous polynomials), and whenever its velocity bi-vector (Formula presented.)., also homogeneous w.r.t. (Formula presented.). whenever (Formula presented.). is obtained using the orientation morphism (Formula presented.). from a graph cocycle (Formula presented.). vertices and 2n - 2 edges, then the (Formula presented.). is a Poisson cocycle. Its construction is uniform for all Poisson bi-vectors (Formula presented.). satisfying the above assumptions, on all finite-dimensional affine manifolds M. Still, if the bi-vector (Formula presented.). is exact in the respective Poisson cohomology, so there exists a vector field (Formula presented.). such that (Formula presented.)., then the universal cocycle (Formula presented.). does not belong to the coset of (Formula presented.). We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the R-matrices for the Lie algebra (Formula presented.).

Original languageEnglish
Pages (from-to)707-713
Number of pages7
JournalPhysics of Particles and Nuclei Letters
Volume17
Issue number5
DOIs
Publication statusPublished - 1-Sept-2020

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