TY - JOUR

T1 - Universal Cocycles and the Graph Complex Action on Homogeneous Poisson Brackets by Diffeomorphisms

AU - Buring, R.

AU - Kiselev, A. V.

N1 - Funding Information:
A part of this research was done while R.B. was visiting at the IHÉS, supported by MI JGU project 5020.
Funding Information:
Supported by BI RUG project 135110 (Groningen) and the IHÉS (in part, by the Nokia Fund).
Funding Information:
A.V.K. thanks the organizers of international workshop SQS?19 (August 26?31, 2019 in Yerevan, Armenia) for a warm atmosphere during the event. A part of this research was done at the IH?S (Bures-sur-Yvette, France); the authors are grateful to the RATP and STIF for setting up stimulating working conditions. The authors thank G.H.E. Duchamp and M.?Kontsevich for helpful discussions.
Publisher Copyright:
© 2020, Pleiades Publishing, Ltd.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - 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.).

AB - 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.).

UR - http://www.scopus.com/inward/record.url?scp=85092407365&partnerID=8YFLogxK

U2 - 10.1134/S1547477120050088

DO - 10.1134/S1547477120050088

M3 - Article

AN - SCOPUS:85092407365

SN - 1547-4771

VL - 17

SP - 707

EP - 713

JO - Physics of Particles and Nuclei Letters

JF - Physics of Particles and Nuclei Letters

IS - 5

ER -