Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex

Abstract

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}}=\mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ on all affine manifolds $M^r$; every such deformation is encoded by oriented graphs on $n+2$ vertices and $2n$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs $\gamma$ on $n$ vertices and $2n-2$ edges. The bi-vector flow $\dot{\mathcal{P}}=\text{Or}(\gamma )(\mathcal{P})$ preserves the space of Poisson structures if $\gamma$ is a cocycle with respect to the vertex-expanding differential in the graph complex. A class of such cocycles ${\mathbf{\gamma }}_{2\ell +1}$ is known to exist: marked by $\ell \in \mathbb{N}$, each of them contains a $(2\ell +1)$-gon wheel with a nonzero coefficient. At $\ell =1$ the tetrahedron ${\mathbf{\gamma }}_3$ itself is a cocycle; at $\ell =2$ the Kontsevich--Willwacher pentagon-wheel cocycle ${\mathbf{\gamma }}_5$ consists of two graphs. We reconstruct the symmetry ${\mathcal{Q}}_5(\mathcal{P})=\text{Or}({\mathbf{\gamma }}_5)(\mathcal{P})$ and verify that ${\mathcal{Q}}_5$ is a Poisson cocycle indeed: $[[\mathcal{P},{\mathcal{Q}}_5(\mathcal{P})]]\doteq 0$ via $[[\mathcal{P},\mathcal{P}]]=0$.

Int. workshop "Supersymmetries and quantum symmetries -- SQS'17" (July 31 -- August 5, 2017 at JINR Dubna, Russia), 4+v pages, 2 figures, 1 table

References in corpus (4)

• The Kontsevich tetrahedral flow revisited
• The heptagon-wheel cocycle in the Kontsevich graph complex
• Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
• On the Kontsevich $\star$-product associativity mechanism

Cited by in corpus (7)

• Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• The Kontsevich graph orientation morphism revisited
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants
• Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms
• The defining properties of the Kontsevich unoriented graph complex