The Kontsevich tetrahedral flow revisited

Abstract

We prove that the Kontsevich tetrahedral flow $\dot{\mathcal{P}}={\mathcal{Q}}_{a:b}(\mathcal{P})$, the right-hand side of which is a linear combination of two differential monomials of degree four in a bi-vector $\mathcal{P}$ on an affine real Poisson manifold $N^n$, does infinitesimally preserve the space of Poisson bi-vectors on $N^n$ if and only if the two monomials in ${\mathcal{Q}}_{a:b}(\mathcal{P})$ are balanced by the ratio $a:b=1:6$. The proof is explicit; it is written in the language of Kontsevich graphs.

arXiv commentTalk given by AVK on 26 July 2016 at the Algebra, Geometry and Physics seminar in the Max Planck Institute for Mathematics (Bonn, Germany). 29 pages, 16 figures

References in corpus (6)

• Deformation quantization of Poisson manifolds, I
• Exotic automorphisms of the Schouten algebra of polyvector fields
• Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?
• The calculus of multivectors on noncommutative jet spaces
• The geometry of variations in Batalin-Vilkovisky formalism
• The deformation quantization mapping of Poisson- to associative structures in field theory

Cited by in corpus (16)

• Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?
• The calculus of multivectors on noncommutative jet spaces
• Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex
• The heptagon-wheel cocycle in the Kontsevich graph complex
• The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets
• Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus
• The Kontsevich graph orientation morphism revisited
• The deformation quantization mapping of Poisson- to associative structures in field theory
• The Kontsevich tetrahedral flow in 2D: a toy model
• The orientation morphism: from graph cocycles to deformations of Poisson structures
• Open problems in the Kontsevich graph construction of Poisson bracket symmetries
• The expansion $\star$ mod $\bar{o}({\hslash }^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization
• Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants
• Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D
• The tower of Kontsevich deformations for Nambu-Poisson structures on ${\mathbb{R}}^d$: dimension-specific micro-graph calculus
• Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms