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.

Talk 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