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.