The Kontsevich tetrahedral flow revisited

TL;DR

This paper revisits the Kontsevich tetrahedral flow, proving it preserves Poisson structures only under a specific ratio of its defining monomials, using explicit graph-based methods.

Contribution

It provides an explicit proof that the Kontsevich tetrahedral flow preserves Poisson structures only at a specific ratio of its monomials.

Findings

Preservation occurs only at ratio a:b=1:6.

The proof is explicit and graph-based.

The flow infinitesimally preserves Poisson bi-vectors under this ratio.

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.