Arthemy Kiselev (RuG)
The Kontsevich tetrahedral flows on the spaces of Poisson bi-vectors
In the preprint IHES/M/16/12 [joint with A.Bouisaghouane] we examine two claims from the paper "Formality Conjecture" by M.Kontsevich (1996): specifically, that 1) a certain tetrahedral graph flow preserves the class of (real-analytic) Poisson structures, and that 2) another tetrahedral graph flow vanishes at every such Poisson structure.
By using twelve Poisson structures with high-degree polynomial coefficients as explicit counterexamples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at the Poisson bi-vectors.
The counterexamples at hand themselves suggest a correction to the formula for the "exotic" flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance (1:6) for which the flow does preserve the space of Poisson bi-vectors. We explore the nontrivial mechanism of that preservation by virtue of the Jacobi identity or its implications; here we use the standard generators of nonlinear Poisson brackets on finite-dimensional real vector spaces.