Tetrahedral symmetry of the Jacobi identity for Poisson structures

Abstract:
On the space of smooth functions on an affine manifold, a Poisson structure $P$ is a Lie bracket satisfying a Leibniz rule. First-order deformations of $P$ are given by bi-vector fields $Q$ such that $P + \varepsilon Q$ enjoys the Jacobi identity modulo $\varepsilon^2$. Every Poisson structure admits deformations of "trivial" type, where $Q = L_X P$ is the Lie derivative of $P$ with respect to a vector field $X$. Kontsevich's tetrahedral flow is a symmetry of the Jacobi identity: given any Poisson structure $P$ it provides a deformation $Q(P)$. We study whether or not this deformation can be nontrivial in concrete examples.