Abstract. We prove that the Kontsevich tetrahedral flowṖ = Q a:b (P), the righthand side of which is a linear combination of two differential monomials of degree four in a bi-vector 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 Q a:b (P) are balanced by the ratio a : b = 1 : 6. The proof is explicit; it is written in the language of Kontsevich graphs.Introduction. The main question which we address in this paper is how Poisson structures can be deformed in such a way that they stay Poisson. We reveal one such method that works for all Poisson structures on affine real manifolds; the construction of that flow on the space of bi-vectors was proposed in [11]: the formula is derived from two differently oriented tetrahedral graphs on four vertices. The flow is a linear combination of two terms, each quartic-nonlinear in the Poisson structure. By using several examples of Poisson brackets with high polynomial degree coefficients, the first and last authors demonstrated in [1] that the ratio 1 : 6 is the only possible balance at which the tetrahedral flow can preserve the property of the Cauchy datum to be Poisson. But does the Kontsevich tetrahedral flowṖ = Q 1:6 (P) with ratio 1 : 6 actually preserve the space of all Poisson bi-vectors?We prove the infinitesimal version of this claim: namely, we show that [[P, Q 1:6 (P)]] = 0 for every bi-vector P satisfying the master-equation [[P, P]] = 0 for Poisson structures. The proof is graphical: to prove that equation (2) holds, we find an operator ♦, encoded by using the Kontsevich graphs, that solves equation (10). We also show that there is no universal mechanism (that would involve the language of Kontsevich graphs) for the tetrahedral flow to be trivial in the respective Poisson cohomology.The text is structured as follows. In section 1 we recall how oriented graphs can be used to encode differential operators acting on the space of multivectors. In particular, differential polynomials in a given Poisson structure are obtained as soon as a copy of that Poisson bi-vector is placed in every internal vertex of a graph. Specifically, the right-hand side Q a:b = a · Γ 1 + b · Γ 2 of the Kontsevich tetrahedral flowṖ = Q a:b (P) on the space of bi-vectors on an affine Poisson manifold N n , P is a linear combination

The deformation quantization by Kontsevich is a way to construct an associative noncommutative star-product ⋆ = × + { , } P +ō( ) in the algebra of formal power series in on a given finite-dimensional affine Poisson manifold: here × is the usual multiplication, { , } P = 0 is the Poisson bracket, and is the deformation parameter. The product ⋆ is assembled at all powers k 0 via summation over a certain set of weighted graphs with k+2 vertices; for each k>0, every such graph connects the two co-multiples of ⋆ using k copies of { , } P . Cattaneo and Felder interpreted these topological portraits as genuine Feynman diagrams in the Ikeda-Izawa model for quantum gravity.By expanding the star-product up toō( 3 ), i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = ♦ (Poisson) that converts the Jacobi identity for the bracket { , } P into the associativity of ⋆.

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter . Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, moduloō( 4 ), for the newly built ⋆-product expansion.Assoc ⋆ (f, g, h) = ♦ P, Jac(P), Jac(P) modō( 4 ), is quadratic and has differential order two with respect to the Jacobiator. For all Poisson brackets {·, ·} P on finite-dimensional affine manifolds N n our ten-parameter expression of the ⋆-product does agree up toō( 4 ) with previously known results about the values of Kontsevich graph weights at some fixed values of the ten master-parameters and about the linear relations between those weights at all values of the master-parameters. 4The software implementation [5] consists of a C++ library and a set of command-line programs. The latter are specified in what follows; a full list of new C++ subroutines and their call syntaxis is contained in Appendix B. Whenever a command-line program refers to just one particular function in C++, we indicate that in the text. The current text refers to version 0.16 of the software.

We recall the construction of the Kontsevich graph orientation morphism γ → O r(γ) which maps cocycles γ in the non-oriented graph complex to infinitesimal symmetriesṖ = O r(γ)(P) of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition [[P, O r(γ)(P)]] . = 0 through the differential consequences of the Jacobi identity [[P, P]] = 0 for Poisson bi-vectors P. To illustrate the reasoning, we use the Kontsevich tetrahedral flowṖ = O r(γ 3 )(P), as well as the flow produced from the Kontsevich-Willwacher pentagon-wheel cocycle γ 5 and the new flow obtained from the heptagonwheel cocycle γ 7 in the unoriented graph complex.

Kontsevich designed a scheme to generate infinitesimal symmetriesṖ = Q(P) of Poisson brackets P on all affine manifolds M r ; every such deformation is encoded by oriented graphs on n + 2 vertices and 2n edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs γ on n vertices and 2n − 2 edges. The bi-vector flowṖ = O r(γ)(P) preserves the space of Poisson structures if γ is a cocycle with respect to the vertex-expanding differential in the graph complex.A class of such cocycles γ 2ℓ+1 is known to exist: marked by ℓ ∈ N, each of them contains a (2ℓ + 1)-gon wheel with a nonzero coefficient. At ℓ = 1 the tetrahedron γ 3 itself is a cocycle; at ℓ = 2 the Kontsevich-Willwacher pentagon-wheel cocycle γ 5 consists of two graphs. We reconstruct the symmetry Q 5 (P) = O r(γ 5 )(P) and verify that Q 5 is a Poisson cocycle indeed: [[P, Q 5 (P)]] . = 0 via [[P, P]] = 0. . Partially supported by JBI RUG project 103511 (Groningen). A part of this research was done while R. B. and A.V.K. were visiting at the IHÉS (Bures-sur-Yvette, France) and A.V.K. was visiting at the University of Mainz. 1 The dilationṖ = P is an example of symmetry for Jacobi identity; we study nonlinear flowsṖ = Q(P) which are universal w.r.t. all affine manifolds and should persist under the quantization i {·, ·} P → [·, ·].