The space Gra of unoriented finite graphs with unlabelled vertices and

wedge ordering on the set of edges carries the structure of a complex

with respect to the vertex-expanding differential d. In fact, this

space is a differential graded Lie algebra such that the differential

d = [*−*, •] is taking the Lie bracket with a single edge. Under the

Kontsevich orientation morphism, d-cocycle graphs γ on n vertices and

2n−2 edges yield infinitesimal symmetries dP/dt = Or(γ)(P) of Poisson

bivectors P on finite-dimensional affine manifolds M. Namely, every

oriented graph built of the decorated wedges ←i−*−j→ determines a

differential-polynomial expression Q(P) in the coefficients P(ij)

(x(1), ... , x(d)) of a bivector P whenever the arrows −a→ denote

derivatives ∂/∂ x(a) in a local coordinate chart, each vertex * at the

top of a wedge contains a copy of P, and one takes the product of

vertex contents and sums up over all the indexes.

We recall the construction of the Kontsevich graph orientation

morphism, revealing in particular why there always exists a

factorization of the Poisson cocycle condition [[P, Q(P)]] = 0 through

the differential consequences of the Jacobi identity [[P,P]] = 0

for Poisson bivectors. To illustrate the reasoning, we use the

Kontsevich tetrahedral flow dP/dt = Or(γ3)(P), as well as the flow

produced from the Kontsevich–Willwacher pentagon-wheel cocycle γ5 and

the new flow obtained from the heptagon-wheel cocycle γ7 in the

unoriented graph complex.

(This is joint work with R.Buring.)

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