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Orientation morphism: from graph cocycles to deformations of Poisson structures

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Arthemy Kiselev
University of Groningen
Tue, 2018-11-20 14:00 - 15:00
MPIM Lecture Hall

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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