Can we deform a given Poisson bracket --universally for all

finite-dimensional Poisson manifolds-- in such a way that it stays

Poisson at least infinitesimally and there is no a priori mechanism

for the deformation to be trivial in the respective Poisson

cohomology? Although the question might sound too general, Kontsevich

answered it in the affirmative (1996) by finding a source of

solutions: built from suitable cocycles in the Kontsevich graph

complex, these deformations are encoded by directed graphs. Willwacher

(2010-15) established that there are infinitely many generators of

nonlinear proper infinitesimal symmetries of the Jacobi identity;

namely, countably many graph cocycles are obtained from the generators

of the Grothendieck--Teichmueller Lie algebra grt (in turn, introduced

by Drinfeld around 1990). The tetrahedral graph cocycle gives an

example of degree-four order-three differential-polynomial flow on the

spaces of Poisson bi-vectors.

In this talk, we discover new properties of such flows'

restrictions --for the Kontsevich tetrahedral graph cocycle and for

the Kontsevich--Willwacher pentagon-wheel cocycle-- to the spaces of

Nambu--Poisson ``determinant'' brackets in dimensions 3 and 4. We

examine the analytic and combinatorial structures now arising from the

graph formula, and we detect a hidden symmetry of the Poisson cocycles

for this class of highly-nonlinear Poisson brackets. We establish that

for the class of Nambu-determinant Poisson bi-vectors, the tetrahedral

graph flow on the affine space R3 appears to be trivial in the second

Poisson cohomology; we examine the combinatorics and symmetry of the

highly-nonlinear trivialising vector field.

(The talk is based on recent joint work with R.Buring (IM JGU Mainz).)

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### Deformation Theory Seminar

Monday, March 14, 2022 - 2:00pm

#### Arthemy Kiselev

Bernouli Inst