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Deformation Theory Seminar

Monday, April 11, 2022 - 2:00pm

Arthemy Kiselev

Bernoulli Inst


University of Pennsylvania

Arthemy's Part II Part II mostly independent of part I The intersection will be minimal. That is, it is possible to join anew and be satisfied. This is how these three talks are given. In brief, no pre-requisite from Part 1, Notes/slides from Part I are available from jds

   Can we deform a given Poisson bracket --universally for allfinite-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).)