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The heptagon-wheel cocycle in the Kontsevich graph complex

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Arthemy Kiselev
University of Groningen
Tue, 2017-11-28 14:00 - 15:00
MPIM Lecture Hall

The real vector space of finite non-oriented graphs is known to
carry a structure of differential graded Lie algebra (dgLa). We recall
that construction and we illustrate it by using the tetrahedral
cocycle (Kontsevich 1996) and the Kontsevich--Willwacher
pentagon-wheel cocycle, which consists of two graphs with real
coefficients. (Under the orientation mapping, either of the two
examples yields a flow on the spaces of Poisson structures that is
universal with respect to all finite-dimensional affine Poisson

  The existence of an infinite sequence of nontrivial cocycles in the
non-oriented graph complex has been predicted by Willwacher: every
such cocycle contains a (2m+1)-gon wheel with a nonzero coefficient at
some integer m>0 (e.g., see above for m=1 and m=2, respectively).
These cocycles generate a noncommutative Lie algebra, the properties
of which are largely unexplored; the dgLa at hand is isomorphic to a
subalgebra in the Lie algebra of the Grothendieck--Teichmueller group,
which was introduced by Drinfel'd. In this talk, the heptagon-wheel
cocycle at m=3 will be presented; it consists of 46 graphs on 8
vertices and 14 edges (see [1710.00658]).

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