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

manifolds.)

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]).

Posted in

Speaker:

Arthemy Kiselev
Affiliation:

University of Groningen
Date:

Tue, 2017-11-28 14:00 - 15:00
Location:

MPIM Lecture Hall
Parent event:

Seminar on Algebra, Geometry and Physics