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IMPRS Minicourse "The Kontsevich (un)oriented graph complex"

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Arthemy Kiselev
University of Groningen
Fri, 07/12/2018 - 15:30 - 17:00
MPIM Seminar Room
Parent event: 
IMPRS Minicourse

 This one-day course is about the differential graded Lie algebra
structure on the space of graphs, and its morphism to the space of
endomorphisms of multi-vector fields on affine manifolds. The
orientation morphism produces universal symmetries of Poisson brackets
from cocycles in the unoriented graph comples..
   This knowledge and skills can be important to those working in
Poisson geometry, deformation quantisation,
Grothendieck--Teichmueller, Drinfeld associators, or quantum groups
and Hopf algebras.

   The course will contain both theory and exercises. An approximate
schedule is this:

 * 10:00 -- 12:00 (Aula): DGLA structure on the vector space of graphs
with wedge ordering of edges. "Zero graphs" = minus themselves. The
differential, Lie bracket, Jacobi identity. Defining properties of the
complex (identities with zero graphs).
   -- Cocycles: tetrahedron, pentagon wheel (exercises in graph calculus).

 * 13:00 -- 15:00 (Aula): The Kontsevich oriented graphs with
decorated edges and ordering of outgoing edges. The Poisson complex.
Universal symmetries of Poisson brackets.
  -- Leibniz graphs. Star-product, associator. Factorisation problem
for the associator and for Poisson cocycles.

  -- (tea)

 * 15:30 -- 17:00 (Seminarraum): The orientation morphism: from graph
cocycles to symmetries of Poisson brackets. Endomorphisms, Schouten
bracket, its Jacobi identity. Nijenhuis--Richardson bracket, its
Jacobi identity. Master-equation.
  -- The edge. Morphism Or: edge to Schouten. Verifying the Poisson
cocycle condition. Canonical vs non-canonical solution of the graph
factorisation problem, topological identities in the space of Leibniz

   No special pre-requisites are expected from the MPIM visitors.
Literature references will be provided during the lectures.

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