Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Arthemy Kiselev
Affiliation:

University of Groningen
Date:

Fri, 2018-12-07 13:30 - 15:00 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

graphs.

No special pre-requisites are expected from the MPIM visitors.

Literature references will be provided during the lectures.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/4234

[3] https://www.mpim-bonn.mpg.de/node/2761