Open problems in the Kontsevich graph construction of Poisson bracket symmetries

TL;DR

This paper discusses open problems related to Kontsevich's graph-based construction of Poisson bracket symmetries, focusing on combinatorial, topological, integrability, and quantum aspects.

Contribution

It formulates several open research questions about the properties and extensions of Kontsevich's graph construction for Poisson symmetries.

Findings

Identifies key open problems in graph combinatorics and topology

Highlights challenges in integrability and analytic properties

Explores cohomological and quantum aspects of the theory

Abstract

Poisson brackets admit infinitesimal symmetries which are encoded using oriented graphs; this construction is due to Kontsevich (1996). We formulate several open problems about combinatorial and topological properties of the graphs involved, about integrability and analytic properties of such symmetry flows (in particular, for known classes of Poisson brackets), and about cohomological, differential geometric, and quantum aspects of the theory. Keywords: Poisson bracket, Jacobi identity, infinitesimal symmetry, affine manifold, Poisson cohomology, graph complex.