Poisson geometry and non-commutative differential calculus

Abstract

Poisson brackets of special type on n-tuples of N by N matrices may be encoded by double brackets in the sense of van den Bergh. Interesting examples include constant and linear (KKS) Poisson brackets. In particular, these brackets admit moment maps for the GL(N) action by simultaneous conjugation of matrices in the n-tuple. Surprisingly, there are instances where the theory of double brackets deviates from the standard wisdoms of Poisson geometry. For instance, KKS brackets turn out to be non-degenerate, and under some assumptions a moment map uniquely determines the double bracket. These observations give rise to a new proof of the theorem by L. Jeffrey on symplectomorphisms between moduli of flat connections and reduced spaces of products of coadjoint orbits. The talk is mostly based on the work by F. Naef. If time permits, I’ll sketch how these results are related to the Kashiwara-Vergne theory and to the Goldman-Turaev Lie bialgebra.Non UBCUnreviewedAuthor affiliation: University of GenevaFacult