Non-commutative (quasi-)Poisson structure for marked surfaces

The notion of a double Poisson algebra was introduced independently in 2007-08 by M. Van den Bergh and by W. Crawley-Boevey, P. Etingof, and V. Ginzburg. A double Poisson bracket on a non-commutative algebra generalizes the classical Poisson bracket for the algebra of smooth functions on a manifold.

Poisson structures on spaces of surface group representations into Lie groups were introduced by W. Goldman in 1986 and have since been studied by many researchers. In particular, V. Fock and A. Goncharov described a Poisson structure on spaces of surface group representations into split Lie groups using their X- and A-coordinates.

In my talk, I will generalize this construction by introducing a double quasi-Poisson structure on the space of representations of the fundamental group of a marked surface S into GL_n(A) for a non-commutative algebra A without polynomial identities. I will demonstrate that this bracket arises from the so-called partial non-abelianization procedure and a double bracket that is naturally associated with an appropriate ramified covering of S.

This is joint work with M. Gekhtman.