By a fundamental theorem of D. Quillen, there is a natural duality - an instance of general Koszul duality - between differential graded (DG) Lie algebras and DG cocommutative coalgebras defined over a field k of characteristic 0. A cyclic pairing (i.e., an inner product satisfying a natural cyclicity condition) on the cocommutative coalgebra gives rise to an interesting structure on the universal enveloping algebra U_a of the Koszul dual Lie algebra a called the derived Poisson bracket. Interesting special cases of the derived Poisson bracket include the Chas-Sullivan bracket on string topology. We study the derived Poisson brackets on universal enveloping algebras U_a, and their relation to the classical Poisson brackets on the derived moduli spaces DRep_g(a) of representations of a in a finite dimensional reductive Lie algebra g. More specifically, we show that certain derived character maps of a intertwine the derived Poisson bracket with the classical Poisson structure on the representation homology HR_•(a,g) related to DRep_g(a).

We would like to thank Yuri Berest, Ayelet Lindenstrauss, Tony Pantev and Vladimir Turaev for interesting discussions. We also thank the referee for valuable comments that helped improve the presentation of this paper. The first author is grateful to the Department of Mathematics, University of Pennsylvania for conducive working conditions during his visit in the summer of 2018. The work of the first author was partially supported by NSF grant DMS 1702323.