Formal noncommutative symplectic geometry (generalizing symplectic geometry to the context of noncommutative geometry) has been introduced by Maxim Kontsevich, motivated by several constructions in geometry and mathematical physics including the cohomology of (compactifications of) certain moduli spaces, cohomology of foliations and perturbation expansions of Chern-Simons theory. He noticed that there is a parallel between computations in such seemingly distant problems, and conjectured a generalized version of Lie theory which should apply to those, and which should be at formal level reflected in some version of duality theory. The underlying theory has been constructed by Kapranov and Ginzburg following his advice as a Koszul duality for operads.

Some of the structures are now viewed from the point of view of double derivations and double Poisson structures and are closed to derived picture.

References

Kontsevich gave lectures at Harvard in 1991-1992 on aspects related to the expansions of Chern-Simons theory.

Maxim Kontsevich, Formal (non)-commutative symplectic geometry, The Gelfand Mathematical. Seminars, 1990-1992, Ed. L.Corwin, I.Gelfand, J.Lepowsky, Birkhauser 1993, 173-187, pdf

Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, Volume II, Progress in Mathematics 120, Birkhauser 1994, 97-121, pdfMR1247289

Victor Ginzburg, Mikhail Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272; reprint arxiv/0709.1228; Erratum to: Koszul duality for operads, Duke Math. J. 80 (1995), no. 1, 293.

Victor Ginzburg, Non-commutative symplectic geometry, quiver varieties, and operads, Math. Res. Lett. 8 (2001), no. 3, 377–400, math.QA/0005165; Lectures on noncommutative geometry, math.AG/0506603