Non-commutative Symplectic Geometry, Quiver varieties,$\,$ and$\,$ Operads

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions of Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the Calogero-Moser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich’s formalism of `non-commutative symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of $\PP$-geometry, a `non-commutative geometry' for an algebra over an arbitrary cyclic Koszul operad.

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Published 1 January 2001