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 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
P-geometry, a `non-commutative geometry' for an algebra over an arbitrary
cyclic Koszul operad.