Noncommutative Symplectic Geometry, Quiver varieties, and Operads
Abstract
Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of KacMoody algebras and quantum groups, instantons on 4manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the CalogeroMoser 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 `noncommutative Symplectic geometry'. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of Pgeometry, a `noncommutative geometry' for an algebra over an arbitrary cyclic Koszul operad.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2000
 arXiv:
 arXiv:math/0005165
 Bibcode:
 2000math......5165G
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  KTheory and Homology;
 Mathematics  Mathematical Physics
 EPrint:
 minor corrections made