Necklace Lie algebras and noncommutative symplectic geometry

Recently, V. Ginzburg proved that Calogero phase space is a coadjoint orbit for some infinite dimensional Lie algebra coming from non-commutative symplectic geometry, [12]. In this note we generalize his argument to specific quotient varieties of representations of (deformed) preprojective algebras. This result was also obtained independently by V. Ginzburg [13]. Using results of W. Crawley-Boevey and M. Holland [10], [8] and [9] we give a combinatorial description of all the relevant couples (alpha, lambda) which are coadjoint orbits. We give a conjectural explanation for this coadjoint orbit result and relate it to different noncommutative notions of smoothness.