Noncommutative smoothness and coadjoint orbits

Abstract

In [ math.AG/0010030 ; Math. Z., to appear] R. Bocklandt and the author proved that certain quotient varieties of representations of deformed preprojective algebras are coadjoint orbits for the necklace Lie algebra N Q of the corresponding quiver Q → . A conjectural ring-theoretical explanation of these results was given in terms of noncommutative smoothness in the sense of C. Procesi [J. Algebra 107 (1987) 63–74]. In this paper we prove these conjectures. The main tool in the proof is the étale local description due to W. Crawley-Boevey [ math.AG/0105247 ]. Along the way we determine the smooth locus of the Marsden–Weinstein reductions for quiver representations.