  We develop a new framework for noncommutative differential geometry based on
double derivations. This leads to the notion of moment map and of Hamiltonian
reduction in noncommutative symplectic geometry. For any smooth associative
algebra B, we define its noncommutative cotangent bundle T^*B, which is a basic
example of noncommutative symplectic manifold. Applying Hamiltonian reduction
to noncommutative cotangent bundles gives an interesting class of associative
algebras, P=P(B), that includes preprojective algebras associated with quivers.
Our formalism of noncommutative Hamiltonian reduction provides the space
P/[P,P] with a Lie algebra structure, analogous to the Poisson bracket on the
zero fiber of the moment map. In the special case where P is the preprojective
algebra associated with a quiver of non-Dynkin type, we give a complete
description of the Gerstenhaber algebra structure on the Hochschild cohomology
of P in terms of the Lie algebra P/[P,P].
