Noncommutative geometry and quiver algebras

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, Pi = Pi(B), that includes preprojective algebras associated with quivers. Our formalism of noncommutative Hamiltonian reduction provides the space Pi/[Pi, Pi] with a Lie algebra structure, analogous to the Poisson bracket on the zero fiber of the moment map. In the special case where Pi 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 Pi in terms of the Lie algebra Pi/[Pi, Pi]. (c) 2006 Elsevier Inc. All rights reserved.