Noncommutative Geometry and Quiver algebras Export

@misc{citeulike:461729, abstract = {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].}, archivePrefix = {arXiv}, author = {Crawley-Boevey, William and Etingof, Pavel and Ginzburg, Victor}, citeulike-article-id = {461729}, citeulike-linkout-0 = {http://arxiv.org/abs/math.AG/0502301}, citeulike-linkout-1 = {http://arxiv.org/pdf/math.AG/0502301}, day = {11}, eprint = {math.AG/0502301}, keywords = {bimodules, differentials, geometry, necklaces, noncommutative, quivers, symplectic}, month = {Jul}, posted-at = {2006-01-11 07:39:04}, priority = {0}, title = {Noncommutative Geometry and Quiver algebras}, url = {http://arxiv.org/abs/math.AG/0502301}, year = {2005} }

TY - ELEC ID - citeulike:461729 L3 - citeulike-article-id:461729 N2 - 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]. TI - Noncommutative Geometry and Quiver algebras KW - bimodules KW - differentials KW - geometry KW - necklaces KW - noncommutative KW - quivers KW - symplectic AU - Crawley-Boevey, William AU - Etingof, Pavel AU - Ginzburg, Victor PY - 2005/Jul/11/ UR - http://arxiv.org/abs/math.AG/0502301 ER -