We investigate the geometric, algebraic and homological properties of Poisson structures on smooth manifolds and introduce noncommutative foundations of these structures for associative Poisson algebras. Noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold, symplectic foliation and symplectic leaf for associative Poisson algebras are given. These structures are considered for the case of the endomorphism algebra of a vector bundle, and a full description of the family of Poisson structures for this algebra is given. An algebraic construction of the reduction procedure for degenerate noncommutative Poisson structures is developed. A noncommutative generalization of Bott connection on foliated manifolds is introduced using the notions of a noncommutative submanifold and a quotient manifold. This definition is applied to degenerate noncommutative Poisson algebras, which allows us to consider Bott connection not only for regular but also for singular Poisson structures.
Noncommutative geometry, Poisson structure, endomorphism algebra, Schouten-Nijenhuis bracket, Bott connection.
MSC 2000: 46L87, 17B63, 46L55.