**Z. Giunashvili**

## Noncommutative Symplectic Foliation, Bott
Connection and Phase Space Reduction

**Abstract:**

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.

**Keywords:**

Noncommutative geometry, Poisson structure, endomorphism algebra,
Schouten-Nijenhuis bracket, Bott connection.

**MSC 2000:** 46L87, 17B63, 46L55.