We investigate the geometric, algebraic and homologic structures related with Poisson structure on a smooth manifold. Introduce a noncommutative foundations of these structures for a Poisson algebra. Introduce and investigate noncommutative Bott connection on a foliated manifold using the algebraic definition of submanifold and quotient manifold. Develop an algebraic construction for the reduction of a degenerated Poisson algebra.
Differential geometric structures such as the principal bundle for the canonical vector bundle on a complex Grassmann manifold, the canonical connection form on this bundle, the canonical symplectic form on a complex Grassmann manifold and the corresponding dynamical systems are investigated. The Grassmann manifold is considered as an orbit of the co-adjoint action and the symplectic form is described as the restriction of the canonical Poisson structure on a Lie coalgebra. The holonomy of the connection on the principal bundle over Grassmannian and its relation with Berry phase is considered and investigated for the integral curves of Hamiltonian dynamical systems.
We give an algebraic construction of connection on the symplectic leaves of Poisson manifold, introduced in [6]. This construction is suitable for the definition of the linearized holonomy on a regular symplectic foliation.
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