Noncommutative Hamiltonian dynamics on foliated manifolds

TL;DR

This paper explores noncommutative Hamiltonian dynamics on foliated manifolds, introducing Hamiltonian vector fields in noncommutative Poisson algebras and analyzing their structure in the context of transversely symplectic foliations.

Contribution

It extends the concept of Hamiltonian vector fields to noncommutative Poisson algebras related to foliated manifolds, providing new insights into their structure and properties.

Findings

Defined Hamiltonian vector fields on noncommutative Poisson algebras

Described Poisson structures associated with transversely symplectic foliations

Constructed classes of Hamiltonian vector fields in this setting

Abstract

First, we review the notion of a Poisson structure on a noncommutative algebra due to Block-Getzler and Xu and introduce a notion of a Hamiltonian vector field on a noncommutative Poisson algebra. Then we describe a Poisson structure on a noncommutative algebra associated with a transversely symplectic foliation and construct a class of Hamiltonian vector fields associated with this Poisson structure.