Invariant Poisson Realizations and the Averaging of Dirac Structures

TL;DR

This paper develops an averaging method for Dirac manifolds under compact Lie group actions, leading to invariant Poisson realizations and insights into coupling Dirac structures and gauge transformations.

Contribution

It introduces an averaging procedure for Dirac structures and establishes new theorems on invariant Poisson realizations around symplectic leaves.

Findings

Existence of invariant realizations of Poisson structures proved.

Connection between coupling Dirac structures and gauge transformations established.

Averaging procedure applicable to Dirac manifolds with compact Lie group actions.

Abstract

We describe an averaging procedure on a Dirac manifold, with respect to a class of compatible actions of a compact Lie group. Some averaging theorems on the existence of invariant realizations of Poisson structures around (singular) symplectic leaves are derived. We show that the construction of coupling Dirac structures (invariant with respect to locally Hamiltonian group actions) on a Poisson foliation is related with a special class of exact gauge transformations.