Invariant Poisson Realizations and the Averaging of Dirac Structures

@article{Vallejo2014InvariantPR,
  title={Invariant Poisson Realizations and the Averaging of Dirac Structures},
  author={Jos{\'e} Antonio Vallejo and Yu. Vorobiev},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2014},
  volume={10},
  pages={096}
}
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. 

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References

SHOWING 1-10 OF 32 REFERENCES

Coupling Tensors and Poisson Geometry near a Single Symplectic Leaf

In the framework of the connection theory, a contravariant analog of the Sternberg coupling procedure is developed for studying a natural class of Poisson structures on fiber bundles, called coupling

Induced Dirac structures on isotropy-type manifolds

A new method of singular reduction is extended from Poisson to Dirac manifolds. Then it is shown that the Dirac structures on the strata of the quotient coincide with those of the only other known

B.L. Davis and A. Wade DIRAC STRUCTURES AND GAUGE SYMMETRIES OF PHASE SPACES

We study the geometry of the phase space of a particle in a Yang -Mills-Higgs field in the context of the theory of Dirac structures. Several kno wn constructions are merged into the framework of

A note on equivariant normal forms of Poisson structures

We prove an equivariant version of the local splitting theorem for tame Poisson structures and Poisson actions of compact Lie groups. As a consequence, we obtain an equivariant linearization result

Poisson Geometry with a 3-Form Background

We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable

Singular reduction of Dirac structures

The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about G-invariant vector fields and one-forms are

Averaging of Poisson Structures

The averaging procedure is applied to a class of compatible Poisson structures on the total space of a Poisson fiber bundle equipped with a family of Hamiltonian group actions.

Poisson fiber bundles and coupling Dirac structures

Poisson fiber bundles are studied. We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also

Quantum Algebras and Poisson Geometry in Mathematical Physics

Noncommutative algebras, nanostructures, and quantum dynamics generated by resonances by M. Karasev Algebras with polynomial commutation relations for a quantum particle in electric and magnetic

Differential geometry of singular spaces and reduction of symmetry

Preface 1. Introduction Part I. Differential Geometry of Singular Spaces: 2. Differential structures 3. Derivations 4. Stratified spaces 5. Differential forms Part II. Reduction of Symmetries: 6.