# 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.

## 6 Citations

### The method of averaging for Poisson connections on foliations and its applications

- MathematicsJournal of Geometric Mechanics
- 2020

On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous…

### G-Invariant Deformations of Almost-Coupling Poisson Structures

- Mathematics
- 2017

On a foliated manifold equipped with an action of a compact Lie group $G$, we study a class of almost-coupling Poisson and Dirac structures, in the context of deformation theory and the method of…

### Unimodularity criteria for Poisson structures on foliated manifolds

- Mathematics
- 2017

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results…

### Generalized Hannay-Berry Connections on Foliated Manifolds and Applications

- Mathematics
- 2019

In this paper, we discuss some aspects of the averaging method for Poisson connections on foliated manifolds with symmetry generalizing the previous results on the Hannay-Berry connections on…

### The Poisson linearization problem for $\mathfrak{sl}_2(\mathbb{C})$. Part I: Poisson cohomology

- Mathematics
- 2022

This is the first of two papers, in which we prove a version of Conn’s linearization theorem for the Lie algebra sl2(C) ≃ so(3, 1). Namely, we show that any Poisson structure whose linear…

### Unimodularity criteria for Poisson structures on foliated manifolds

- MathematicsLetters in Mathematical Physics
- 2017

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results…

## References

SHOWING 1-10 OF 32 REFERENCES

### Coupling Tensors and Poisson Geometry near a Single Symplectic Leaf

- Mathematics
- 2000

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

- Mathematics
- 2010

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

- Physics, Mathematics
- 2007

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

- Mathematics
- 2005

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

- Mathematics
- 2001

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

- Mathematics
- 2009

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

- Mathematics
- 2008

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

- Mathematics
- 2005

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

- Mathematics, Physics
- 2005

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

- Mathematics
- 2013

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.…