Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 096, 20 pages      arXiv:1405.0574      https://doi.org/10.3842/SIGMA.2014.096

Invariant Poisson Realizations and the Averaging of Dirac Structures

José A. Vallejo a and Yurii Vorobiev b
a) Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, México
b) Departamento de Matemáticas, Universidad de Sonora, México

Received May 19, 2014, in final form September 09, 2014; Published online September 15, 2014

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.

Key words: Poisson structures; Dirac structures; geometric data; averaging operators.

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