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Coupling symmetries with Poisson structures

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Abstract

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein’s splitting theorem for the integrable system is also studied giving some examples in which such a splitting does not exist, i.e. when the integrable system is not, locally, a product of an integrable system on the symplectic leaf and an integrable system on a transversal. The problem of splitting for integrable systems with additional symmetries is also considered.

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Notes

  1. We could also call this theorem equivariant Carathéodory–Weinstein theorem as its equivalent for symplectic forms is often called the Darboux–Carathéodory theorem. But we prefer to stick to the old denomination Carathéodory–Jacobi–Lie since Lie already worked partial aspects of this result (see Satz 3 on p. 198 in [13]).

  2. One could envisage to implement here the ideas of Crainic and Fernandes in [5] and [4] that use the idea of stability to give a different proof of this equivariant Conn’s linearization theorem. The authors did not try this approach.

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Acknowledgements

Eva Miranda has been partially supported by the DGICYT/FEDER project MTM2009-07594: Estructuras Geometricas: Deformaciones, Singularidades y Geometria Integral until December 2012. Her research will be partially supported by the MINECO project GEOMETRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES with reference: MTM2012-38122-C03-01 starting in January 2013.

The second author of this paper is thankful to the Hanoi National University of Education for their warm hospitality during her visit in occasion of the conference GEDYTO that she co-organized. She is particularly thankful to Professor Do Duc Thai. She also wants to thank the ESF network Contact and Symplectic Topology (CAST) for providing financial support to organize this conference. The collaboration that led to this paper started longtime ago. The authors acknowledge financial support from the Marie Curie postdoctoral EIF project GEASSIS FP6-MOBILITY24513.

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Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, Université de Lorraine, UMR 7122, site de Metz, Metz, France

    Camille Laurent-Gengoux

  2. Departament de Matemàtica Aplicada I, EPSEB, Universitat Politècnica de Catalunya, Barcelona, Spain

    Eva Miranda

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  1. Camille Laurent-Gengoux
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  2. Eva Miranda
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Correspondence to Eva Miranda.

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Laurent-Gengoux, C., Miranda, E. Coupling symmetries with Poisson structures. Acta Math Vietnam. 38, 21–32 (2013). https://doi.org/10.1007/s40306-013-0008-1

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