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

SIGMA 13 (2017), 022, 13 pages      arXiv:1610.09898      https://doi.org/10.3842/SIGMA.2017.022
Contribution to the Special Issue “Gone Fishing”

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

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

Received October 31, 2016, in final form March 28, 2017; Published online April 02, 2017

Abstract
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 averaging.

Key words: Poisson geometry; Dirac structures; deformation; averaging.

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References

  1. Arnold V.I., Kozlov V.V., Neishtadt A.I., Dynamical systems. III, Encyclopaedia of Mathematical Sciences, Vol. 3, Springer-Verlag, Berlin, 1988.
  2. Avendaño Camacho M., Vallejo J.A., Vorobiev Yu., Higher order corrections to adiabatic invariants of generalized slow-fast Hamiltonian systems, J. Math. Phys. 54 (2013), 082704, 15 pages, arXiv:1305.3974.
  3. Avendaño Camacho M., Vallejo J.A., Vorobiev Yu., A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows, J. Phys. A: Math. Theor. 46 (2013), 395201, 12 pages.
  4. Avendaño Camacho M., Vorobiev Yu., On deformations of Poisson structures on fibered manifolds and adiabatic slow-fast systems, Int. J. Geom. Methods Mod. Phys. 14 (2017), 1750086, 15 pages.
  5. Brahic O., Fernandes R.L., Poisson fibrations and fibered symplectic groupoids, in Poisson Geometry in Mathematics and Physics, Contemp. Math., Vol. 450, Amer. Math. Soc., Providence, RI, 2008, 41-59, math.DG/0702258.
  6. Bursztyn H., Radko O., Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble) 53 (2003), 309-337, math.SG/0202099.
  7. Courant T.J., Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), 631-661.
  8. Courant T.J., Weinstein A., Beyond Poisson structures, in Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986), Travaux en Cours, Vol. 27, Hermann, Paris, 1988, 39-49.
  9. Crainic M., Fernandes R.L., Stability of symplectic leaves, Invent. Math. 180 (2010), 481-533, arXiv:0810.4437.
  10. Crainic M., Mărcuţ I., A normal form theorem around symplectic leaves, J. Differential Geom. 92 (2012), 417-461, arXiv:1009.2090.
  11. Frejlich P., Mărcuţ I., The normal form theorem around Poisson transversals, arXiv:1306.6055.
  12. Guillemin V., Lerman E., Sternberg S., Symplectic fibrations and multiplicity diagrams, Cambridge University Press, Cambridge, 1996.
  13. Kolář I., Michor P.W., Slovák J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993.
  14. Ševera P., Weinstein A., Poisson geometry with a 3-form background, Progr. Theoret. Phys. Suppl. (2001), 145-154, math.SG/0107133.
  15. Vaisman I., Lectures on the geometry of Poisson manifolds, Progress in Mathematics, Vol. 118, Birkhäuser Verlag, Basel, 1994.
  16. Vaisman I., Coupling Poisson and Jacobi structures on foliated manifolds, Int. J. Geom. Methods Mod. Phys. 1 (2004), 607-637, math.SG/0402361.
  17. Vaisman I., Foliation-coupling Dirac structures, J. Geom. Phys. 56 (2006), 917-938, math.SG/0412318.
  18. Vallejo J.A., Vorobiev Yu., Invariant Poisson realizations and the averaging of Dirac structures, SIGMA 10 (2014), 096, 20 pages, arXiv:1405.0574.
  19. Vorobiev Yu., Averaging of Poisson structures, in Geometric Methods in Physics, AIP Conf. Proc., Vol. 1079, Amer. Inst. Phys., Melville, NY, 2008, 235-240.
  20. Vorobjev Yu., Coupling tensors and Poisson geometry near a single symplectic leaf, in Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publ., Vol. 54, Polish Acad. Sci. Inst. Math., Warsaw, 2001, 249-274, math.SG/0008162.
  21. Wade A., Poisson fiber bundles and coupling Dirac structures, Ann. Global Anal. Geom. 33 (2008), 207-217, math.SG/0507594.

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