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

SIGMA 14 (2018), 005, 29 pages      arXiv:1506.03216      https://doi.org/10.3842/SIGMA.2018.005

Poisson Geometry Related to Atiyah Sequences

Kirill Mackenzie a, Anatol Odzijewicz b and Aneta Sliżewska b
a) School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK
b) Institute of Mathematics, University in Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland

Received July 05, 2017, in final form January 06, 2018; Published online January 10, 2018

Abstract
We construct and investigate a short exact sequence of Poisson $\mathcal{VB}$-groupoids which is canonically related to the Atiyah sequence of a $G$-principal bundle $P$. Our results include a description of the structure of the symplectic leaves of the Poisson groupoid $\frac{T^*P\times T^*P}{G}\rightrightarrows \frac{T^*P}{G}$. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed.

Key words: Atiyah sequence; $\mathcal{VB}$-groupoid; Poisson groupoid; dualization of $\mathcal{VB}$-groupoid.

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References

  1. Cannas da Silva A., Weinstein A., Geometric models for noncommutative algebras, Berkeley Mathematics Lecture Notes, Vol. 10, Amer. Math. Soc., Providence, RI, Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1999.
  2. Coste A., Dazord P., Weinstein A., Groupoï des symplectiques, in Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A, Vol. 87, University Claude-Bernard, Lyon, 1987, 1-62.
  3. Gay-Balmaz F., Ratiu T.S., The geometric structure of complex fluids, Adv. in Appl. Math. 42 (2009), 176-275, arXiv:0903.4294.
  4. Holm D.D., Marsden J.E., Ratiu T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), 1-81, chao-dyn/9801015.
  5. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.
  6. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  7. Marsden J.E., Ratiu T.S., Weinstein A., Semidirect products and reduction in mechanics, Trans. Amer. Math. Soc. 281 (1984), 147-177.
  8. Pradines J., Remarque sur le groupoïde cotangent de Weinstein-Dazord, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 557-560.
  9. Reyman A.G., Semenov-Tian-Shansky M.A., Group-theoretical methods in the theory of finite-dimensional integrable systems, in Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, Vol. 16, Springer, Berlin - Heidelberg, 1994, 116-225.
  10. Sternberg S., Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field, Proc. Nat. Acad. Sci. USA 74 (1977), 5253-5254.
  11. Weinstein A., A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys. 2 (1977), 417-420.
  12. Weinstein A., Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101-104.

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