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


SIGMA 13 (2017), 059, 22 pages      arXiv:1507.05405      https://doi.org/10.3842/SIGMA.2017.059

Remarks on Contact and Jacobi Geometry

Andrew James Bruce a, Katarzyna Grabowska b and Janusz Grabowski c
a) Mathematics Research Unit, University of Luxembourg, Luxembourg
b) Faculty of Physics, University of Warsaw, Poland
c) Institute of Mathematics, Polish Academy of Sciences, Poland

Received January 16, 2017, in final form July 17, 2017; Published online July 26, 2017

Abstract
We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontrivial line bundles and drastically simplifies the picture of Jacobi and contact geometry. Our results easily reduce to various basic theorems of Jacobi and contact geometry when the principal bundle structure is trivial, while giving new insights into the theory.

Key words: symplectic structures; contact structures; Poisson structures; Jacobi structures; principal bundles; Lie groupoids; symplectic groupoids.

pdf (476 kb)   tex (35 kb)

References

  1. Brown R., Spencer C.B., $G$-groupoids, crossed modules and the fundamental groupoid of a topological group, Nederl. Akad. Wetensch. Proc. Ser. A 79 (1976), 296-302.
  2. Bruce A.J., Grabowska K., Grabowski J., Graded bundles in the category of Lie groupoids, SIGMA 11 (2015), 090, 25 pages, arXiv:1502.06092.
  3. Bruce A.J., Grabowska K., Grabowski J., Higher order mechanics on graded bundles, J. Phys. A: Math. Theor. 48 (2015), 205203, 32 pages, arXiv:1412.2719.
  4. Bruce A.J., Grabowska K., Grabowski J., Linear duals of graded bundles and higher analogues of (Lie) algebroids, J. Geom. Phys. 101 (2016), 71-99, arXiv:1409.0439.
  5. Bruce A.J., Grabowski J., Rotkiewicz M., Polarisation of graded bundles, SIGMA 12 (2016), 106, 30 pages, arXiv:1512.02345.
  6. Bruce A.J., Tortorella A.G., Kirillov structures up to homotopy, Differential Geom. Appl. 48 (2016), 72-86, arXiv:1507.00454.
  7. Bursztyn H., Cabrera A., del Hoyo M., Vector bundles over Lie groupoids and algebroids, Adv. Math. 290 (2016), 163-207, arXiv:1410.5135.
  8. Crainic M., Fernandes R.L., Integrability of Lie brackets, Ann. of Math. 157 (2003), 575-620, math.DG/0105033.
  9. Crainic M., Salazar M.A., Jacobi structures and Spencer operators, J. Math. Pures Appl. 103 (2015), 504-521, arXiv:1309.6156.
  10. Crainic M., Zhu C., Integrability of Jacobi and Poisson structures, Ann. Inst. Fourier (Grenoble) 57 (2007), 1181-1216, math.DG/0403268.
  11. Dazord P., Intégration d'algèbres de Lie locales et groupoïdes de contact, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 959-964.
  12. Dazord P., Sur l'intégration des algèbres de Lie locales et la préquantification, Bull. Sci. Math. 121 (1997), 423-462.
  13. Fernandes R.L., Iglesias Ponte D., Integrability of Poisson-Lie group actions, Lett. Math. Phys. 90 (2009), 137-159, arXiv:0902.3558.
  14. Grabowski J., Modular classes of skew algebroid relations, Transform. Groups 17 (2012), 989-1010, arXiv:1108.2366.
  15. Grabowski J., Graded contact manifolds and contact Courant algebroids, J. Geom. Phys. 68 (2013), 27-58, arXiv:1112.0759.
  16. Grabowski J., Marmo G., Jacobi structures revisited, J. Phys. A: Math. Gen. 34 (2001), 10975-10990, math.DG/0111148.
  17. Grabowski J., Marmo G., The graded Jacobi algebras and (co)homology, J. Phys. A: Math. Gen. 36 (2003), 161-181, math.DG/0207017.
  18. Grabowski J., Rotkiewicz M., Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. 59 (2009), 1285-1305, math.DG/0702772.
  19. Grabowski J., Rotkiewicz M., Graded bundles and homogeneity structures, J. Geom. Phys. 62 (2012), 21-36, arXiv:1102.0180.
  20. Ida C., Popescu P., Contact structures on Lie algebroids, arXiv:1507.01110.
  21. Iglesias D., Marrero J.C., Generalized Lie bialgebroids and Jacobi structures, J. Geom. Phys. 40 (2001), 176-199, math.DG/0008105.
  22. Iglesias-Ponte D., Marrero J.C., Jacobi groupoids and generalized Lie bialgebroids, J. Geom. Phys. 48 (2003), 385-425, math.DG/0208032.
  23. Iglesias-Ponte D., Marrero J.C., Locally conformal symplectic groupoids, in Proceedings of the XI Fall Workshop on Geometry and Physics, Publ. R. Soc. Mat. Esp., Vol. 6, R. Soc. Mat. Esp., Madrid, 2004, 93-102, math.DG/0301108.
  24. Jóźwikowski M., Rotkiewicz M., A note on actions of some monoids, Differential Geom. Appl. 47 (2016), 212-245, arXiv:1602.02028.
  25. Karasev M.V., Analogues of the objects of Lie group theory for nonlinear Poisson brackets, Math. USSR-Izv. 28 (1987), 497-527.
  26. Kerbrat Y., Souici-Benhammadi Z., Variétés de Jacobi et groupoïdes de contact, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 81-86.
  27. Kirillov A.A., Local Lie algebras, Russian Math. Surveys 31 (1976), no. 4, 55-75.
  28. Lê H.V., Oh Y.-G., Tortorella A.G., Vitagliano L., Deformations of coisotropic submanifolds in abstract Jacobi manifolds, arXiv:1410.8446.
  29. Libermann P., On symplectic and contact groupoids, in Differential Geometry and its Applications (Opava, 1992), Math. Publ., Vol. 1, Silesian University Opava, Opava, 1993, 29-45.
  30. Libermann P., On contact groupoids and their symplectification, in Analysis and Geometry in Foliated Manifolds (Santiago de Compostela, 1994), World Sci. Publ., River Edge, NJ, 1995, 153-176.
  31. Lichnerowicz A., Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. 57 (1978), 453-488.
  32. Mackenzie K.C.H., Integrability obstructions for extensions of Lie algebroids, Cahiers Topologie Géom. Différentielle Catég. 28 (1987), 29-52.
  33. Mackenzie K.C.H., On extensions of principal bundles, Ann. Global Anal. Geom. 6 (1988), 141-163.
  34. 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.
  35. Mackenzie K.C.H., Xu P., Integration of Lie bialgebroids, Topology 39 (2000), 445-467, dg-ga/9712012.
  36. Marle C.-M., On Jacobi manifolds and Jacobi bundles, in Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., Vol. 20, Springer, New York, 1991, 227-246.
  37. Mehta R.A., Differential graded contact geometry and Jacobi structures, Lett. Math. Phys. 103 (2013), 729-741, arXiv:1111.4705.
  38. Moerdijk I., Mrčun J., On integrability of infinitesimal actions, Amer. J. Math. 124 (2002), 567-593, math.DG/0006042.
  39. Stachura P., $C^*$-algebra of a differential groupoid (with an appendix by S. Zakrzewski), in Poisson Geometry (Warsaw, 1998), Banach Center Publ., Vol. 51, Polish Acad. Sci. Inst. Math., Warsaw, 2000, 263-281.
  40. Stefanini L., On the integration of $\mathcal{LA}$-groupoids and duality for Poisson groupoids, Trav. Math., Vol. 17, Fac. Sci. Technol. Commun. Univ. Luxemb., Luxembourg, 2007, 39-59, math.DG/0701231.
  41. Stefanini L., On morphic actions and integrability of $\mathcal{LA}$-groupoids, arXiv:0902.2228.
  42. Vitagliano L., Dirac-Jacobi bundles, arXiv:1502.05420.
  43. Vitagliano L., Wade A., Generalized contact bundles, C. R. Math. Acad. Sci. Paris 354 (2016), 313-317, arXiv:1507.03973.
  44. Weinstein A., Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101-104.
  45. Weinstein A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705-727.
  46. Zakrzewski S., Quantum and classical pseudogroups. I. Union pseudogroups and their quantization, Comm. Math. Phys. 134 (1990), 347-370.
  47. Zakrzewski S., Quantum and classical pseudogroups. II. Differential and symplectic pseudogroups, Comm. Math. Phys. 134 (1990), 371-395.

Previous article  Next article   Contents of Volume 13 (2017)