Skip to main content
SpringerLink
Log in

Geometric structures encoded in the lie structure of an Atiyah algebroid

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of the Lie algebraic approach to space. First we show that if the Lie algebras of smooth sections of two Atiyah algebroids are isomorphic, then the corresponding base manifolds are necessarily diffeomorphic. Further, we give two characterizations of the isomorphisms of the Lie algebras of sections for Atiyah algebroids associated to principal bundles with semisimple structure groups. For instance we prove that in the semisimple case the Lie algebras of sections are isomorphic if and only if the corresponding Lie algebroids are, or, as well, if and only if the integrating principal bundles are locally isomorphic. Finally, we apply these results to describe the isomorphisms of sections in the case of reductive structure groups—surprisingly enough they are no longer determined by vector bundle isomorphisms and involve dive rgences on the base manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Almeida, P. Molino, Suites d'Atiyah et feuilletages transversalement complets, C. R. Acad. Sci. Paris Ser. I, Math. 300 (1985), 13–15.

    MATH  MathSciNet  Google Scholar 

  2. I. Amemiya, K. Masuda, K. Shiga, Lie algebras of differential operators, Osaka J. Math. 12 (1975), 139–172.

    MATH  MathSciNet  Google Scholar 

  3. M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. J. Duistermaat, I. M. Singer, Order-preserving isomorphisms between algebras of pseudo-differential operators, Comm. Pure Appl. Math. 29 (1976), no. 1, 39–47.

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Ehresmann, Catégories topologiques et catégories différentiables, Colloq. Géom. Diff. Globale, Bruxelles, 1959, pp. 137–150.

    Google Scholar 

  6. J. Grabowski, Isomorphisms and ideals of the Lie algebras of vector fields, Invent. math. 50 (1978), 13–33.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Grabowski, Lie algebras of vector fields and generalized foliations, Publ. Math. 37 (1993), 359–367.

    MATH  MathSciNet  Google Scholar 

  8. J. Grabowski, Isomorphisms of algebras of smooth functions revisited, Arch. Math. (Basel) 85 (2005), 190–196.

    MATH  MathSciNet  Google Scholar 

  9. K. Grabowska, J. Grabowski, Lie algebras of Lie algebroids, in: Lie Algebroids and Related Topics in Differential Geometry, J. Kubarski et al. (eds.), Banach Center Publ., Warsaw, Vol. 54, 2001, pp. 43–50.

  10. K. Grabowska, J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor. 41 (2008), 175204, 25 pp.

    Google Scholar 

  11. K. Grabowska, J. Grabowski, P. Urbański, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys. 3 (2006), 559–575.

    Article  MathSciNet  Google Scholar 

  12. J. Grabowski, N. Poncin, Automorphisms of quantum and classical Poisson algebras, Compos. Math. 140 (2004), 511–527.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Grabowski, N. Poncin, Derivations of the Lie algebras of differential operators, Indag. Math. 16 (2005), no. 2, 181–200.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Grabowski, N. Poncin, On quantum and classical Poisson algebras, Banach Center Publ., Warsaw, Vol. 76, 2007, pp. 313–324.

    Article  MathSciNet  Google Scholar 

  15. J. Kubarski, Lie algebroid of a principal bundle, Publ. Dép. Math. Univ. C. Bernard, Nouvelle Ser. A 89-1 (1989), 1–66.

    MathSciNet  Google Scholar 

  16. M. de Léon, J. C. Marrero, E. Martinez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math Gen. 38 (2005), no. 24, 241–308.

    Article  Google Scholar 

  17. P. Libermann, Sur les prolongements des fibrés principaux et groupoïdes différentiables, Analyse Globale (Sem. Math. Superieures, No. 42, Univ. Montreal, Montreal, Que., 1969), Presses Univ. Montreal, Montreal, Que., 1971, pp. 7–108.

  18. K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.

  19. C.-M. Marle, Calculus on Lie algebroids, Lie groupoids and Poisson manifolds, Dissertationes Math. 457 (2008), 57 pp.

    Article  MathSciNet  Google Scholar 

  20. E. Martínez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math. 67 (2001), 295–320.

    Article  MATH  MathSciNet  Google Scholar 

  21. T. Mestdag, A Lie algebroid approach to Lagrangian systems with symmetry, in: Differential Geometry and its Applications, Matfyzpress, Prague, 2005, pp. 523–535.

  22. J. Mrcun, On isomorphisms of algebras of smooth functions, Proc. Amer. Math. Soc. 133 (2005), 3109–3113 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  23. H. Omori, Infinite-Dimensional Lie Transformation Groups, Lecture Notes in Mathematics, Vol. 427, Springer-Verlag, New York, 1976.

  24. N. Poncin, Premier et deuxième espaces de cohomologie de l'algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions, Bull. Soc. Roy. Sci. Liège 67 (1998), no. 6, 291–337.

    MATH  MathSciNet  Google Scholar 

  25. J. Pradines, Théorie de Lie pour les groupoïdes différentiables, Calcul différentiel dans la catégorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris Sér. A 264 (1967), 246–248.

    MathSciNet  Google Scholar 

  26. L. E. Pursell, M. E. Shanks, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc. 5 (1954), 468–472.

    Article  MATH  MathSciNet  Google Scholar 

  27. С. М. Скрябин, Регулярные алгебры Лu дuфференцuрованuй коммуmаmuвных колец, препринт ВИНИТИ 4403–W87, 1987. [S. M. Skryabin, The regular Lie rings of derivations of commutative rings, preprint VINITI 4403-W87, 1987 (Russian)].

  28. A. Weinstein, Groupoids: unifying internal and external symmetry, Amer. Math. Soc. Not. 43 (1996), 744–752.

    MATH  Google Scholar 

  29. A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Commun. 7 (1996), 207–231.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, P.O. Box 21 00-956, Warsaw, Poland

    Janusz Grabowski

  2. University of Luxembourg, Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg City, Grand-Duchy of Luxembourg

    Alexei Kotov & Norbert Poncin

Authors
  1. Janusz Grabowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexei Kotov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Norbert Poncin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Janusz Grabowski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Grabowski, J., Kotov, A. & Poncin, N. Geometric structures encoded in the lie structure of an Atiyah algebroid. Transformation Groups 16, 137–160 (2011). https://doi.org/10.1007/s00031-011-9126-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-011-9126-9

Keywords

Search

Navigation