Annals of Global Analysis and Geometry

, Volume 48, Issue 1, pp 87–123 | Cite as

The Lie group of bisections of a Lie groupoid

  • Alexander Schmeding
  • Christoph Wockel


In this article, we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor.


Global analysis Lie groupoid Infinite-dimensional Lie group  Mapping space Local addition Bisection Regularity of Lie groups 

Mathematics Subject Classification

Primary: 22E65 Secondary: 58H05 46T10 58D05 



The research on this paper was partially supported by the DFG Research Training group 1670 Mathematics inspired by String Theory and Quantum Field Theory, the Scientific Network String Geometry (DFG project code NI 1458/1-1) and the project Topology in Norway (Norwegian Research Council project 213458).


  1. 1.
    Amann, H.: Ordinary Differential Equations. Studies in Mathematics, vol. 13. de Gruyter, Berlin (1990)CrossRefzbMATHGoogle Scholar
  2. 2.
    Alzaareer, H., Schmeding, A.: Differentiable mappings on products with different degrees of differentiability in the two factors. Expo. Math. (2014). doi: 10.1016/j.exmath.2014.07.002. arXiv:1208.6510
  3. 3.
    Bertram, W., Glöckner, H., Neeb, K.-H.: Differential calculus over general base fields and rings. Expo. Math. 22(3), 213–282 (2004). doi: 10.1016/S0723-0869(04)80006-9. arXiv:math/0303300
  4. 4.
    Dahmen, R.: Direct Limit Constructions in Infinite Dimensional Lie Theory. Ph.D. thesis (2012).
  5. 5.
    Engelking, R.: General Topology. Sigma Series in Pure Mathematics, vol. 6. Heldermann, Berlin (1989)zbMATHGoogle Scholar
  6. 6.
    Fiorenza, D., Rogers, C.L., Schreiber, U.: Higher geometric prequantum theory 2013. arXiv:1304.0236
  7. 7.
    Fiorenza, D., Rogers, C.L., Schreiber, U.: L-infinity algebras of local observables from higher prequantum bundles 2013. arXiv:1304.6292
  8. 8.
    Glöckner, H.: Infinite-dimensional Lie groups without completeness restrictions. In: Strasburger, A., Hilgert, J., Neeb, K., Wojtyński, W. (eds.) Geometry and Analysis on Lie Groups, vol. 55, pp. 43–59. Banach Center Publications, Warsaw (2002)Google Scholar
  9. 9.
    Glöckner, H.: Regularity properties of infinite-dimensional Lie groups. Oberwolfach Rep. 13, 791–794 (2013). doi: 10.4171/OWR/2013/13 Google Scholar
  10. 10.
    Glöckner, H.: Regularity properties of infinite-dimensional Lie groups, and semiregularity 2015. arXiv:1208.0715
  11. 11.
    Haller, S., Teichmann, J.: Smooth perfectness through decomposition of diffeomorphisms into fiber preserving ones. Ann. Glob. Anal. Geom. 23(1), 53–63 (2003)Google Scholar
  12. 12.
    Huebschmann, J.: Poisson cohomology and quantization. J. Reine Angew. Math. 408, 57–113 (1990). doi: 10.1515/crll.1990.408.57 zbMATHMathSciNetGoogle Scholar
  13. 13.
    Kriegl, A., Michor, P.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)Google Scholar
  14. 14.
    Kosmann-Schwarzbach, Y., Magri, F.: Poisson–Nijenhuis structures. Ann. Inst. H. Poincaré Phys. Théor. 53(1), 35–81 (1990).
  15. 15.
    Lang, S.: Differential and Riemannian Manifolds, 3rd edn. Graduate Texts in Mathematics, vol. 160. Springer, New York (1995)Google Scholar
  16. 16.
    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)CrossRefzbMATHGoogle Scholar
  17. 17.
    Michor, P.W.: Manifolds of Differentiable Mappings. Shiva Mathematics Series, vol. 3. Shiva Publishing Ltd., Nantwich (1980).
  18. 18.
    Milnor, J.: On Infinite-Dimensional Lie Groups. Institute for Advanced Study, Princeton (1982, preprint)Google Scholar
  19. 19.
    Milnor, J.: Remarks on infinite-dimensional Lie groups. In: Relativity, Groups and Topology, II (Les Houches, 1983), pp. 1007–1057. North-Holland, Amsterdam (1984)Google Scholar
  20. 20.
    Neeb, K.-H.: Towards a Lie theory of locally convex groups. Jpn. J. Math. 1(2), 291–468 (2006)Google Scholar
  21. 21.
    Nishimura, H.: The lie algebra of the group of bisections 2006. arXiv:math/0612053
  22. 22.
    Neeb, K.-H., Salmasian, H.: Differentiable vectors and unitary representations of frechet-lie supergroups 2012. arXiv:1208.2639
  23. 23.
    Nikolaus, T., Sachse, C., Wockel, C.: A smooth model for the string group. Int. Math. Res. Not. IMRN (16), 3678–3721 (2013). doi: 10.1093/imrn/rns154. arXiv:1104.4288
  24. 24.
    Rybicki, T.: On the group of Lagrangian bisections of a symplectic groupoid. In: Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000). Banach Center Publ., vol. 54, pp. 235–247. Polish Academy of Sciences, Warsaw (2001)Google Scholar
  25. 25.
    Rybicki, T.: A Lie group structure on strict groups. Publ. Math. Debr. 61(3–4), 533–548 (2002)Google Scholar
  26. 26.
    Schreiber, U.: Differential cohomology in a cohesive infinity-topos 2013. arXiv:1310.7930
  27. 27.
    da Silva, A.C., Weinstein, A.: Geometric Models for Noncommutative algebras. Berkeley Mathematics Lecture Notes, vol. 10. American Mathematical Society/Berkeley Center for Pure and Applied Mathematics, Providence/Berkeley (1999)Google Scholar
  28. 28.
    Wockel, C.: Lie group structures on symmetry groups of principal bundles. J. Funct. Anal. 251(1), 254–288 (2007). doi: 10.1016/j.jfa.2007.05.016. arXiv:math/0612522
  29. 29.
    Wockel, C.: Infinite-Dimensional and Higher Structures in Differential Geometry. Lecture Notes for a Course Given at the University of Hamburg 2013.
  30. 30.
    Xu, P.: Flux homomorphism on symplectic groupoids. Math. Z. 226(4), 575–597 (1997). doi: 10.1007/PL00004355. arXiv:dg-ga/9605003

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.NTNU TrondheimTrondheimNorway
  2. 2.University of HamburgHamburgGermany

Personalised recommendations