Transformation Groups

, Volume 16, Issue 1, pp 137–160

Geometric structures encoded in the lie structure of an Atiyah algebroid

Authors

    • Polish Academy of SciencesInstitute of Mathematics
  • Alexei Kotov
    • University of Luxembourg
  • Norbert Poncin
    • University of Luxembourg
Article

DOI: 10.1007/s00031-011-9126-9

Cite this article as:
Grabowski, J., Kotov, A. & Poncin, N. Transformation Groups (2011) 16: 137. doi:10.1007/s00031-011-9126-9

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.

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Copyright information

© Springer Science+Business Media, LLC 2011