Geometric structures encoded in the Lie structure of an Atiyah algebroid

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dc.creatorGrabowski, Janusz
dc.creatorKotov, Alexei
dc.creatorPoncin, Norbert
dc.date2009-05-08
dc.date.accessioned2026-07-07T13:13:02Z
dc.date.available2026-07-07T13:13:02Z
dc.descriptionWe investigate Atiyah algebroids, i.e. the infinitesimal objects of principal bundles, from the viewpoint of 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 principle 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 diffeomorphic. 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 divergences on the base manifolds.
dc.description22 pages, LaTeX
dc.identifierhttps://arxiv.org/abs/0905.1226
dc.identifierhttp://arxiv.org/abs/0905.1226
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/229761
dc.subjectDifferential Geometry
dc.subjectRings and Algebras
dc.subject17B40, 55R10, 57R50, 58H05 (Primary), 16S60, 17B20, 53D17 (Secondary)
dc.titleGeometric structures encoded in the Lie structure of an Atiyah algebroid
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