caa a22 4500 445862289 CHVBK 20180317145447.0 cr unu---uuuuu 170323e20110301xx s 000 0 eng 10.1007/s00031-011-9126-9 doi (NATIONALLICENCE)springer-10.1007/s00031-011-9126-9 Geometric structures encoded in the lie structure of an Atiyah algebroid [Elektronische Daten] [Janusz Grabowski, Alexei Kotov, Norbert Poncin] 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. Springer Science+Business Media, LLC, 2011 Grabowski Janusz Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, P.O. Box21 00-956, Warsaw, Poland aut Kotov Alexei University of Luxembourg, Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg City, Grand-Duchy of Luxembourg aut Poncin Norbert University of Luxembourg, Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg City, Grand-Duchy of Luxembourg aut Transformation Groups SP Birkhäuser Verlag Boston 16/1(2011-03-01), 137-160 1083-4362 16:1<137 2011 16 31 https://doi.org/10.1007/s00031-011-9126-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00031-011-9126-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Grabowski Janusz Polish Academy of Sciences, Institute of Mathematics, Śniadeckich 8, P.O. Box21 00-956, Warsaw, Poland aut NATIONALLICENCE 700 1- Kotov Alexei University of Luxembourg, Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg City, Grand-Duchy of Luxembourg aut NATIONALLICENCE 700 1- Poncin Norbert University of Luxembourg, Mathematics Research Unit 6, rue Richard Coudenhove-Kalergi, L-1359, Luxembourg City, Grand-Duchy of Luxembourg aut NATIONALLICENCE 773 0- Transformation Groups SP Birkhäuser Verlag Boston 16/1(2011-03-01), 137-160 1083-4362 16:1<137 2011 16 31 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer