Geometric structures encoded in the lie structure of an Atiyah algebroid |
| |
Authors: | Janusz Grabowski Alexei Kotov Norbert Poncin |
| |
Institution: | (1) Centre de Math?matiques Laurent Schwartz, ?cole Polytechnique, 91128 Palaiseau, France |
| |
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. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|