Geometric structures encoded in the Lie structure of an Atiyah algebroid

TL;DR

This paper explores the geometric and algebraic structures of Atiyah algebroids, showing how Lie algebra isomorphisms relate to the underlying manifolds and principal bundles, especially for semisimple and reductive groups.

Contribution

It provides new characterizations of Lie algebra isomorphisms of Atiyah algebroids and links these to the geometry of the base manifolds and principal bundles.

Findings

Lie algebra isomorphisms imply diffeomorphic base manifolds

Characterizations for semisimple structure groups

Reductive groups involve divergences in isomorphisms

Abstract

We 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.