Transitive Lie algebroids and \textbf{Q}-manifolds

Abstract

We introduce the notions of locally trivial \textbf{Q}-groupoid and principal \textbf{Q}-bundle, which are the appropriate versions of locally trivial Lie groupoids and principal fibre bundles in the framework of R. Barre's \textbf{Q}-manifolds. As an application, we prove that every transitive Lie algebroid over a second countable, smooth manifold arises from the Atiyah sequence of a certain principal \Q-bundle. Consequently, transitive Lie algebroids over second countable, smooth manifolds are integrated to locally trivial \textbf{Q}-groupoids.