Transitive Lie algebroids and $\mathbf{\text{Q}}$-manifolds

We introduce the notion of $\mathbf{\text{Q}}$-principal bundle, which is the appropriate version of principal fibre bundles in the setting of R. Barre’s $\mathbf{\text{Q}}$-manifolds. As an application, we prove that every transitive Lie algebroid arises from the Atiyah sequence of a certain $\mathbf{\text{Q}}$-principal bundle, and we give the interpretation of this result in terms of groupoids.