Cartan connections and Atiyah Lie algebroids

This work extends both classical results on Atiyah Lie algebroids and previous developments carried out by some of the authors on Ehresmann connections on Atiyah Lie algebroids in their algebraic version. In this paper, we study Cartan connections in a framework relying on two Atiyah Lie algebroids based on a H-principal fiber bundle P and its associated G-principal fiber bundle Q≔P×HG, where H⊂G defines the model for a Cartan geometry. Completion of a known commutative and exact diagram relating these two Atiyah Lie algebroids allows to completely characterize Cartan connections on P as a fresh standpoint. Furthermore, in the context of gravity and mixed anomalies, our construction answers a long standing mathematical question about the correct geometrico-algebraic setting in which to combine inner gauge transformations and infinitesimal diffeomorphisms.