Jeremy Attard, Jordan François, Serge Lazzarini, Thierry Masson
Published 2019-04-09Version 1
This work extends previous developments carried out by some of the authors on Ehresmann connections on Atiyah Lie algebroids. In this paper, we study Cartan connections in a framework relying on two Atiyah Lie algebroids based on a $H$-principal fiber bundle $\mathcal{P}$ and its associated $G$-principal fiber bundle $\mathcal{Q} := \mathcal{P} \times_H G$, where $H \subset G$ defines the model for a Cartan geometry. The first main result of this study is a commutative and exact diagram relating these two Atiyah Lie algebroids, which allows to completely characterize Cartan connections on $\mathcal{P}$. 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.
Path integrals on a manifold that is a product of the total space of the principal fiber bundle and the vector space
Transition to the case of "resolved gauge" in the Lagrange-Poincaré equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle
Coordinate representation of the Lagrange-Poincaré equations for a mechanical system with symmetry on the total space of a principal fiber bundle whose base is the bundle space of the associated bundle
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