Cartan Connections and Atiyah Lie Algebroids

Abstract : 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.
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Contributor : Serge Lazzarini <>
Submitted on : Friday, April 12, 2019 - 11:44:35 AM
Last modification on : Saturday, April 13, 2019 - 1:24:46 AM

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  • HAL Id : hal-02097864, version 1
  • ARXIV : 1904.04915



Jérémy Attard, Jordan François, Serge Lazzarini, Thierry Masson. Cartan Connections and Atiyah Lie Algebroids. 2019. ⟨hal-02097864⟩



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