A Discrete Theory of Connections on Principal Bundles

 

MELVIN LEOK
Control and Dynamical Systems
California Institute of Technology
Pasadena, CA 91125-8100, USA
mleok@cds.caltech.edu

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Abstract:

Motivated by applications to Discrete Lagrangian Reduction, we consider the discrete analogue of the Atiyah seq uence of a principal bundle, and relate a splitting of the discrete Atiyah sequence with discrete horizontal lifts and discrete connection forms. Continuous connections can be obtained by taking the limit of discrete connections in a natural way.

This yields an isomorphism between $(Q\times Q)/G$ and $\tilde{G}\oplus(S\times S)$. Both the discrete connection and the associated continuous connection are necessary to express the Discrete Lagrange-Poincare operator in coordinates.

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