Publication: Higher order jet bundels of lie group-valued functions
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2020
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Rodríguez Abella, Álvaro
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For each positive integer k, the bundle of k-jets of functions from a smooth manifold, X, to a Lie group, G, is denoted by Jk(X,G) and it is canonically endowed with a Lie groupoid structure over X. In this work, we utilize a linear connection to trivialize this bundle, i.e., to build an injective bundle morphism from Jk(X,G) into a vector bundle over G. Afterwards, we give the explicit expression of the groupoid multiplication on the trivialized space, as well as the formula for the inverse element. In the last section, a coordinated chart on X is considered and the local expression of the trivialization is computed.
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[1] L. Colombo and D. Martín de Diego. Higher-order variational problems on Lie groups and
optimal control applications. Journal of Geometric Mechanics, 6(4):451–478, 2014.
[2] O. Esen, M. Kudeyt, and S. Sütlü. Second order Lagrangian dynamics on double
cross product groups. Journal of Geometry and Physics, 159:103934, 2021.
ISSN 0393-0440. doi: https://doi.org/10.1016/j.geomphys.2020.103934. URL
https://www.sciencedirect.com/science/article/pii/S0393044020302217.
[3] F. Gay-Balmaz, D. Holm, D. Meier, T. Ratiu, and F.-X. Vialard. Invariant higher-order variational
problems. Commun. Math. Phys., 309:413–458, 2012. doi: 10.1007/s00220-011-1313-y.
[4] B. Hall. Lie Groups, Lie Algebras, and Representations. Graduate Texts in Mathematics.
Springer, Cham, second edition, 2015. doi: https://doi.org/10.1007/978-3-319-13467-3.
[5] L. Mangiarotti and G. Sardanashvily. Connections in Classical and Quan-tum Field Theory. World Scientific, 2000. doi: 10.1142/2524. URL
https://www.worldscientific.com/doi/abs/10.1142/2524.
[6] D. J. Saunders. The geometry of jet bundles. Cambridge University Press, 1989.
[7] V. S. Varadarajan. Lie Groups, Lie Algebras, and Their Representations. Graduate Texts in Mathematics. Springer, New York, NY, 1 edition, 1984. ISBN 978-1-4612-1126-6. doi: https://doi.org/10.1007/978-1-4612-1126-6.
[8] C. Vizman. The group structure for jet bundles over Lie groups. Journal of Lie Theory, 23(3),
2013.