2024-02-21T17:20:49Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/716992023-08-27T11:55:22Zcom_20.500.14352_14col_20.500.14352_15
00925njm 22002777a 4500
dc
Castrillón López, Marco
author
Rodríguez Abella, Álvaro
author
2022
In this work, generalized principal bundles modelled by Lie group bundle actions are investigated.In particular, definition of equivariant connections in these bundles, associated to Lie group bundle connections, is provided, together with the analysis of their existence and their main properties. The final part gives some examples. In particular, since this research was initially originated by some problems on geometric reduction of gauge field theories, we revisit the classical Utiyama Theorem from the perspective investigated in the article.
[1] K. Ajaykumar and B.S. Kiranagi. Lie algebra bundles and a generalization of Lie group bundles. Proc. Jangjeon Math. Soc., 22(4):529–542, 2019.
[2] F. Casas, A. Murua, and M. Nadinic. Efficient computation of the Zassenhaus formula. Computer Physics Communications, 183(11):2386 – 2391, 2012.
[3] M. Castrillón López, P.L. García Pérez, and T.S. Ratiu. Euler–Poincaré reduction on principal bundles. Letters in Mathematical Physics, 58, 11 2001.
[4] M. Castrillón López and T.S. Ratiu. Reduction in principal bundles: Covariant Lagrange-Poincaré equations. Communications in Mathematical Physics, 236, 01 2003.
[5] H. Cendra, J.E. Marsden, and T.S. Ratiu. Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems, pages 221–273. Springer Berlin Heidelberg, 2001.
[6] H. Cendra, J.E. Marsden, and T.S. Ratiu. Lagrangian reduction by stages. Mem. Amer. Math. Soc. V, 152(722), 07 2001.
[7] B. Costa, M. Forger, and L. Pêgas. Lie groupoids in classical field theory I: Noether’s theorem. Journal of Geometry and Physics, 131, 08 2015.
[8] M. del Hoyo and R. Loja Fernandes. Riemannian metrics on Lie groupoids. J. Reine und angewandte Mathematik, 2018(735):143–173, 2018.
[9] A. Douady and M. Lazard. Espaces fibrés en algèbres de Lie et en groupes. Invent. Math., 1:133–151, 1966.
[10] D.C.P. Ellis, F. Gay-Balmaz, D.D. Holm, and T.S. Ratiu. Lagrange-Poincaré field equations. Journal of Geometry and Physics, 61(11):2120 – 2146, 2011.
[11] M. Forger and B.L. Soares. Local symmetries in gauge theories in a finite-dimensional setting. Journal of Geometry and Physics, 62(9):1925 – 1938, 2012.
[12] P.L. García Pérez. Gauge algebras, curvature and symplectic structure. J. Differential Geom., 12(2):209–227, 1977.
[13] S. Kobayashi and K. Nomizu. Foundations of Differential Geometry. Number v. 1 in Foundations of Differential Geometry. Interscience Publishers, 1963.
[14] I. Kolář, P.W. Michor, and J. Slovák. Natural Operations in Differential Geometry. Springer-Verlag Berlin Heidelberg, 1st edition, 1993.
[15] J. Lee. Introduction to Smooth Manifolds. Springer-Verlag New York, 2nd edition, 2012.
[16] K.H.C. Mackencie. General Theory of Lie Groupoids and Lie Algebroids. Cambridge niversity Press, Cambridge, 20052.
[17] K.H.C. Mackenzie. Classification of principal bundles and Lie groupoids with prescribed gauge group bundle. J. Pure Appl. Algebra, 58(2):181–208, 1989.
[18] J.E. Marsden and J. Scheurle. The reduced Euler-Lagrange equations. Fields Institute Communications, 1, 07 1993.
[19] Á. Rodríguez Abella. Covariant reduction by fiberwise actions in classical field theory. TEMat monográficos, 2: 211–214, 2021.
[20] G. Sardanashvily. Advanced Differential Geometry for Theoreticians. Lap Lambert cademic Publishing GmbH KG, 2013.
[21] D.J. Saunders. The Geometry of Jet Bundles. Cambridge University Press, 1989.
https://hdl.handle.net/20.500.14352/71699
Principal bundles and connections modelled by Lie group bundles