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Weak Lie symmetry and extended Lie algebra

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### Abstract

The concept of weak Lie motion (weak Lie symmetry) is introduced. Applications given exhibit a reduction of the usual symmetry, e.g., in the case of the rotation group. In this context, a particular generalization of Lie algebras is found (“extended Lie algebras”) which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).

© 2013 American Institute of Physics

Received 11 December 2012
Accepted 04 March 2013
Published online 01 April 2013

Acknowledgments:
My sincere thanks go to A. Papadopoulos for inviting me to the Strasbourg conference at IRMA. Remarks by P. Cartier and Y. Kosmann-Schwarzbach, participants at the conference, were quite helpful. I am also grateful for advice on mathematical concepts like algebroids by H. Seppänen and Ch. Zhu of the Mathematical Institute of the University of Göttingen as well as to my colleague F. Müller-Hoissen, Max Planck Institute for Dynamics and Self-Organization, Göttingen, for a clarifying conceptual discussion. As a historian of mathematics, E. Scholz, University of Wuppertal, did guide me to the relevant historical literature. The presentation did also profit much from the suggestions of a referee.

Article outline:

I. INTRODUCTION
II. LIE-DRAGGING
A. Lie algebra for physicists
B. Lie-dragging
III. MOTIONS AND COLLINEATIONS
IV. WEAK LIE MOTIONS (WEAK SYMMETRIES)
A. Weak symmetries
B. Complete sets of weak Lie motions
V. WEAKLY STATIC AND SPHERICALLY SYMMETRIC METRICS
A. Weakly static metrics
B. Weak spherical symmetry
C. The group *G*_{3} acting as a group of weak Lie motions
VI. A NEW ALGEBRASTRUCTURE
A. Lie-dragging for vector fields not forming Lie algebras
VII. EXTENDED LIE ALGEBRAS
VIII. EXTENDED MOTIONS AND EXTENDED WEAK (LIE) MOTIONS
IX. DISCUSSION AND CONCLUSION

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2013-04-01

2016-02-14

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