SIGMA 5 (2009), 072, 18 pages      arXiv:0907.2394      https://doi.org/10.3842/SIGMA.2009.072

Clifford Fibrations and Possible Kinematics

Alan S. McRae
Department of Mathematics, Washington and Lee University, Lexington, VA 24450-0303, USA

Received April 10, 2009, in final form June 19, 2009; Published online July 14, 2009

Abstract
Following Herranz and Santander [Herranz F.J., Santander M., Mem. Real Acad. Cienc. Exact. Fis. Natur. Madrid 32 (1998), 59-84, physics/9702030] we will construct homogeneous spaces based on possible kinematical algebras and groups [Bacry H., Levy-Leblond J.-M., J. Math. Phys. 9 (1967), 1605-1614] and their contractions for 2-dimensional spacetimes. Our construction is different in that it is based on a generalized Clifford fibration: Following Penrose [Penrose R., Alfred A. Knopf, Inc., New York, 2005] we will call our fibration a Clifford fibration and not a Hopf fibration, as our fibration is a geometrical construction. The simple algebraic properties of the fibration describe the geometrical properties of the kinematical algebras and groups as well as the spacetimes that are derived from them. We develop an algebraic framework that handles all possible kinematic algebras save one, the static algebra.

Key words: Clifford fibration; Hopf fibration; kinematic.

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References

1. Bacry H., Levy-Leblond J.-M., Possible kinematics, J. Math. Phys. 9 (1967), 1605-1614.
2. Ballesteros A., Herranz F.J., Superintegrability on three-dimensional Riemannian and relativistic spaces of constant curvature, SIGMA 2 (2006), 010, 22 pages, math-ph/0512084.
3. Harkin A.A., Harkin J.B., Geometry of generalized complex numbers, Math. Mag. 77 (2004), 118-129.
4. Herranz F.J., Ortega R., Santander M., Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry, J. Phys. A: Math. Gen. 33 (2000), 4525-4551, math-ph/9910041.
5. Herranz F.J., Santander M., Homogeneous phase spaces: the Cayley-Klein framerwork, in Geometría y Física (Madrid, Real Academia de Ciencias, 1996), Editors J.F. Cariñena et al., Mem. Real Acad. Cienc. Exact. Fis. Natur. Madrid 32 (1998), 59-84, physics/9702030.
6. Herranz F.J., Santander M., Conformal symmetries of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6601-6618, math-ph/0110019.
7. McRae A.S., The Gauss-Bonnet theorem for Cayley-Klein geometries of dimension two, New York J. Math. 12 (2006), 143-155.
8. McRae A.S., Clifford algebras and possible kinematics, SIGMA 3 (2007), 079, 29 pages, arXiv:0707.2869.
9. Penrose R., The road to reality. A complete guide to the laws of the universe, Alfred A. Knopf, Inc., New York, 2005.
10. Urbantke H.K., The Hopf fibration - seven times in physics, J. Geom. Phys. 46 (2003), 125-150.
11. Van der Waerden B.L., Algebra, Vol. 2, Springer-Verlag, New York, 1991.
12. Yaglom I.M., A simple non-Euclidean geometry and its physical basis: an elementary account of Galilean geometry and the Galilean principle of relativity, Heidelberg Science Library, Translated from the Russian by Abe Shenitzer, With the editorial assistance of Basil Gordon, Springer-Verlag, New York - Heidelberg, 1979.