Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 072, 18 pages      arXiv:0907.2394

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

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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