K-Loops, gyrogroups and symmetric spaces

Abstract

In this paper we present a new class of K-Loops consisting of homogeneous symmetric spaces of the noncompact type. The most famous physical example of a representative of this class is found in special relativity: The set of all relativistically admissible velocities, R c 3:= {υ ∈ R3: ¦v¦< c} (c the speed of light), together with the relativistic velocity composition law forms a K-Loop, as was already shown by A. Ungar in 1988. The set of the corresponding boosts turns out to be the effective homogeneous symmetric space SO (1,3)0/SO(3), where SO(1,3)0 denotes the proper orthochronous Lorentz group and SO(3) the subgroup of proper rotations in three dimensions, which is also found to be the set of fixed points of an automorphism σ: SO(1,3)0 → SO(1,3)0, whose action on a matrix A in the defining representation is given by σ(A) = (AT)−1. Moreover, the triple (so(l,3),so(3),s) consisting of the Lie algebras of SO(1,3) and SO(3) and the induced automorphism s = (dσ)e is an orthogonal symmetric Lie algebra of the noncompact type. This property of the Lie algebra makes SO(l, 3)/SO(3) an (effective) homogeneous symmetric space of the noncompact type. The method of exact decomposition allows endowing each homogeneous symmetric space of the noncompact type with the structure of a K-Loop in a similar manner. Finally, we treat some alternative approaches which lead to K-Loops that are isomorphic to the one described above.