Abstract
In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.
Highlights
Gyrogroups are a suitable generalization of groups, arising from the study of the parametrization of the Lorentz transformation group made by Ungar in [1]
The three main gyrogroups associated with analytic hyperbolic geometry are Möbius, Einstein, and Proper Velocity gyrogroups
Factorizations of Möbius gyrogroups were first studied in [4,5]. These factorizations were used for defining continuous wavelet transforms on the unit sphere, associated to appropriate sections in the quotient Möbius gyrogroup [6]
Summary
Gyrogroups are a suitable generalization of groups, arising from the study of the parametrization of the Lorentz transformation group made by Ungar in [1]. The aim of this paper is to give a general theory for real inner product gyrogroups that encode a real gyrovector space structure imposing only two conditions (see Section 3). In this way, we unify the factorization theory for some well-known gyrogroups in the literature that are examples of real inner product gyrogroups such as Einstein, Möbius, Proper Velocity, and Chen’s gyrogroups.
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