Open Access
2004 Lie Gyrovector Spaces
Azniv Kasparian, Abraham A. Ungar
J. Geom. Symmetry Phys. 1: 3-53 (2004). DOI: 10.7546/jgsp-1-2004-3-53


An arbitrary section of the canonical projection of a group onto the cosets modulo a subgroup is associated with a binary operation on the cosets. We provide sufficient conditions for obtaining a left loop, a left gyrogroup or a gyrocommutative gyrogroup in such a way. The non-positively curved sections in Lie groups allow a scalar multiplication, which turns them into quasi left Lie gyrovector spaces. The left invariant metrics on homogeneous spaces turn out to be compatible with the gyro-structure. For instance, their geodesics are gyro-lines; the associated distance to the origin is a gyro-homogeneous norm, satisfying gyro-triangle inequality; etc. The work establishes infinitesimal criteria for a homogeneous space to bear a left Lie gyrovector space or a Lie gyrovector space structure. It characterizes the Cartan gyrovector spaces and works out explicitly the example of the upper half-plane.


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Azniv Kasparian. Abraham A. Ungar. "Lie Gyrovector Spaces." J. Geom. Symmetry Phys. 1 3 - 53, 2004.


Published: 2004
First available in Project Euclid: 23 May 2017

zbMATH: 1071.83001
MathSciNet: MR2096566
Digital Object Identifier: 10.7546/jgsp-1-2004-3-53

Rights: Copyright © 2004 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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