Journal of Geometry and Symmetry in Physics

Lie Gyrovector Spaces

Azniv Kasparian and Abraham A. Ungar

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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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J. Geom. Symmetry Phys., Volume 1 (2004), 3-53.

First available in Project Euclid: 23 May 2017

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

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