Möbius gyrogroups: A Clifford algebra approach

M Ferreira,G Ren

doi:10.1016/j.jalgebra.2010.05.014

Abstract

Using the Clifford algebra formalism we study the Möbius gyrogroup of the ball of radius t of the paravector space R ⊕ V , where V is a finite-dimensional real vector space. We characterize all the gyro-subgroups of the Möbius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyro-subgroup for the paravector ball. The geometric and algebraic properties of the equivalence classes are investigated. We show that the equivalence classes locate in a k-dimensional sphere, where k is the dimension of the gyro-subgroup, and the resulting quotient spaces are again Möbius gyrogroups. With the algebraic structure of the factorizations we study the sections of Möbius fiber bundles inherited by the Möbius projectors.

Highlights

The Möbius gyrogroup plays an important role in the theory of grogroups since it provides a concrete model for the abstract theory
Its study leads to a better understanding of Lorentz transformations from the special relativity theory since the Lorentz group acts on ball of all possible symmetric velocities via conformal maps [19, 9]
The Möbius gyrogroup is associated with the Poincaré model of conformal geometry known as the rapidity space [15], because the Poincaré distance from the origin of the ball to any point on the ball coincides with the rapidity of a boost

Summary

Introduction

The Möbius gyrogroup plays an important role in the theory of grogroups since it provides a concrete model for the abstract theory. Employing analogies shared by complex numbers and linear transformations of vector spaces Ungar extended in [21] the Möbius addition in the complex disk to the ball of an arbitrary real inner product space. The advantage of our approach lies in the fact that Möbius gyrogroups of the ball of a real inner product space is analogous to the corresponding theory in the unit disc by an algebraic formalism. Starting from an arbitrary real inner product space of dimension n we embed it into the Cliord algebra Cl0,n and we construct the paravector space R ⊕ V of Cl0,n, the direct sum of scalars and vectors This paravector space will be the environment for studying the Möbius gyrogroup on the ball. In [5] the author used the approach of gyrogroups encoded in the conformal group of the unit sphere in Rn, the so called proper Lorentz group, to dene spherical continuous wavelet transforms on the unit sphere via sections of a quotient Möbius gyrogroup

Gyrogroups

Cliord algebras

Möbius addition in the paravector ball

Möbius orthogonal projectors

Quotient Möbius gyrogroups

Möbius ber bundles