On Möbius gyrogroup and Möbius gyrovector space

Abstract

Gyrogroups are new algebraic structures that appeared in 1988 in the study of Einstein’s velocity addition in the special relativity theory. These new algebraic structures were studied intensively by Abraham Ungar. The first gyrogroup that was considered into account is the unit ball of Euclidean space \mathbb{R}^3ℝ3 endowed with Einstein’s velocity addition. The second geometric example of a gyrogroup is the complex unit disk \mathbb{D}𝔻={z ∈ \mathbb{C}: |z|<1ℂ:|z|<1}. To construct a gyrogroup structure on \mathbb{D}𝔻, we choose two elements z_1, z_2 ∈\mathbb{D}z1,z2∈𝔻 and define the Möbius addition by z_1\oplus z_2 = \frac{z_1+z_2}{1+\bar{z_1}z_2}z1⊕z2=z1+z21+z1‾z2. Then (\mathbb{D},\oplus)(𝔻,⊕) is a gyrocommutative gyrogroup. If we define r \odot xr⊙x==\frac{(1+|x|)^r - (1-|x|)^r}{(1+|x|)^r + (1-|x|)^r}\frac{x}{|x|}(1+|x|)r−(1−|x|)r(1+|x|)r+(1−|x|)rx|x|, where x ∈ \mathbb{D}x∈𝔻 and r ∈ \mathbb{R}r∈ℝ, then (\mathbb{D},\oplus,\odot)(𝔻,⊕,⊙) will be a real gyrovector space. This paper aims to survey the main properties of these Möbius gyrogroup and Möbius gyrovector space.