TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhysProc.14.041
TI  - On Möbius gyrogroup and Möbius gyrovector space
PY  - 2023/11/24
UR  - https://scipost.org/SciPostPhysProc.14.041
JF  - SciPost Physics Proceedings
JA  - SciPost Phys. Proc.
VL  - 
IS  - 14
SP  - 041
A1  - Mavaddat Nezhaad, Kurosh
AU  - Ashrafi, Ali Reza
AB  - 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 $\mathbb{R}^3$ endowed with Einstein's velocity addition. The second geometric example of a gyrogroup is the complex unit disk $\mathbb{D}$={z ∈ $\mathbb{C}: |z|<1$}. To construct a gyrogroup structure on $\mathbb{D}$, we choose two elements $z_1, z_2 ∈ \mathbb{D}$ and define the Möbius addition by $z_1\oplus z_2 = \frac{z_1+z_2}{1+\bar{z_1}z_2}$. Then $(\mathbb{D},\oplus)$ is a gyrocommutative gyrogroup. If we define $r \odot x$ $=$ $\frac{(1+|x|)^r - (1-|x|)^r}{(1+|x|)^r + (1-|x|)^r}\frac{x}{|x|}$, where $x ∈ \mathbb{D}$ and $r ∈ \mathbb{R}$, then $(\mathbb{D},\oplus,\odot)$ will be a real gyrovector space. This paper aims to survey the main properties of these Möbius gyrogroup and Möbius gyrovector space.
ER  -

@Article{10.21468/SciPostPhysProc.14.041,
	title={{On Möbius gyrogroup and Möbius gyrovector space}},
	author={Kurosh Mavaddat Nezhaad and Ali Reza Ashrafi},
	journal={SciPost Phys. Proc.},
	pages={041},
	year={2023},
	publisher={SciPost},
	doi={10.21468/SciPostPhysProc.14.041},
	url={https://scipost.org/10.21468/SciPostPhysProc.14.041},
}