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On Möbius Gyrogroup and Möbius Gyrovector Space
by Kurosh Mavaddat Nezhaad and Ali Reza Ashrafi
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Submission summary
Authors (as registered SciPost users):  Kurosh Mavaddat Nezhaad 
Submission information  

Preprint Link:  scipost_202212_00042v2 (pdf) 
Date accepted:  20230811 
Date submitted:  20230228 17:43 
Submitted by:  Mavaddat Nezhaad, Kurosh 
Submitted to:  SciPost Physics Proceedings 
Proceedings issue:  34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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 $\mathbb{R}^3$ endowed with Einstein's velocity addition. The second geometric example of a gyrogroup is the complex unit disk $\mathbb{D}=\{z \in \mathbb{C}: z<1\}$. To construct a gyrogroup structure on $\mathbb{D}$, we choose two elements $z_1, z_2 \in \mathbb{D}$ 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 $r \odot x$ $=$ $\frac{(1+x)^r  (1x)^r}{(1+x)^r + (1x)^r}\frac{x}{x}$, where $x \in \mathbb{D}$ and $r \in \mathbb{R}$, then $(\mathbb{D},\oplus,\odot)$ will be a real gyrovector space. This paper aims to survey the main properties of these M\"{o}bius gyrogroup and M\"{o}bius gyrovector space.
Author comments upon resubmission
Following the review, all typos and corrections are considered, corrected, and I read the whole again.
With kind regards,
Kurosh Mavaddat Nezhaad
Published as SciPost Phys. Proc. 14, 041 (2023)