关于莫比乌斯陀螺群和莫比乌斯陀螺向量空间

SciPost Physics Proceedings Pub Date : 2023-11-24 DOI:10.21468/scipostphysproc.14.041

Kurosh Mavaddat Nezhaad, A. Ashrafi

引用次数: 0

摘要

陀螺群是 1988 年在研究狭义相对论中爱因斯坦速度加法时出现的新代数结构。亚伯拉罕-温加尔对这些新的代数结构进行了深入研究。第一个被考虑的陀螺群是欧几里得空间的单位球 \mathbb{R}^3ℝ3，它被赋予了爱因斯坦速度加法。陀螺群的第二个几何例子是复数单位盘 \mathbb{D}𝔻={z∈ \mathbb{C}：|z|ℂ:|z|1}。为了在 \mathbb{D}𝔻 上构建陀螺群结构，我们选择两个元素 z_1, z_2 ∈\mathbb{D}z1、z2∈𝔻 并用 z_1oplus z_2 = \frac{z_1+z_2}{1+\bar{z_1}z_2}z1⊕z2=z1+z21+z1‾z2 定义莫比乌斯加法。那么 (\mathbb{D},\oplus)(𝔻,⊕) 是一个陀螺交换陀螺群。如果我们定义 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|、其中 x∈ \mathbb{D}x∈𝔻，r∈ \mathbb{R}r∈ℝ，那么 (\mathbb{D},\oplus,\odot)(𝔻,⊕,⊙) 将是一个实陀螺向量空间。本文旨在考察这些莫比乌斯陀螺群和莫比乌斯陀螺矢量空间的主要性质。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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On Möbius gyrogroup and Möbius gyrovector space

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

ℝ3 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|<1

ℂ:|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}