Extension of the unit disk gyrogroup into the unit ball of any real inner product space
Extension of the unit disk gyrogroup into the unit ball of any real inner product space
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DOI：
10.1006/jmaa.1996.0359
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发表时间：
1996-09-15
影响因子：
1.3
通讯作者：
Ungar, AA
中科院分区：
文献类型：
--
作者：
Ungar, AA
复单位圆盘的所有全纯自同构群由涉及类平移全纯自同构和旋转的莫比乌斯变换组成。前者称为陀螺翻译。与复平面的平移形成群相反，复平面的平移是结合-累积运算（即，它们的组合律是结合律和交换律）形成群，而复数单位圆盘的旋转平移无法形成群。相反，左陀螺平移是陀螺关联-陀螺交换运算（即，它们的组合定律是陀螺关联和陀螺交换），形成陀螺群。作者先前研究过复单位圆盘陀螺组(Aequationes Math. 47, 1994, 240-254)。利用复数和向量空间线性变换所共享的类比，我们在本文中将复数盘陀螺组及其莫比乌斯变换扩展到任何实数内积空间的球及其广义莫比乌斯变换。陀螺群是一种数学对象，首先出现在相对论速度的研究中，相对论速度在速度加法下形成非群陀螺群，这与前相对论速度相反，前相对论速度在速度加法下形成群。人们发现，数学规律似乎在从前相对论速度到相对论速度的转变中消失了，但却隐藏在称为托马斯进动的相对论效应中。在其抽象背景下，托马斯进动被称为托马斯回转，从而产生了我们的“陀螺术语”。我们的陀螺术语由作者开发（Amer. J. Phys. 59, 1991, 824-834），涉及陀螺群、陀螺关联-陀螺交换律和陀螺自同构等术语，其中我们广泛使用前缀“陀螺仪”。(C) 1996 年学术出版社
The group of all holomorphic automorphisms of the complex unit disk consists of Mobius transformations involving translation-like holomorphic automorphisms and rotations. The former are called gyrotranslations. As opposed to translations of the complex plane, which are associative-commulative operations (i.e., their composition law is associative and commutative) forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroassociative-gyrocommutative operations (i.e., their composition law is gyroassociative and gyrocommutative) forming a gyrogroup. The complex unit disk gyrogroup has previously been studied by the author (Aequationes Math. 47, 1994, 240-254). Employing analogies shared by complex numbers and linear transformations of vector spaces, we extend in this article the complex disk gyrogroup and its Mobius transformations into the ball of any real inner product space and its generalized Mobius transformations. A gyrogroup is a mathematical object which first arose in the study of relativistic velocities which, under velocity addition, form a nongroup gyrogroup, as opposed to prerelativistic velocities, which form a group under velocity addition. It has been discovered that the mathematical regularity, seemingly lost in the transition from prerelativistic to relativistic velocities, is concealed in a relativistic effect known as Thomas precession. In its abstract context, Thomas precession is called Thomas gyration, giving rise to our ''gyroterminology.'' Our gyroterminology, developed by the author (Amer. J. Phys. 59, 1991, 824-834), involves terms like gyrogroups, gyroassociative-gyrocommutative laws, and gyroautomorphisms, in which we extensively use the prefix ''gyro.'' (C) 1996 Academic Press, Inc.