Extension of the Unit Disk Gyrogroup into the Unit Ball of Any Real Inner Product Space
✍ Scribed by Abraham A. Ungar
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 172 KB
- Volume
- 202
- Category
- Article
- ISSN
- 0022-247X
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✦ Synopsis
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 Ž complex plane, which are associative-commutative operations i.e., their composi-. tion law is associative and commutative forming a group, gyrotranslations of the complex unit disk fail to form a group. Rather, left gyrotranslations are gyroasso-Ž ciative-gyrocommutative operations i.e., their composition law is gyroassociative . and gyrocommutative forming a gyrogroup. The complex unit disk gyrogroup has Ž . 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 ẗransformations into the ball of any real inner product space and its generalized Mobius transformations. A gyrogroup is a mathematical object which first arose in ẗhe 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 Ž . 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. '' ᮊ 1996 Academic Press, Inc.