Clifford algebras, Möbius transformations, Vahlen matrices, and B-loops

Commentationes mathematicae Universitatis Carolinae

Jimmie Lawson¹ 

关键词: Bruck loop;    Clifford algebra;    gyrogroup;    Möbius transformations;    Vahlen matrices;    involutive group;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball B in n-dimensional euclidean space \mathbb R^n. One notable achievement is a compact, convenient formula for the Möbius loop operation a\ast b=(a+b)(1-ab)^{-1}, where the operations on the right are those arising from the Clifford algebra (a formula comparable to (w+z)(1+\overline wz)^{-1} for the Möbius loop multiplication in the unit complex disk).

【 授权许可】

CC BY