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

Document Type

Article

Publication Date

1-1-2010

Abstract

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 ℝn. One notable achievement is a compact, convenient formula for the Möbius loop operation a * 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+wz)-1 for the Möbius loop multiplication in the unit complex disk).

Publication Source (Journal or Book title)

Commentationes Mathematicae Universitatis Carolinae

First Page

319

Last Page

331

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