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 \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).
Keywords: Bruck loop, Clifford algebra, gyrogroup, Möbius transformations, Vahlen matrices, involutive group
AMS Subject Classification: 20N05 51B10 15A66