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

Jimmie Lawson (2010)

Commentationes Mathematicae Universitatis Carolinae

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 n . One notable achievement is a compact, convenient formula for the Möbius loop operation a * b = ( a + b ) ( 1 - a b ) - 1 , where the operations on the right are those arising from the Clifford algebra (a formula comparable to ( w + z ) ( 1 + w ¯ z ) - 1 for the Möbius loop multiplication...

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