Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail:
[email protected]
December 19, 2002
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Contents 1 Clifford Algebras, Clifford Groups, Pin and Spin 1.1 Introduction: Rotations As Group Actions . . . . . . 1.2 Clifford Algebras . . . . . . . . . . . . . . . . . . . . 1.3 Clifford Groups . . . . . . . . . . . . . . . . . . . . . 1.4 The Groups Pin(n) and Spin(n) . . . . . . . . . . . 1.5 The Groups Pin(p, q) and Spin(p, q) . . . . . . . . . 1.6 Periodicity of the Clifford Algebras Clp,q . . . . . . . 1.7 The Complex Clifford Algebras Cl(n, C) . . . . . . . 1.8 The Groups Pin(p, q) and Spin(p, q) as double covers 1.9 More on the Topology of O(p, q) and SO(p, q) . . . .
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CONTENTS
Chapter 1 Clifford Algebras, Clifford Groups, and the Groups Pin(n) and Spin(n) 1.1
Introduction: Rotations As Group Actions
One of the main goals of these notes is to explain how rotations in Rn are induced by the action of a certain group, Spin(n), on Rn , in a way that generalizes the action of the unit comp