**Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups**

by Jean Gallier

Department of Computer and Information Science

University of Pennsylvania

Philadelphia, PA 19104, USA

e-mail: jean_at_saul.cis.upenn.edu

2 May 2008

**Abstract: **One of the main goals of these notes is to explain how rotations in R^n are induced by the action of a certain group, Spin(n), on R^n, in a way that generalizes the action of the unit complex numbers, U(1), on R^2, and the action of the unit quaternions, SU(2), on R^3 (i.e., the action is deﬁned in terms of multiplication in a larger algebra containing both the group Spin(n) and R^n). The group Spin(n), called a spinor group, is deﬁned as a certain subgroup of units of an algebra, Cl_n, the Clifford algebra associated with R^n. Since the spinor groups are certain well chosen subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a ﬁrm understanding of spinor groups. These notes provide a tutorial on Clifford algebra and the groups Spin and Pin, including a study of the structure of the Clifford algebra Cl_{p,q} associated with a nondegenerate symmetric bilinear form of signature (p, q) and culminating in the beautiful “8-periodicity theorem” of Elie Cartan and Raoul Bott (with proofs).

PDF, 46 pages, free download: http://www.cis.upenn.edu/~cis610/clifford.pdf

**Contents**

1 Clifford Algebras, Clifford Groups, Pin and Spin . . . . . . . . . . . . . . . . . . . 7

1.1 Introduction: Rotations As Group Actions . . . . . . . . . . . . . . . . . . . 7

1.2 Clifford Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Clifford Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 The Groups Pin(n) and Spin(n) . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 The Groups Pin(p, q) and Spin(p, q) . . . . . . . . . . . . . . . . . . . . . . 30

1.6 Periodicity of the Clifford Algebras Clp,q . . . . . . . . . . . . . . . . . . . . 33

1.7 The Complex Clifford Algebras Cl(n,C) . . . . . . . . . . . . . . . . . . . . 36

1.8 The Groups Pin(p, q) and Spin(p, q) as double covers . . . . . . . . . . . . . 37

1.9 More on the Topology of O(p, q) and SO(p, q) . . . . . . . . . . . . . . . . . 41

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