Clifford Algebras and the Classical Groups (Cambridge Studies in Advanced Mathematics, Series Number 50)

This book reflects the growing interest in the theory of Clifford algebras and their applications. The author has reworked his previous book on this subject, Topological Geometry, and has expanded and added material. As in the previous version, the author includes an exhaustive treatment of all the generalizations of the classical groups, as well as an excellent exposition of the classification of the conjugation anti-involution of the Clifford algebras and their complexifications. Toward the end of the book, the author introduces ideas from the theory of Lie groups and Lie algebras. This treatment of Clifford algebras will be welcomed by graduate students and researchers in algebra. Contents......Page 8 Foreword......Page 10 1 Linear spaces......Page 12 2 Real and complex algebras......Page 20 3 Exact sequences......Page 29 4 Real quadratic spaces......Page 33 5 The classification of real quadratic spaces......Page 43 6 Anti-involutions of R(n)......Page 54 7 Anti-involutions of C(n)......Page 60 8 Quaternions......Page 68 9 Quaternionic linear spaces......Page 78 10 Anti-involutions of H(n)......Page 83 11 Tensor products of algebras......Page 92 12 Anti-involutions of 2K(n)......Page 102 13 The classical groups......Page 111 14 Quadric Grassmannians......Page 121 15 Clifford algebras......Page 134 16 Spin groups......Page 151 17 Conjugation......Page 159 18 2 x 2 Clifford matrices......Page 178 19 The Cayley algebra......Page 189 20 Topological spaces......Page 202 21 Manifolds......Page 213 22 Lie groups......Page 236 23 Conformal groups......Page 256 24 Triality......Page 267 References......Page 296 Index......Page 300 The Clifford Algebras Of Real Quadratic Forms And Their Complexifications Are Studied Here In Detail, And Those Parts Which Are Immediately Relevant To Theoretical Physics Are Seen In The Proper Broad Context. Central To The Work Is The Classification Of The Conjugation And Reversion Anti-involutions That Arise Naturally In The Theory. It Is Of Interest That All The Classical Groups Play Essential Roles In This Classification. Other Features Include Detailed Sections On Conformal Groups, The Eight-dimensional Non-associative Cayley Algebra, Its Automorphism Group, The Exceptional Lie Group G2, And The Triality Automorphism Of Spin 8. The Book Is Designed To Be Suitable For The Last Year Of An Undergraduate Course Or The First Year Of A Postgraduate Course. Ian R. Porteous. Includes Bibliographical References (p. 285-288) And Index. "The Clifford algebras of real quadratic forms and their complexifications are studied here in detail, and those parts which are immediately relevant to theoretical physics are seen in the proper broad context. Central to the work is the classification of the conjugation and reversion anti-involutions that arise naturally in the theory. It is of interest that all the classical groups play essential roles in this classification. Other features include detailed sections on conformal groups, the eight-dimensional non-associative Cayley algebra, its automorphism group, the exceptional Lie group G[subscript 2], and the triality automorphism of Spin 8. The book is designed to be suitable for the last year of an undergraduate course or the first year of a postgraduate course."--Cover A treatment of the theory of Clifford algebras that will be welcomed for its clarity and detail