Clifford Algebras and the Classical GroupsCambridge University Press, 1995-10-05 - 295 psl. 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. |
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Rezultatai 1–5 iš 43
i psl.
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vii psl.
... Conjugation 148 18 2 x 2 Clifford matrices 167 19 The Cayley algebra 178 20 Topological spaces 191 21 22 Manifolds Lie groups 202 225 23 Conformal groups 245 24 Triality 256 References 285 Index 289 Foreword This book's parent ...
... Conjugation 148 18 2 x 2 Clifford matrices 167 19 The Cayley algebra 178 20 Topological spaces 191 21 22 Manifolds Lie groups 202 225 23 Conformal groups 245 24 Triality 256 References 285 Index 289 Foreword This book's parent ...
ix psl.
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x psl.
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13 psl.
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Turinys
1 Linear spaces | 1 |
2 Real and complex algebras | 9 |
3 Exact sequences | 18 |
4 Real quadratic spaces | 22 |
5 The classification of real quadratic spaces | 32 |
6 Antiinvolutions of Rn | 43 |
7 Antiinvolutions of Cn | 49 |
8 Quaternions | 57 |
15 Clifford algebras | 123 |
16 Spin groups | 140 |
17 Conjugation | 148 |
18 2 x 2 Clifford matrices | 167 |
19 The Cayley algebra | 178 |
20 Topological spaces | 191 |
21 Manifolds | 202 |
22 Lie groups | 225 |
9 Quaternionic linear spaces | 67 |
10 Antiinvolutions of Hn | 72 |
11 Tensor products of algebras | 81 |
12 Antiinvolutions of 2Kn | 91 |
13 The classical groups | 100 |
14 Quadric Grassmannians | 110 |
23 Conformal groups | 245 |
24 Triality | 256 |
285 | |
289 | |
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0-triad 2K-linear adjoint anti-automorphism anti-involution automorphism bijective Cayley algebra chart Clifford algebra commutative compact complex numbers conjugation Corollary defined denoted diagram differentiable dimension division algebra embedding endomorphism equivalent example finite finite-dimensional real linear follows G₂ GL(n Grassmannian group map H-correlation homeomorphic induced injective invertible involution irreducible isomorphic Let f Let G Lie group linear map linear subspace map f map ƒ matrix non-zero null subspace orthogonal automorphism paravector positive-definite projective Proof Let Proposition Prove pure quaternion quadratic form quadric quadric Grassmannian quaternion real algebra real linear space real number real quadratic space rotation Rp,q scalar product semi-linear map smooth manifolds smooth map smooth submanifold SO(n Spin(n standard subalgebra subgroup surjective symplectic space Table Theorem topological group topological space triality unit element universal Clifford algebra y₁