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 15 iš 77
vii psl.
... linear spaces 10 Anti-involutions of H(n) 11 Tensor products of algebras 12 Anti-involutions of 2K(n) 13 The classical groups 14 Quadric Grassmannians 15 Clifford algebras 16 Spin groups 17 Conjugation 18 2 >< 2 Clifford matrices 19 The ...
... linear spaces 10 Anti-involutions of H(n) 11 Tensor products of algebras 12 Anti-involutions of 2K(n) 13 The classical groups 14 Quadric Grassmannians 15 Clifford algebras 16 Spin groups 17 Conjugation 18 2 >< 2 Clifford matrices 19 The ...
1 psl.
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2 psl.
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8 psl.
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41 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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action adjoint analogous anti-involution assigned associative associative algebra automorphism basis called Cayley Chapter chart Clifford algebra closed column commutative compact complex conjugation connected Consider consisting continuous Corollary correlation defined definition denoted determinant diagram dimension element embedding EndX equal equation equivalent example Exercise exists field finite first follows given H-linear homeomorphic identified identity implying induced injective invertible involution isomorphic Lie group linear map linear space linear subspace map f matrix Moreover namely non-degenerate non-zero norm null orthogonal orthonormal paravector particular plane positive projective Proof Proposition Prove quadric quaternion real algebra real linear space real quadratic space regarded remark representation represented respectively restriction rotation scalar product semi-linear sequence skew smooth space X sphere standard structure subalgebra subgroup subset suppose surjective symmetric symplectic Table tangent Theorem unique universal