The geometry of the octonions:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2015
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 210 S. Ill., graph. Darst. 23 cm |
| ISBN: | 9814401811 9789814401814 9789811218187 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804175011037052928 |
|---|---|
| adam_text | Titel: The geometry of the octonions
Autor: Dray, Tevian
Jahr: 2015
Contents Preface vii Acknowledgments ix List of Figures xv List of Tables xvii 1. Introduction 1 1. Number Systems 3 2. The Geometry of the Complex Numbers 5 2.1 Complex Numbers...................... 5 2.2 History............................. 5 2.3 Algebra............................ 6 2.4 Geometry.................. 7 3. The Geometry of the Quaternions 9 3.1 Quaternions.......................... 9 3.2 History............................. 10 3.3 Algebra............................ 11 3.4 Geometry........................... 13 xi
The Geometry of the Octonions xi i 4. The Geometry of the Octonions 15 4.1 Octonions........................... 15 4.2 History............................. 16 4.3 Algebra............................ 17 4.4 Geometry........................... 19 5. Other Number Systems 23 5.1 The Cayley-Dickson Process................. 23 5.2 Sedenions........................... 24 5.3 The Hurwitz Theorem.................... 25 5.4 Split Complex Numbers................... 26 5.5 Split Quaternions....................... 27 5.6 Split Octonions........................ 28 5.7 Subalgebras of the Split Octonions..... 29 II. Symmetry Groups 31 6. Some Orthogonal Groups 33 6.1 Rotations........................... 33 6.2 The Geometry of SO(2) ................... 35 6.3 The Geometry of SO(3) ................... 36 6.4 The Geometry of SO(4) ................... 37 6.5 Lorentz Transformations................... 38 6.6 The Geometry of SO(3,1).................. 40 6.7 The Geometry of SO(4,2).................. 42 7. Some Unitary Groups 45 7.1 Unitary Transformations................... 45 7.2 The Geometry of U(l).................... 46 7.3 The Geometry of SU(2) ................... 47 7.4 The Geometry of SU(3) ................... 49 7.5 The Geometry of SU(2, 2).................. 52 8. Some Symplectic Groups 57 8.1 Symplectic Transformations................. 57 8.2 The Geometry of Sp(4, M).................. 58 8.3 The Geometry of Sp(6, M).................. 59
Contents xiii 9. Symmetry Groups over Other Division Algebras 61 9.1 Some Orthogonal Groups over Other Division Algebras . . 61 9.2 Some Unitary Groups over Other Division Algebras .... 66 9.3 Some Lorentz Groups over Other Division Algebras .... 68 9.4 Some Symplectic Groups over Other Division Algebras . . 74 10. Lie Groups and Lie Algebras 77 10.1 Lie Groups .......................... 77 10.2 Lie Algebras........ 78 10.3 The Classification of Lie Groups .............. 80 10.4 Real Forms.......................... 82 11. The Exceptional Groups 85 11.1 The Geometry of G i ..................... 85 11.2 The Albert Algebra...................... 88 11.3 The Geometry of F 4 ..................... 90 11.4 The Geometry of Eg..................... 93 11.5 The Geometry of E 7 ..................... 95 11.6 The Geometry of Es ..................... 105 III. Applications 107 12. Division Algebras in Mathematics 109 12.1 The Hopf Bundles ...................... 109 12.2 The Octonionic Projective Line............... 113 12.3 Spinors............................. 114 12.4 Môbius Transformations................... 115 12.5 The Octonionic Projective Plane..... 119 12.6 Quaternionic Integers..................... 125 12.7 Octonionic Integers...................... 127 12.8 The Geometry of the es Lattice............... 129 13. Octonionic Eigenvalue Problems 133 13.1 The Eigenvalue Problem................... 133 13.2 The 2x2 Real Eigenvalue Problem............. 135 13.3 The 2x2 Non-real Eigenvalue Problem.......... 137 13.4 The 3x3 Real Eigenvalue Problem............. 144
The Geometry of the Octonions 13.5 The 3x3 Non-real Eigenvalue Problem .......... 152 13.6 The Jordan Eigenvalue Problem............... 154 13.7 Diagonalizing Jordan Matrices with F 4 ........... 160 14. The Physics of the Octonions 163 14.1 Spin.............................. 163 14.2 Quaternionic Spin ...................... 166 14.3 Introduction to the Dirac Equation............. 169 14.4 Gamma Matrices....................... 171 14.5 The Dirac Equation ..................... 174 14.6 The Weyl Equation...................... 177 14.7 Leptons............................ 178 14.8 Cayley Spinors........................ 182 14.9 The Jordan Formulation of Quantum Mechanics ..... 184 14.10 The 3-$ Rule................ 186 15. Magic Squares 189 15.1 The 2x2 Magic Square................... 189 15.2 The Geometry of SU(2, K (g) K)............... 190 15.3 The 3x3 Magic Square................... 198 Further Reading 199 Bibliography 201 Index 205
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| author | Dray, Tevian Manogue, Corinne A. 1955- |
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| id | DE-604.BV042781819 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:09:30Z |
| institution | BVB |
| isbn | 9814401811 9789814401814 9789811218187 |
| language | English |
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| physical | XVII, 210 S. Ill., graph. Darst. 23 cm |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | World Scientific |
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| spelling | Dray, Tevian Verfasser (DE-588)102623753X aut The geometry of the octonions Tevian Dray ; Corinne A. Manogue New Jersey [u.a.] World Scientific 2015 XVII, 210 S. Ill., graph. Darst. 23 cm txt rdacontent n rdamedia nc rdacarrier Oktave Mathematik (DE-588)4595894-4 gnd rswk-swf Oktave Mathematik (DE-588)4595894-4 s DE-604 Manogue, Corinne A. 1955- Verfasser (DE-588)1043326898 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028211838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dray, Tevian Manogue, Corinne A. 1955- The geometry of the octonions Oktave Mathematik (DE-588)4595894-4 gnd |
| subject_GND | (DE-588)4595894-4 |
| title | The geometry of the octonions |
| title_auth | The geometry of the octonions |
| title_exact_search | The geometry of the octonions |
| title_full | The geometry of the octonions Tevian Dray ; Corinne A. Manogue |
| title_fullStr | The geometry of the octonions Tevian Dray ; Corinne A. Manogue |
| title_full_unstemmed | The geometry of the octonions Tevian Dray ; Corinne A. Manogue |
| title_short | The geometry of the octonions |
| title_sort | the geometry of the octonions |
| topic | Oktave Mathematik (DE-588)4595894-4 gnd |
| topic_facet | Oktave Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028211838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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