Quadratic algebras, Clifford algebras, and arithmetic Witt groups:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1994
|
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 286 S. graph. Darst. |
| ISBN: | 354094110X 038794110X |
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MARC
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| 245 | 1 | 0 | |a Quadratic algebras, Clifford algebras, and arithmetic Witt groups |c Alexander J. Hahn |
| 264 | 1 | |a New York u.a. |b Springer |c 1994 | |
| 300 | |a XI, 286 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
Preface vii
Introduction 1
Notation and Terminology 3
Chapter 1. Fundamental Concepts in the Theory of Algebras 5
A. Free Quadratic Algebras 5
B. Involutions on Algebras 9
C. Gradings on Algebras 10
D. Tensor Products and Graded Tensor Products 11
E. Exercises 13
Chapter 2. Separable Algebras 18
A. Separability of Algebras 18
B. Separability Idempotents 20
C. Separable Free Quadratic Algebras 21
D. Properties of Conjugation 24
E. Exercises 27
Chapter 3. Groups of Free Quadratic Algebras 29
A. The Group Quf(R) 29
B. The Discriminant 8 32
C. The Group QUf(R) 33
D. Another Look at (a, b)6 * (b, c)1! 36
E. Exercises 40
Chapter 4. Bilinear and Quadratic Forms 45
A. Localization 45
B. Bilinear Forms 48
C. The Group Dis(R) 52
D. Quadratic Forms 56
E. Exercises 61
ix
Chapter 5. Clifford Algebras: The Basics 65
A. Definition and Existence 65
B. Generation, Grading, and Involutions 66
C. Graded Tensor Product 69
D. Exterior Algebras 73
E. Exercises 75
Chapter 6. Algebras with Standard Involution 77
A. Standard Involutions 77
B. Free Quaternion Algebras 81
C. Separability of Free Quaternion Algebras 84
D. Nonsingular Algebras 86
E. Exercises 88
Chapter 7. Arf Algebras and Special Elements 92
A. The Arf Algebra 92
B. The Arf Algebra of an Orthogonal Sum 96
C. Special Elements 99
D. Exercises 104
Chapter 8. Consequences of the Existence of Special Elements 106
A. Connections between C(M) and Cq(M) 106
B. Gradings Defined by Roots of X2 aX b 108
C. Linear Maps with Polynomial X2 aX b 109
D. Graded Properties of Representations 114
E. Comparing the Tensor and Graded Tensor Products 118
F. Exercises 121
Chapter 9. Structure of Clifford and Arf Algebras 123
A. More on Separable Algebras 123
B. The Separability of C(M) and Q)(M) 126
C. The Even Odd Splitting of C(M) 127
D. The Structures ofCenC(M),CenC0(M), and A(M) 131
E. Exercises 133
Chapter 10. The Existence of Special Elements 137
A. Separable Quadratic Algebras 137
B. The Discriminant Module of S 142
C. Criteria for the Existence of Special Elements 144
D. Special Elements and the Discriminant 146
E. Exercises 150
Chapter 11. Matrix Theory of Clifford Algebras over Fields 153
A. Matrix Connections between C(M) and Cq(M) 153
B. Basics about Quadratic Spaces 157
C. Quaternion Algebras 159
D. Periodicity Phenomena 162
E. Local and Global Fields 164
F. Exercises 169
Chapter 12. Dis(R) and Qu(R) 172
A. The Quadratic Group Qu(R) 172
B. More about Dis(R) 175
C. Connecting Qu(R) with Dis(R) 177
D. The Case of an Integrally Closed Domain 183
E. The Classical Discriminant 187
F. Exercises 190
Chapter 13. Brauer Groups and Witt Groups 194
A. Brauer and Brauer Wall Groups 194
B. The Graded Quadratic Group QU(R) 199
C. The Witt Group of Quadratic Forms 204
D. The Witt Group of Symmetric Bilinear Forms 209
E. The Classical Situations 214
F. Exercises 218
Chapter 14. The Arithmetic of Wq(R) 223
A. Arithmetic Dedekind Domains 223
B. The Arithmetic of Br(R)2 225
C. Analyzing Wq(R) 229
D. Computing Qu(RJ and Wq(R«) 233
E. Connections between W(R) and Wq(R) 237
F. Exercises 242
Chapter 15. Applications of Clifford Modules 249
A. Clifford Modules 249
B. Vector Fields on Spheres 251
C. Connections with Topological K Theory 253
D. Lie Groups and Lie Algebras 257
E. Dirac Operators 258
F. Spin Manifolds 260
G. Isoparametric Hypersurfaces 263
Bibliography 267
Index 281
|
| any_adam_object | 1 |
| author | Hahn, Alexander J. 1943- |
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| author_facet | Hahn, Alexander J. 1943- |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.57 |
| dewey-search | 512/.57 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:31:29Z |
| institution | BVB |
| isbn | 354094110X 038794110X |
| language | English |
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| physical | XI, 286 S. graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
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| publisher | Springer |
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| series2 | Universitext |
| spelling | Hahn, Alexander J. 1943- Verfasser (DE-588)120421909 aut Quadratic algebras, Clifford algebras, and arithmetic Witt groups Alexander J. Hahn New York u.a. Springer 1994 XI, 286 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Anneaux commutatifs ram Clifford, algèbres de ram Formes quadratiques ram Clifford algebras Commutative rings Forms, Quadratic Clifford-Algebra (DE-588)4199958-7 gnd rswk-swf Witt-Gruppe (DE-588)4190128-9 gnd rswk-swf Quadratische Algebra (DE-588)4332091-0 gnd rswk-swf Kommutativer Ring (DE-588)4164825-0 gnd rswk-swf Quadratische Algebra (DE-588)4332091-0 s DE-604 Clifford-Algebra (DE-588)4199958-7 s Witt-Gruppe (DE-588)4190128-9 s Kommutativer Ring (DE-588)4164825-0 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006051536&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hahn, Alexander J. 1943- Quadratic algebras, Clifford algebras, and arithmetic Witt groups Anneaux commutatifs ram Clifford, algèbres de ram Formes quadratiques ram Clifford algebras Commutative rings Forms, Quadratic Clifford-Algebra (DE-588)4199958-7 gnd Witt-Gruppe (DE-588)4190128-9 gnd Quadratische Algebra (DE-588)4332091-0 gnd Kommutativer Ring (DE-588)4164825-0 gnd |
| subject_GND | (DE-588)4199958-7 (DE-588)4190128-9 (DE-588)4332091-0 (DE-588)4164825-0 |
| title | Quadratic algebras, Clifford algebras, and arithmetic Witt groups |
| title_auth | Quadratic algebras, Clifford algebras, and arithmetic Witt groups |
| title_exact_search | Quadratic algebras, Clifford algebras, and arithmetic Witt groups |
| title_full | Quadratic algebras, Clifford algebras, and arithmetic Witt groups Alexander J. Hahn |
| title_fullStr | Quadratic algebras, Clifford algebras, and arithmetic Witt groups Alexander J. Hahn |
| title_full_unstemmed | Quadratic algebras, Clifford algebras, and arithmetic Witt groups Alexander J. Hahn |
| title_short | Quadratic algebras, Clifford algebras, and arithmetic Witt groups |
| title_sort | quadratic algebras clifford algebras and arithmetic witt groups |
| topic | Anneaux commutatifs ram Clifford, algèbres de ram Formes quadratiques ram Clifford algebras Commutative rings Forms, Quadratic Clifford-Algebra (DE-588)4199958-7 gnd Witt-Gruppe (DE-588)4190128-9 gnd Quadratische Algebra (DE-588)4332091-0 gnd Kommutativer Ring (DE-588)4164825-0 gnd |
| topic_facet | Anneaux commutatifs Clifford, algèbres de Formes quadratiques Clifford algebras Commutative rings Forms, Quadratic Clifford-Algebra Witt-Gruppe Quadratische Algebra Kommutativer Ring |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006051536&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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