Holdings: Quadratic mappings and Clifford algebras

Quadratic mappings and Clifford algebras:

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Bibliographic Details
Main Authors:	Helmstetter, Jacques (Author), Micali, Artibano (Author)
Format:	Book
Language:	English
Published:	Basel [u.a.] Birkhäuser 2008
Subjects:	Clifford algebras Forms, Quadratic Quadratischer Modul Clifford-Algebra Quadratische Form
Online Access:	Inhaltsverzeichnis
Physical Description:	XIII, 504 S.
ISBN:	3764386053 9783764386054

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MARC


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adam_text	JACQUES HELMSTETTER ARTIBANO MICALI QUADRATIC MAPPINGS AND CLIFFORD ALGEBRAS BIRKHAUSER BASEL * BOSTON * BERLIN INTRODUCTION IX 1 ALGEBRAIC PRELIMINARIES 1.1 SOME GENERAL NOTATION AND DEFINITIONS 1 1.2 UNIVERSAL OBJECTS IN A CATEGORY 2 1.3 EXAMPLES OF UNIVERSAL OBJECTS 4 1.4 TENSOR ALGEBRAS AND SYMMETRIC ALGEBRAS 8 1.5 FUNCTORS 10 1.6 EXACT SEQUENCES 12 1.7 PROJECTIVE MODULES AND FLAT MODULES 14 1.8 FINITELY PRESENTED MODULES 15 1.9 CHANGES OF BASIC RINGS 17 1.10 RINGS AND MODULES OF FRACTIONS 21 1.11 LOCALIZATION AND GLOBALIZATION 25 1.12 FINITELY GENERATED MODULES 30 1.13 SOME APPLICATIONS 37 EXERCISES 42 2 QUADRATIC MAPPINGS 2.1 GENERALITIES 53 2.2 CHANGES OF BASIC RING AND LOCALIZATIONS 57 2.3 NONDEGENERATE QUADRATIC MAPPINGS 59 2.4 OPERATIONS ON QUADRATIC MAPPINGS AND SYMMETRIC BILINEAR MAPPINGS 63 2.5 HYPERBOLIC AND METABOLIC SPACES 67 2.6 ORTHOGONAL DECOMPOSITIONS OF QUADRATIC SPACES 72 2.7 WITT RINGS 78 2.8 EXAMPLES OF WITT RINGS 83 EXERCISES 93 VI CONTENTS 3 CLIFFORD ALGEBRAS 3.1 DEFINITIONS AND ELEMENTARY PROPERTIES 105 3.2 THE PARITY GRADING OF CLIFFORD ALGEBRAS 112 3.3 CLIFFORD ALGEBRAS OF FREE MODULES OF RANK 2 117 3.4 GRADED QUADRATIC EXTENSIONS 122 3.5 GRADED AZUMAYA ALGEBRAS 131 3.6 TRACES AND DETERMINANTS 144 3.7 CLIFFORD ALGEBRAS OF QUADRATIC SPACES 149 3.8 DISCRIMINANT MODULES, QUADRATIC EXTENSIONS AND QUATERNION ALGEBRAS 154 EXERCISES 164 4 COMULTIPLICATIONS. EXPONENTIALS. DEFORMATIONS 4.1 COALGEBRAS AND COMODULES 175 4.2 ALGEBRAS AND COALGEBRAS GRADED BY PARITIES 181 4.3 EXTERIOR ALGEBRAS 185 4.4 INTERIOR PRODUCTS IN CLIFFORD ALGEBRAS 190 4.5 EXPONENTIALS IN EVEN EXTERIOR SUBALGEBRAS 195 4.6 SYSTEMS OF DIVIDED POWERS 199 4.7 DEFORMATIONS OF CLIFFORD ALGEBRAS 201 4.8 APPLICATIONS OF DEFORMATIONS 210 EXERCISES 219 5 ORTHOGONAL GROUPS AND LIPSCHITZ GROUPS 5.1 TWISTED INNER AUTOMORPHISMS AND ORTHOGONAL GROUPS 232 5.2 FILTRATIONS OF CLIFFORD ALGEBRAS 243 5.3 LIPSCHITZ MONOIDS AND DERIVED GROUPS 249 5.4 THE INVARIANCE PROPERTY 258 5.5 ASSOCIATED LIE ALGEBRAS 261 5.6 FIRST RESULTS ABOUT ORTHOGONAL TRANSFORMATIONS 267 5.7 PRODUCTS OF REFLECTIONS WHEN K IS A LOCAL RING 272 5.8 FURTHER RESULTS ABOUT ORTHOGONAL TRANSFORMATIONS 281 5.9 MORE INFORMATION ABOUT EXTERIOR ALGEBRAS 287 5.10 THE LIPSCHITZ MONOID LIP(M) WHEN K IS A FIELD 296 EXERCISES 303 6 FURTHER ALGEBRAIC DEVELOPMENTS 6.1 MODULES OVER A NONCOMMUTATIVE ALGEBRA 323 6.2 GRADED MODULES OVER A GRADED ALGEBRA 327 6.3 GRADED SEMI-SIMPLE MODULES 334 6.4 GRADED MORITA THEORY 337 6.5 GRADED SEPARABLE ALGEBRAS 344 CONTENTS VII 6.6 GRADED CENTRAL SIMPLE ALGEBRAS OVER A FIELD 353 6.7 MORE INFORMATION ABOUT GRADED AZUMAYA ALGEBRAS 358 6.8 INVOLUTIONS ON GRADED CENTRAL SIMPLE ALGEBRAS 365 EXERCISES 376 7 HYPERBOLIC SPACES 7.1 SOME REPRESENTATIONS OF CLIFFORD ALGEBRAS 391 7.2 THE CARTAN-CHEVALLEY MAPPING 395 7.3 THE BIJECTION C(V) / MAX (U) C(V) - C(M) 403 7.4 THE CARTAN-CHEVALLEY CRITERION 409 7.5 APPLICATIONS TO LIPSCHITZ MONOIDS 414 7.6 APPLICATIONS TO TOTALLY ISOTROPIC DIRECT SUMMANDS OF MAXIMAL RANK 421 EXERCISES 425 8 COMPLEMENTS ABOUT WITT RINGS AND OTHER TOPICS 8.1 WITT RINGS OVER LOCAL RINGS WHEN 2 IS INVERTIBLE 439 8.2 CONTINUATION WHEN 2 IS NOT INVERTIBLE 445 8.3 WITT RINGS OF PRIIFER RINGS 453 8.4 QUADRATIC FORMS IN CHARACTERISTIC 2 461 8.5 CLIFFORD ALGEBRAS IN CHARACTERISTIC 2 464 8.6 THE GROUP OF CLASSES OF CLIFFORD ALGEBRAS 468 EXERCISES 478 BIBLIOGRAPHY BOOKS AND BOOKLETS 489 OTHER PUBLICATIONS 492 INDEX OF DEFINITIONS 499 INDEX OF NOTATION 503
adam_txt	JACQUES HELMSTETTER ARTIBANO MICALI QUADRATIC MAPPINGS AND CLIFFORD ALGEBRAS BIRKHAUSER BASEL * BOSTON * BERLIN INTRODUCTION IX 1 ALGEBRAIC PRELIMINARIES 1.1 SOME GENERAL NOTATION AND DEFINITIONS 1 1.2 UNIVERSAL OBJECTS IN A CATEGORY 2 1.3 EXAMPLES OF UNIVERSAL OBJECTS 4 1.4 TENSOR ALGEBRAS AND SYMMETRIC ALGEBRAS 8 1.5 FUNCTORS 10 1.6 EXACT SEQUENCES 12 1.7 PROJECTIVE MODULES AND FLAT MODULES 14 1.8 FINITELY PRESENTED MODULES 15 1.9 CHANGES OF BASIC RINGS 17 1.10 RINGS AND MODULES OF FRACTIONS 21 1.11 LOCALIZATION AND GLOBALIZATION 25 1.12 FINITELY GENERATED MODULES 30 1.13 SOME APPLICATIONS 37 EXERCISES 42 2 QUADRATIC MAPPINGS 2.1 GENERALITIES 53 2.2 CHANGES OF BASIC RING AND LOCALIZATIONS 57 2.3 NONDEGENERATE QUADRATIC MAPPINGS 59 2.4 OPERATIONS ON QUADRATIC MAPPINGS AND SYMMETRIC BILINEAR MAPPINGS 63 2.5 HYPERBOLIC AND METABOLIC SPACES 67 2.6 ORTHOGONAL DECOMPOSITIONS OF QUADRATIC SPACES 72 2.7 WITT RINGS 78 2.8 EXAMPLES OF WITT RINGS 83 EXERCISES 93 VI CONTENTS 3 CLIFFORD ALGEBRAS 3.1 DEFINITIONS AND ELEMENTARY PROPERTIES 105 3.2 THE PARITY GRADING OF CLIFFORD ALGEBRAS 112 3.3 CLIFFORD ALGEBRAS OF FREE MODULES OF RANK 2 117 3.4 GRADED QUADRATIC EXTENSIONS 122 3.5 GRADED AZUMAYA ALGEBRAS 131 3.6 TRACES AND DETERMINANTS 144 3.7 CLIFFORD ALGEBRAS OF QUADRATIC SPACES 149 3.8 DISCRIMINANT MODULES, QUADRATIC EXTENSIONS AND QUATERNION ALGEBRAS 154 EXERCISES 164 4 COMULTIPLICATIONS. EXPONENTIALS. DEFORMATIONS 4.1 COALGEBRAS AND COMODULES 175 4.2 ALGEBRAS AND COALGEBRAS GRADED BY PARITIES 181 4.3 EXTERIOR ALGEBRAS 185 4.4 INTERIOR PRODUCTS IN CLIFFORD ALGEBRAS 190 4.5 EXPONENTIALS IN EVEN EXTERIOR SUBALGEBRAS 195 4.6 SYSTEMS OF DIVIDED POWERS 199 4.7 DEFORMATIONS OF CLIFFORD ALGEBRAS 201 4.8 APPLICATIONS OF DEFORMATIONS 210 EXERCISES 219 5 ORTHOGONAL GROUPS AND LIPSCHITZ GROUPS 5.1 TWISTED INNER AUTOMORPHISMS AND ORTHOGONAL GROUPS 232 5.2 FILTRATIONS OF CLIFFORD ALGEBRAS 243 5.3 LIPSCHITZ MONOIDS AND DERIVED GROUPS 249 5.4 THE INVARIANCE PROPERTY 258 5.5 ASSOCIATED LIE ALGEBRAS 261 5.6 FIRST RESULTS ABOUT ORTHOGONAL TRANSFORMATIONS 267 5.7 PRODUCTS OF REFLECTIONS WHEN K IS A LOCAL RING 272 5.8 FURTHER RESULTS ABOUT ORTHOGONAL TRANSFORMATIONS 281 5.9 MORE INFORMATION ABOUT EXTERIOR ALGEBRAS 287 5.10 THE LIPSCHITZ MONOID LIP(M) WHEN K IS A FIELD 296 EXERCISES 303 6 FURTHER ALGEBRAIC DEVELOPMENTS 6.1 MODULES OVER A NONCOMMUTATIVE ALGEBRA 323 6.2 GRADED MODULES OVER A GRADED ALGEBRA 327 6.3 GRADED SEMI-SIMPLE MODULES 334 6.4 GRADED MORITA THEORY 337 6.5 GRADED SEPARABLE ALGEBRAS 344 CONTENTS VII 6.6 GRADED CENTRAL SIMPLE ALGEBRAS OVER A FIELD 353 6.7 MORE INFORMATION ABOUT GRADED AZUMAYA ALGEBRAS 358 6.8 INVOLUTIONS ON GRADED CENTRAL SIMPLE ALGEBRAS 365 EXERCISES 376 7 HYPERBOLIC SPACES 7.1 SOME REPRESENTATIONS OF CLIFFORD ALGEBRAS 391 7.2 THE CARTAN-CHEVALLEY MAPPING 395 7.3 THE BIJECTION C(V) /\ MAX (U) C(V) - C(M) 403 7.4 THE CARTAN-CHEVALLEY CRITERION 409 7.5 APPLICATIONS TO LIPSCHITZ MONOIDS 414 7.6 APPLICATIONS TO TOTALLY ISOTROPIC DIRECT SUMMANDS OF MAXIMAL RANK 421 EXERCISES 425 8 COMPLEMENTS ABOUT WITT RINGS AND OTHER TOPICS 8.1 WITT RINGS OVER LOCAL RINGS WHEN 2 IS INVERTIBLE 439 8.2 CONTINUATION WHEN 2 IS NOT INVERTIBLE 445 8.3 WITT RINGS OF PRIIFER RINGS 453 8.4 QUADRATIC FORMS IN CHARACTERISTIC 2 461 8.5 CLIFFORD ALGEBRAS IN CHARACTERISTIC 2 464 8.6 THE GROUP OF CLASSES OF CLIFFORD ALGEBRAS 468 EXERCISES 478 BIBLIOGRAPHY BOOKS AND BOOKLETS 489 OTHER PUBLICATIONS 492 INDEX OF DEFINITIONS 499 INDEX OF NOTATION 503
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spelling	Helmstetter, Jacques Verfasser aut Quadratic mappings and Clifford algebras Jacques Helmstetter ; Artibano Micali Basel [u.a.] Birkhäuser 2008 XIII, 504 S. txt rdacontent n rdamedia nc rdacarrier Clifford algebras Forms, Quadratic Quadratischer Modul (DE-588)4819733-6 gnd rswk-swf Clifford-Algebra (DE-588)4199958-7 gnd rswk-swf Quadratische Form (DE-588)4128297-8 gnd rswk-swf Quadratische Form (DE-588)4128297-8 s Clifford-Algebra (DE-588)4199958-7 s Quadratischer Modul (DE-588)4819733-6 s DE-604 Micali, Artibano Verfasser aut Erscheint auch als Online-Ausgabe 978-3-7643-8606-1 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016280692&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Helmstetter, Jacques Micali, Artibano Quadratic mappings and Clifford algebras Clifford algebras Forms, Quadratic Quadratischer Modul (DE-588)4819733-6 gnd Clifford-Algebra (DE-588)4199958-7 gnd Quadratische Form (DE-588)4128297-8 gnd
subject_GND	(DE-588)4819733-6 (DE-588)4199958-7 (DE-588)4128297-8
title	Quadratic mappings and Clifford algebras
title_auth	Quadratic mappings and Clifford algebras
title_exact_search	Quadratic mappings and Clifford algebras
title_exact_search_txtP	Quadratic mappings and Clifford algebras
title_full	Quadratic mappings and Clifford algebras Jacques Helmstetter ; Artibano Micali
title_fullStr	Quadratic mappings and Clifford algebras Jacques Helmstetter ; Artibano Micali
title_full_unstemmed	Quadratic mappings and Clifford algebras Jacques Helmstetter ; Artibano Micali
title_short	Quadratic mappings and Clifford algebras
title_sort	quadratic mappings and clifford algebras
topic	Clifford algebras Forms, Quadratic Quadratischer Modul (DE-588)4819733-6 gnd Clifford-Algebra (DE-588)4199958-7 gnd Quadratische Form (DE-588)4128297-8 gnd
topic_facet	Clifford algebras Forms, Quadratic Quadratischer Modul Clifford-Algebra Quadratische Form
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016280692&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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