Group theoretical methods and their applications:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English German |
| Published: |
Boston u.a.
Birkhäuser
1992
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 296 S. graph. Darst. |
| ISBN: | 0817635270 3764335270 |
Staff View
MARC
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| 100 | 1 | |a Fässler, Albert |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Gruppentheoretische Methoden und ihre Anwendung |
| 245 | 1 | 0 | |a Group theoretical methods and their applications |c A. Fässler ; E. Stiefel |
| 264 | 1 | |a Boston u.a. |b Birkhäuser |c 1992 | |
| 300 | |a XII, 296 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Lie, Algèbres de | |
| 650 | 4 | |a Opérateurs linéaires | |
| 650 | 4 | |a Représentations de groupes | |
| 650 | 4 | |a Lie algebras | |
| 650 | 4 | |a Linear operators | |
| 650 | 4 | |a Representations of groups | |
| 650 | 0 | 7 | |a Gruppentheorie |0 (DE-588)4072157-7 |2 gnd |9 rswk-swf |
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Record in the Search Index
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|---|---|
| adam_text | Contents
1 Preliminaries 1
1.1 The Concept of Groups 1
1.1.1 Transformation Groups 3
1.2 Price Index in Economics 5
1.3 The Realization of Groups 8
1.4 Representation of Groups 11
1.5 Equivalence of Representations 14
1.6 Reducibility of Representations 16
1.7 Complete Reducibility 17
1.8 Basic Conclusions 21
1.9 Representations of Special Finite Groups 22
1.9.1 The Cyclic Group Cg 22
1.9.2 The Dihedral Group Dh 23
1.10 Kronecker Products 25
1.11 Unitary Representations 27
1.11.1 Unitary Representations and Unitary Matrices ... 29
Problems 30
2 Linear Operators with Symmetries 33
2.1 Schur s Lemma 33
2.2 Symmetry of a Matrix 35
2.2.1 Representations of Abelian Groups 36
2.3 The Fundamental Theorem 36
Problems 47
3 Symmetry Adapted Basis Functions 51
3.1 Illustration by Dihedral Groups 51
3.1.1 The Representation dper 52
3.1.2 The Notion of the Orbit 52
3.1.3 Instructions on How to Use Table (3.7) 54
3.1.4 Real Orthonormal Symmetry Adapted Basis .... 56
3.2 Application in Quantum Physics 57
3.3 Application to Finite Element Method 59
3.3.1 Discretization and Symmetry of the Problem .... 59
3.3.2 Elliptic Boundary Value Problem 60
vi CONTENTS
3.3.3 Heat conduction 63
3.4 Perturbed Problems with Symmetry 64
3.5 Fast Fourier Transform on Finite Groups 65
3.5.1 Definitions and Properties 65
3.5.2 Direct and Fast Algorithm 66
3.5.3 The Classical Fast Fourier Transform (FFT) .... 68
3.5.4 Applications and Remarks 70
4 Continuous Groups And Representations 73
4.1 Continuous Matrix Groups 73
4.1.1 Comments on U(n) 74
4.1.2 Comments on SU(n) 75
4.1.3 Connectedness of Continuous Groups 75
4.2 Relationship Between Some Groups 76
4.2.1 Relationship between SL(2, C) and the Lorentz group 77
4.2.2 Relationship Between SU(2) and SO(3) 81
4.3 Constructing Representations 83
4.3.1 Irreducible Representations of SU(2) 84
4.3.2 Irreducible Representations of SO(3) 85
4.3.3 Complete Reduction of Representations of SU(2) and
SO(3) 86
4.4 Clebsch Gordan Coefficients 89
4.4.1 Spherical Functions 90
4.4.2 The Kronecker Product 0t 0 dm 91
4.4.3 Spherical Functions and Laplace Operator 95
4.5 The Lorentz group and SL(2,C) 98
4.5.1 Irreducible Representations 99
4.5.2 Complete Reduction 99
Problems 103
5 Symmetry Ad. Vectors, Characters 105
5.1 Orthogonality of Representations 105
5.2 Algorithm for Symmetry Adapted Bases 108
5.3 Applications 117
5.4 Similarity Classes of Groups 133
5.4.1 Subgroups of SO(3) 134
5.4.2 Permutation Groups 136
5.5 Characters 139
5.5.1 General Properties 139
5.5.2 Orthogonality Relations of Characters 142
5.5.3 A Composition Formula for Characters 143
5.5.4 Fundamental Theorems about Characters 143
5.5.5 Projectors 146
5.5.6 Summary 147
5.6 Representation Theory of Finite Groups 148
5.6.1 The Regular Representation 148
5.6.2 The Character Matrix 149
CONTENTS vii
5.6.3 Completeness 151
5.7 Extension to Compact Lie Groups 151
Problems 157
6 Various Topics of Application 159
6.1 Bifurcation and A New Technique 159
6.1.1 Introduction 159
6.1.2 The Hopf Bifurcation 160
6.1.3 Symmetry Breaking 163
6.1.4 A New Approach 166
6.1.5 The Brusselator 168
6.2 A Diffusion Model in Probability Theory 170
6.2.1 Introduction 170
6.2.2 General Considerations 172
6.2.3 The Bernoulli Laplace Model 176
6.2.4 Discussion and Applications 179
Problems 179
7 Lie Algebras 181
7.1 Infinitesimal Operator and Exponential Map 181
7.1.1 Infinitesimal Operators 181
7.1.2 The Exponential Mapping 185
7.2 Lie Algebra of a Continuous Group 186
7.3 Representation of Lie Algebras 190
7.4 Representations of SU(2) and SO(3) 197
7.4.1 Infinitesimal Aspects of the Kronecker Product . . . 200
7.4.2 Clebsch Gordan Coefficients 201
7.5 Examples from Quantum Mechanics 202
7.5.1 Energy and Angular Momentum 203
7.5.2 Spin Orbit Coupling 204
Problems 206
8 Applications to Solid State Physics 209
8.1 Lattices 209
8.1.1 Reciprocal Lattice 211
8.1.2 Brillouin Zone, Stabilizer 212
8.2 Point Groups and Representations 214
8.2.1 The List of All ( f ) Groups 214
8.2.2 Representations and Characters of (+) Groups . . . 215
8.2.3 (±) Groups of the First Kind 216
8.2.4 (±) Groups of the Second Kind 217
8.2.5 Review 218
8.3 The 32 Crystal Classes 219
8.4 Symmetries and the Ritz Method 220
8.5 Examples of Applications 223
8.6 Crystallographic Space Groups 232
Problems 236
viii CONTENTS
9 Unitary and Orthogonal Groups 239
9.1 The Groups U(n) and SU(n) 239
9.1.1 Similarity Classes, Diagram 240
9.1.2 Characters of Irreducible Representations 242
9.1.3 Division Algorithm 244
9.1.4 Weights 247
9.1.5 Algorithm for the Complete Reduction 248
9.2 The Special Orthogonal Group SO(n) 250
9.2.1 Weyl Group and Diagram of SO(n) 252
9.2.2 Algorithm for the Computation of Weights 253
9.3 Subspaces of Representations of SU(3) 254
A 265
Answers to Selected Problems 265
Bibliography 280
Index 286
|
| any_adam_object | 1 |
| author | Fässler, Albert Stiefel, Eduard 1909-1978 |
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| dewey-raw | 512/.2 |
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| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
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| id | DE-604.BV005596705 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:32:07Z |
| institution | BVB |
| isbn | 0817635270 3764335270 |
| language | English German |
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| physical | XII, 296 S. graph. Darst. |
| publishDate | 1992 |
| publishDateSearch | 1992 |
| publishDateSort | 1992 |
| publisher | Birkhäuser |
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| spelling | Fässler, Albert Verfasser aut Gruppentheoretische Methoden und ihre Anwendung Group theoretical methods and their applications A. Fässler ; E. Stiefel Boston u.a. Birkhäuser 1992 XII, 296 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lie, Algèbres de Opérateurs linéaires Représentations de groupes Lie algebras Linear operators Representations of groups Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Anwendung (DE-588)4196864-5 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 s DE-604 Anwendung (DE-588)4196864-5 s Stiefel, Eduard 1909-1978 Verfasser (DE-588)13818061X aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003506153&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fässler, Albert Stiefel, Eduard 1909-1978 Group theoretical methods and their applications Lie, Algèbres de Opérateurs linéaires Représentations de groupes Lie algebras Linear operators Representations of groups Gruppentheorie (DE-588)4072157-7 gnd Anwendung (DE-588)4196864-5 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4196864-5 |
| title | Group theoretical methods and their applications |
| title_alt | Gruppentheoretische Methoden und ihre Anwendung |
| title_auth | Group theoretical methods and their applications |
| title_exact_search | Group theoretical methods and their applications |
| title_full | Group theoretical methods and their applications A. Fässler ; E. Stiefel |
| title_fullStr | Group theoretical methods and their applications A. Fässler ; E. Stiefel |
| title_full_unstemmed | Group theoretical methods and their applications A. Fässler ; E. Stiefel |
| title_short | Group theoretical methods and their applications |
| title_sort | group theoretical methods and their applications |
| topic | Lie, Algèbres de Opérateurs linéaires Représentations de groupes Lie algebras Linear operators Representations of groups Gruppentheorie (DE-588)4072157-7 gnd Anwendung (DE-588)4196864-5 gnd |
| topic_facet | Lie, Algèbres de Opérateurs linéaires Représentations de groupes Lie algebras Linear operators Representations of groups Gruppentheorie Anwendung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=003506153&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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