Group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have strongly influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced tremendous advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, can be modelled by symmetry groups. Thus group theory and the closely related representation theory have many applications in physics and chemistry. This new and important book gathers the latest research from around the globe in the study of group theory and highlights such topics as: application of symmetry analysis to the description of ordered structures in crystals, a survey of Lie Group analysis, graph groupoids and representations, and others.
Author(s): Charles W. Danellis
Series: Mathematics Research Developments
Publisher: Nova Science Pub Inc
Year: 2010
Language: English
Pages: 343
Tags: Математика;Общая алгебра;Теория групп;Теория представлений;
GROUP THEORY:CLASSES, REPRESENTATION ANDCONNECTIONS, AND APPLICATIONS......Page 3
GROUP THEORY:CLASSES, REPRESENTATION ANDCONNECTIONS, AND APPLICATIONS......Page 5
CONTENTS......Page 7
PREFACE......Page 9
ABSTRACT......Page 13
INTRODUCTION......Page 14
PHYSICAL CHARACTERISTICS OF THE PROBLEM......Page 15
SYMMETRY-ADAPTED DESCRIPTION OF THE ORDERING MODE......Page 16
CONDITIONS IMPOSED ON PHYSICAL SOLUTIONS AND THENUMBER OF FREE PARAMETERS......Page 18
"MODY" PROGRAM - A PRACTICAL IMPLEMENTATION OFSYMMETRY ANALYSIS......Page 21
SCALAR ORDER PARAMETERS: SITE OCCUPATION PROBABILITY......Page 22
VECTOR QUANTITIES - MAGNETIC MOMENTS OR ATOMICDISPLACEMENTS......Page 23
TENSORS - QUADRUPOLAR ORDERING IN SOLIDS......Page 24
EXAMPLE 1: MAGNETIC ORDERING – COMPARISON OF THREEORDERING WAVE-VECTORS IN......Page 26
EXAMPLE 2: ORDERING SITE OCCUPATION PROBABILITIES ANDACCOMPANYING ATOMIC DISPLACEMENTS IN ERMN2D2......Page 37
Hydrogen Ordering......Page 38
EXAMPLE 3: QUADRUPOLAR MOMENT TENSOR ORDERING IN UPD3......Page 43
SYMMETRY ANALYSIS OF THE ACCOMPANYING STRUCTURALDEFORMATIONS......Page 47
REFERENCES......Page 50
INTRODUCTION......Page 53
1.2. Examples......Page 55
2.1. Category of G- Representations......Page 56
2.3. G-Representations as Functor Categories......Page 57
2.4. Relative G- Representations as Functor Categories......Page 58
3.1. Some Generalities on Algebraic Groups......Page 59
3.2. Representations of G in P (F)......Page 62
3.3. G- Modules on G-spaces X......Page 63
1.1. Definition of ( ) n K C......Page 64
1.2. The Plus Construction – Another Definition of ( ( )) ( ) n n K PA =K A n≥ 1......Page 65
1.3. Examples of n K of Ordinary And Equivariant Exact Categories......Page 66
1.4. Mod- l s higher K-theory (ordinary and equivariant)......Page 68
1.5.2.Examples......Page 69
2.1. Mackey functors – Brief Review......Page 70
2.1.1. Definition......Page 71
2.2.2. Theorem [10] [39]......Page 73
2.3.1. Theorem [39] [10]......Page 74
Definition 2.3.4.......Page 75
1.1. On ( ) n K RG , ( ) n G RG , ( ) n SK RG , ( ) n SG RG , G finite......Page 76
1.1.4. Theorem [39] [47] [24]......Page 77
1.1.7. Theorem [39] [77]......Page 78
1.1.9. Theorem [39] [77]......Page 79
1.2. Consequences for Some Infinite Groups......Page 80
1.3. Profinite higher K-theory of RG, RV......Page 81
1.3.4. Remarks......Page 82
1.3.10. Theorem [39] [40]......Page 83
2.1. The representation ring R (H) and the group 0( ( )) G K VB G......Page 84
2.2. The groups ( , ), ( , ) n n K G X G G X , X a G-Scheme......Page 85
2.2.7. Theorem [50]......Page 86
2.3.2. Theorem [55]......Page 87
2.4.3. Theorem. [40]......Page 88
2.5.2. Remarks......Page 89
ACKNOWLEDGMENT......Page 90
REFERENCES......Page 91
ABSTRACT......Page 95
REVALUATED CITIZENSHIP......Page 96
LIBERAL NATIONALISM......Page 98
REVALUATED CITIZENSHIP AS NATION-BUILDING......Page 99
THE CONCEPT OF “INTEGRATION”......Page 101
SOCIETAL CULTURES AND NATIONS......Page 103
NATIONALITY AND THE INSTRUMENTAL ARGUMENTS......Page 104
NATURALIZATION REQUIREMENTS: INCLUSION OR EXCLUSION?......Page 107
CONCLUSION......Page 109
REFERENCES......Page 110
INTRODUCTION......Page 115
Torture......Page 118
Genocide......Page 121
The State, the Individual and Human Rights......Page 125
Rape, Torture, and the Political......Page 127
Rape as Genocide......Page 130
RAPE AS A FORM OF TORTURE: ARRIVING AT ANUNDERSTANDING OF THE ‘POLITICAL’......Page 132
Working Through what is ‘The Political’?......Page 134
What is the Political?......Page 135
Depositing the Political......Page 138
Associating the Political with Rape as a Form of Torture......Page 141
RAPE AS GENOCIDE: THEORETICAL IMPLICATIONS FOR THE GROUP......Page 144
The Group and the Contention of Group Rights......Page 147
Rape as Genocide......Page 154
CONCLUSION......Page 157
REFERENCES......Page 159
ABSTRACT......Page 165
TYPE OF TASK......Page 166
STEINER’S THEORY OF GROUP PROCESSES AND PRODUCTIVITY......Page 167
BION’S THEORY OF THE PROFESSIONAL WORK GROUP......Page 169
A COMBINATION OF THE TWO THEORIES......Page 170
The Work Column......Page 171
The Fight Column......Page 173
The Flight Column......Page 174
The Pairing Column......Page 175
REFERENCES......Page 176
1. Introduction......Page 179
2. The Group Structure......Page 180
3. The Representation......Page 182
4. Retrieval of Geometric Information......Page 184
References......Page 186
1. Introduction......Page 189
2.1. Lie algebras from Lie groups......Page 191
2.2. Adjoint representations and the Killing form......Page 192
2.3. Simple Lie algebras classification......Page 193
2.4. Lie groups from Lie algebras......Page 194
3.1. A toy model......Page 195
3.2. The generalized Euler construction.......Page 199
3.3.1. Geometric identification......Page 200
3.3.2. A topological method......Page 202
4.1. The Lie algebra......Page 203
4.2.1. The SU(3)-Euler parametrization......Page 206
4.2.2. The SO(4)-Euler parametrization......Page 207
5. Generalized Euler Angles for F4......Page 210
5.1.1. The maximal subgroup......Page 212
5.1.2. The whole F4......Page 213
6. The F4- Euler Angles for E6......Page 215
7.1. Analytic continuation of the generalized Euler angles......Page 217
7.2. The Iwasawa decomposition......Page 218
7.3. The coset manifold......Page 219
8. Realizing G2(2) and G2(2)/SO(4)......Page 220
8.1. Euler construction of G2(2)/SO(4)......Page 221
8.2. Iwasawa construction of G2(2)/SO(4)......Page 222
9. Conclusions......Page 224
References......Page 225
A SURVEY OF SOME RESULTS IN THELIE GROUP ANALYSIS......Page 229
4.1 Some Useful Results in the Theory of Ordinary Differential Equations......Page 230
4.2 Some Useful Results in the Theory of Partial Differential Equations......Page 234
5.1 An Overview of the Theory of Lie Groups......Page 243
5.1.2 Tangent Transformations......Page 245
5.1.3 Modern Theory of Lie Groups......Page 248
5.2.1. One-parameter Lie Groups of Transformations......Page 250
5.2.2 Multi - parameter Lie groups of Transformations and Lie Algebras......Page 261
5.3 Lie Group Analytical Approach for Valuation of Financial Derivatives......Page 263
REFERENCES......Page 272
Abstract......Page 275
1.1. Groupoids......Page 277
1.2. Canonical Representations of Groupoids......Page 278
2.1. Graph Groupoids......Page 279
2.2. Canonical Representations......Page 286
2.3. Connected Graphs......Page 288
3. Subgroupoids Induced by Full-Subgraphs......Page 289
4.1. Quotient Graphs......Page 292
4.2. Quotient Groupoids Induced by Graph-Groupoid Inclusions......Page 298
4.3. Representations of Quotient Groupoids......Page 302
5. Finite-Graph Fractaloids and Representations......Page 303
Abstract......Page 311
1. Introduction......Page 312
2. Finding the group of invariance in special and in general relativity......Page 313
3. Applying Kretschmann standpoint to solutions with intermediaterelativity postulate......Page 315
4. The singular border between submanifolds endowed with differentinvariance groups......Page 318
A The infinitesimal Killing vectors......Page 321
References......Page 322
1. Introduction......Page 325
2. The Airault-Malliavin-Baxendale equation......Page 327
4. Haar distribution on a path group......Page 329
References......Page 333
INDEX......Page 335
