Discrete, finite and Lie groups: comprehensive group theory in geometry and analysis
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Boston
De Gruyter
[2023]
|
| Schriftenreihe: | De Gruyter STEM
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| Schlagworte: | |
| Online-Zugang: | https://www.degruyter.com/isbn/9783111200750 Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | XXII, 508 Seiten Illustrationen, Diagramme 24 cm x 17 cm, 862 g |
| ISBN: | 9783111200750 3111200752 |
Internformat
MARC
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| 020 | |a 9783111200750 |c pbk: EUR 99.95 (DE) (freier Preis), EUR 99.95 (AT) (freier Preis) |9 978-3-11-120075-0 | ||
| 020 | |a 3111200752 |9 3-11-120075-2 | ||
| 024 | 3 | |a 9783111200750 | |
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| 100 | 1 | |a Fré, Pietro |d 1952- |e Verfasser |0 (DE-588)1107208297 |4 aut | |
| 245 | 1 | 0 | |a Discrete, finite and Lie groups |b comprehensive group theory in geometry and analysis |c Pietro Giuseppe Fré |
| 264 | 1 | |a Berlin ; Boston |b De Gruyter |c [2023] | |
| 300 | |a XXII, 508 Seiten |b Illustrationen, Diagramme |c 24 cm x 17 cm, 862 g | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a De Gruyter STEM | |
| 650 | 0 | 7 | |a Gruppentheorie |0 (DE-588)4072157-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lie-Algebra |0 (DE-588)4130355-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Analysis |0 (DE-588)4001865-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differentialgeometrie |0 (DE-588)4012248-7 |2 gnd |9 rswk-swf |
| 653 | |a Homogene Räume | ||
| 653 | |a Geodäten | ||
| 653 | |a Dynkin-Diagramme | ||
| 653 | |a Punkt- und Raumgruppen | ||
| 653 | |a Weyl Gruppe | ||
| 653 | |a Homogeneous Spaces | ||
| 653 | |a Geodesics | ||
| 653 | |a Dynkin Diagrams | ||
| 653 | |a Point and Space Groups | ||
| 653 | |a Weyl Group | ||
| 653 | |a Homogene Räume; Geodäten; Dynkin-Diagramme; Punkt- und Raumgruppen; Weyl Gruppe | ||
| 653 | |a Homogeneous Spaces; Geodesics; Dynkin Diagrams; Point and Space Groups; Weyl Group | ||
| 689 | 0 | 0 | |a Gruppentheorie |0 (DE-588)4072157-7 |D s |
| 689 | 0 | 1 | |a Lie-Algebra |0 (DE-588)4130355-6 |D s |
| 689 | 0 | 2 | |a Differentialgeometrie |0 (DE-588)4012248-7 |D s |
| 689 | 0 | 3 | |a Analysis |0 (DE-588)4001865-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 710 | 2 | |a Walter de Gruyter GmbH & Co. KG |0 (DE-588)10095502-2 |4 pbl | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe, PDF |z 978-3-11-120153-5 |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe, EPUB |z 978-3-11-120277-8 |
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| 856 | 4 | 2 | |m DNB Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034365579&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-034365579 | ||
| 883 | 1 | |8 1\p |a vlb |d 20230327 |q DE-101 |u https://d-nb.info/provenance/plan#vlb | |
Datensatz im Suchindex
| _version_ | 1804185451042439168 |
|---|---|
| adam_text | CONTENTS
PREFACE
-
VII
SOME
REMARKS
ABOUT
NOTATIONS
-
XIII
1
1.1
1.2
1.3
1.3.1
GROUPS:
THE
INTUITIVE
NOTION
-
1
EXAMPLES
-
2
GROUPS
AS
TRANSFORMATION
GROUPS
-
3
REPRESENTATIONS
OF
A
GROUP
-
4
VECTOR
SPACES
-
5
2
2.1
2.1.1
2.1.2
2.2
2.3
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
2.3.7
2.3.8
2.3.9
2.3.10
2.4
2.5
2.6
2.6.1
2.6.2
2.7
2.8
2.8.1
2.8.2
2.8.3
2.8.4
2.8.5
2.9
2.9.1
FUNDAMENTAL
NOTIONS
OF
ALGEBRA
-
8
HISTORICAL
REMARKS
-
8
CAYLEY
AND
SYLVESTER:
A
SHORT
ACCOUNT
OF
THEIR
LIVES
-
8
GALOIS
AND
THE
ADVENT
OF
GROUP
THEORY
-
10
SUMMARY
OF
THE
CONTENT
OF
THIS
CHAPTER
-
12
GROUPS
-
12
SOME
EXAMPLES
-
13
ABELIAN
GROUPS
-
13
THE
GROUP
COMMUTATOR
-
13
CONJUGATE
ELEMENTS
-
14
ORDER
AND
DIMENSION
OF
A
GROUP
-
14
ORDER
OF
AN
ELEMENT
-
14
THE
MULTIPLICATION
TABLE
OF
A
FINITE
GROUP
-
15
HOMOMORPHISMS,
ISOMORPHISMS,
AND
AUTOMORPHISMS
-
15
RANK,
GENERATORS,
AND
RELATIONS
(A
FIRST
BIRD
S-EYE
VIEW)
-
16
SUBGROUPS
-
17
RINGS
-
18
FIELDS
-
19
VECTOR
SPACES
-
20
DUAL
VECTOR
SPACES
-
20
INNER
PRODUCTS
-
21
ALGEBRAS
-
22
LIE
ALGEBRAS
-
22
HOMOMORPHISM
-
23
SUBALGEBRAS
AND
IDEALS
-
23
REPRESENTATIONS
-
24
ISOMORPHISM
-
24
ADJOINT
REPRESENTATION
-
24
MODULI
AND
REPRESENTATIONS
-
25
FIELDS,
ALGEBRAIC
CLOSURE,
AND
DIVISION
ALGEBRAS
-
26
XVI
-
CONTENTS
2.10
THERE
ARE
ONLY
TWO
OTHER
DIVISION
ALGEBRAS:
THE
QUATERNIONS
AND
THE
OCTONIONS
-
27
2.10.1
2.10.2
2.11
FROBENIUS
AND
HIS
THEOREM
-
27
GALOIS
AND
FIELD
EXTENSIONS
-
28
BIBLIOGRAPHICAL
NOTE
-
30
3
3.1
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.2.7
3.2.8
3.3
3.3.1
3.4
3.4.1
3.4.2
3.5
3.5.1
3.5.2
3.5.3
3.5.4
3.5.5
3.5.6
3.5.7
3.6
GROUPS:
NOTICEABLE
EXAMPLES
AND
SOME
DEVELOPMENTS
-
31
SURVEY
OF
THE
CONTENTS
OF
THIS
CHAPTER
-
31
GROUPS
OF
MATRICES
-
31
GENERAL
LINEAR
GROUPS
-
31
SPECIAL
LINEAR
GROUPS
-
32
UNITARY
GROUPS
-
32
SPECIAL
UNITARY
GROUPS
-
32
ORTHOGONAL
GROUPS
-
33
SPECIAL
ORTHOGONAL
GROUPS
-
33
SYMPLECTIC
GROUPS
-
33
GROUPS
OF
TRANSFORMATIONS
-
34
SOME
EXAMPLES
OF
FINITE
GROUPS
-
34
THE
PERMUTATION
GROUPS
S
N
-
35
GROUPS
AS
THE
INVARIANCES
OF
A
GIVEN
GEOMETRY
-
36
THE
EUCLIDEAN
GROUPS
-
37
PROJECTIVE
GEOMETRY
-
39
MATRIX
GROUPS
AND
VECTOR
SPACES
-
40
GENERAL
LINEAR
GROUPS
AND
BASIS
CHANGES
IN
VECTOR
SPACES
-
40
TENSOR
PRODUCT
SPACES
-
40
(ANTI)SYMMETRIZED
PRODUCT
SPACES
-
41
EXTERIOR
FORMS
-
42
SPECIAL
LINEAR
GROUPS
AS
VOLUME-PRESERVING
TRANSFORMATIONS
-
44
METRIC-PRESERVING
CHANGES
OF
BASIS
-
44
ISOMETRIES
-
47
BIBLIOGRAPHICAL
NOTE
-
48
4
4.1
4.2
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
BASIC
ELEMENTS
OF
FINITE
GROUP
THEORY
-
49
INTRODUCTION
-
49
BASIC
NOTIONS
AND
STRUCTURAL
THEOREMS
FOR
FINITE
GROUPS
-
50
CAYLEY
S
THEOREM
-
50
LEFT
AND
RIGHT
COSETS
-
51
LAGRANGE
S
THEOREM
-
52
CONJUGACY
CLASSES
-
53
CONJUGATE
SUBGROUPS
-
55
CENTER,
CENTRALIZERS,
AND
NORMALIZERS
-
56
THE
DERIVED
GROUP
-
57
CONTENTS
-
XVII
4.2.8
4.2.9
4.2.10
4.2.11
4.2.12
4.2.13
4.3
4.3.1
4.3.2
4.3.3
4.3.4
4.4
SIMPLE,
SEMI-SIMPLE,
AND
SOLVABLE
GROUPS
-
57
EXAMPLES
OF
SIMPLE
GROUPS
-
58
HOMOMORPHISM
THEOREMS
-
58
DIRECT
PRODUCTS
-
59
ACTION
OF
A
GROUP
ON
A
SET
-
60
SEMI-DIRECT
PRODUCTS
-
62
LINEAR
REPRESENTATIONS
OF
FINITE
GROUPS
-
64
SCHUR
S
LEMMAS
-
66
CHARACTERS
-
69
DECOMPOSITION
OF
A
REPRESENTATION
INTO
IRREDUCIBLE
REPRESENTATIONS
-
70
THE
REGULAR
REPRESENTATION
-
71
STRATEGY
TO
CONSTRUCT
THE
IRREDUCIBLE
REPRESENTATIONS
OF
A
SOLVABLE
GROUP
-
72
4.4.1
4.4.2
4.5
THE
INDUCTIVE
ALGORITHM
FOR
IRREPS
-
72
THE
OCTAHEDRAL
GROUP
024
~
S
4
AND
ITS
IRREPS
-
75
BIBLIOGRAPHICAL
NOTE
-
80
5
5.1
5.2
5.2.1
5.2.2
5.2.3
5.2.4
5.3
FINITE
SUBGROUPS
OF
SO(3):
THE
ADE
CLASSIFICATION
-
81
INTRODUCTION
-
81
ADE
CLASSIFICATION
OF
THE
FINITE
SUBGROUPS
OF
SU(2)
-
82
THE
ARGUMENT
LEADING
TO
THE
DIOPHANTINE
EQUATION
-
83
CASE
R
=
2:
THE
INFINITE
SERIES
OF
CYCLIC
GROUPS
AN
-
88
CASE
R
=
3
AND
ITS
SOLUTIONS
-
88
SUMMARY
OF
THE
ADE
CLASSIFICATION
OF
FINITE
ROTATION
GROUPS
-
91
BIBLIOGRAPHICAL
NOTE
-
93
6
6.1
6.2
6.2.1
6.2.2
6.2.3
6.3
MANIFOLDS
AND
LIE
GROUPS
-
94
INTRODUCTION
-
94
DIFFERENTIABLE
MANIFOLDS
-
95
HOMEOMORPHISMS
AND
THE
DEFINITION
OF
MANIFOLDS
-
97
FUNCTIONS
ON
MANIFOLDS
-
102
GERMS
OF
SMOOTH
FUNCTIONS
-
104
HOLOMORPHIC
FUNCTIONS
REVISITED
AND
THE
CONFORMAL
AUTOMORPHISM
GROUP
OF
CUM
----
105
6.3.1
6.4
6.4.1
6.4.2
6.5
6.6
6.7
6.8
THE
ANNULUS
AND
ITS
PREIMAGE
ON
THE
RIEMANN
SPHERE
-
105
TANGENT
AND
COTANGENT
SPACES
-
116
TANGENT
VECTORS
AT
A
POINT
P
YY
J
-
117
DIFFERENTIAL
FORMS
AT
A
POINT
P
YY
J
-
121
ABOUT
THE
CONCEPT
OF
FIBER
BUNDLE
-
123
THE
NOTION
OF
LIE
GROUP
-
124
DEVELOPING
THE
NOTION
OF
FIBER
BUNDLE
-
125
TANGENT
AND
COTANGENT
BUNDLES
-
131
XVIII
-
CONTENTS
6.8.1
6.8.2
6.8.3
6.8.4
6.9
6.9.1
6.9.2
6.9.3
6.9.4
6.10
SECTIONS
OF
A
BUNDLE
-
133
THE
LIE
ALGEBRA
OF
VECTOR
FIELDS
-
135
THE
COTANGENT
BUNDLE
AND
DIFFERENTIAL
FORMS
-
136
DIFFERENTIAL
K-FORMS
-
139
HOMOTOPY,
HOMOLOGY,
AND
COHOMOLOGY
-
141
HOMOTOPY
-
143
HOMOLOGY
-
146
HOMOLOGY
AND
COHOMOLOGY
GROUPS:
GENERAL
CONSTRUCTION
-
152
RELATION
BETWEEN
HOMOTOPY
AND
HOMOLOGY
-
154
HOLOMORPHIC
FUNCTIONS
AND
THEIR
INTEGRALS
IN
VIEW
OF
HOMOTOPY,
HOMOLOGY,
AND
COHOMOLOGY
-
155
6.10.1
6.10.2
6.10.3
CURVILINEAR
INTEGRALS
IN
THE
IR
2
PLANE
-
155
THE
PRIMITIVE
OF
A
DIFFERENTIAL
1-FORM
-
158
THE
GREEN-RIEMANN
FORMULA,
AN
INSTANCE
OF
THE
GENERAL
STOKES
THEOREM
-
160
6.10.4
6.10.5
6.10.6
6.10.7
6.10.8
6.11
ILLUSTRATIONS
OF
HOMOTOPY
IN
2
-
162
CONSEQUENCES
OF
A
FUNDAMENTAL
THEOREM
-
166
HOLOMORPHICITY
-
167
HOLOMORPHICITY
CONDITION
-
169
CAUCHY
THEOREM
-
170
BIBLIOGRAPHICAL
NOTE
-
176
7
7.1
7.1.1
7.2
7.3
7.4
7.4.1
7.4.2
7.5
THE
RELATION
BETWEEN
LIE
GROUPS
AND
LIE
ALGEBRAS
-
177
THE
LIE
ALGEBRA
OF
A
LIE
GROUP
-
177
LEFT/RIGHT-INVARIANT
VECTOR
FIELDS
-
179
MAURER-CARTAN FORMS
ON
LIE
GROUP
MANIFOLDS
-
184
MAURER-CARTAN
EQUATIONS
-
187
MATRIX
LIE
GROUPS
-
187
SOME
PROPERTIES
OF
MATRICES
-
187
LINEAR
LIE
GROUPS
-
189
BIBLIOGRAPHICAL
NOTE
-
189
8
8.1
8.1.1
8.1.2
8.1.3
8.1.4
CRYSTALLOGRAPHIC
GROUPS
AND
GROUP
EXTENSIONS
-
191
LATTICES
AND
CRYSTALLOGRAPHIC
GROUPS
-
191
LATTICES
-
191
THE
N-TORUST
-
192
CRYSTALLOGRAPHIC
GROUPS
AND
THE
BRAVAIS
LATTICES
FOR
N
=
3
AND
N
=
2
-
193
RIGOROUS
MATHEMATICAL
CLASSIFICATION
OF
THE
POINT
GROUPS
IN
TWO
DIMENSIONS
-
198
8.2
8.2.1
8.2.2
THE
PROPER
POINT
GROUPS:
THE
OCTAHEDRAL
GROUP
024
-
200
THE
CUBIC
LATTICE
AND
THE
OCTAHEDRAL
POINT
GROUP
-
200
IRREDUCIBLE
REPRESENTATIONS
OF
THE
OCTAHEDRAL
GROUP
-
201
CONTENTS
-
XIX
8.3
THE
FULL
TETRAHEDRAL
GROUP
T$2
AND
THE
OCTAHEDRAL
GROUP
024
ARE
ISOMORPHIC
-
203
8.3.1
8.3.2
8.4
8.4.1
8.5
8.5.1
8.5.2
8.5.3
8.5.4
8.5.5
8.6
THE
VIBRATIONS
OF
XY4
MOLECULES
-
203
STRUCTURE
OF
THE
FULL
TETRAHEDRAL
GROUP
-
205
GROUP
EXTENSIONS
AND
SPACE
GROUPS
-
212
GENERAL
THEORY
OF
GROUP
EXTENSIONS
-
212
SPACE
GROUPS
-
215
THE
BASIC
INGREDIENT
OF
THE
CONSTRUCTION:
THE
LIFTING
MAP
-
216
THE
PRODUCT
LAW
AND
THE
CLOSURE
CONDITION
-
217
FINITE
GROUP
COHOMOLOGY
-
219
FROBENIUS
CONGRUENCES
-
220
ANOTHER
EXAMPLE
IN
THE
PLANE:
THE
TETRAGONAL
LATTICE
-
224
BIBLIOGRAPHICAL
NOTE
-
224
9
9.1
9.2
9.3
9.4
9.4.1
9.4.2
MONODROMY
GROUPS
OF
DIFFERENTIAL
EQUATIONS
-
225
THE
ROLE
OF
DIFFERENTIAL
EQUATIONS
IN
ALL
BRANCHES
OF
SCIENCE
-
225
GROUPS
IN
A
NEW
CAPACITY
-
226
ORDINARY
DIFFERENTIAL
EQUATIONS
-
227
SECOND
ORDER
DIFFERENTIAL
EQUATIONS
WITH
SINGULAR
POINTS
-
227
SOLUTIONS
AT
REGULAR
SINGULAR
POINTS
AND
THE
INDICIAL
EQUATION
-
229
FUCHSIAN
EQUATIONS
WITH
THREE
REGULAR
SINGULAR
POINTS
AND
THE
P-SYMBOL
-
232
9.4.3
9.5
THE
HYPERGEOMETRIC
EQUATION
AND
ITS
SOLUTIONS
-
235
AN
EXAMPLE
OF
MONODROMY
GROUP:
DIFFERENTIAL
EQUATIONS
AND
TOPOLOGY
-
241
9.5.1
9.5.2
9.5.3
9.6
9.7
THE
TETRAHEDRAL
GROUP
IN
TWO
DIMENSIONS
AND
THE
TORUS
-
241
AN
ALGEBRAIC
REPRESENTATION
OF
THE
TORUS
BY
MEANS
OF
A
CUBIC
-
244
A
DIFFERENTIAL
EQUATION
ENTERS
THE
PLAY
AND
BRINGS
IN
A
NEW
GROUP
-
245
CONCLUSIVE
REMARKS
ON
THIS
CHAPTER
-
252
BIBLIOGRAPHICAL
NOTE
-
252
10
10.1
10.2
10.3
10.3.1
10.3.2
10.3.3
10.3.4
10.4
10.4.1
10.5
STRUCTURE
OF
LIE
ALGEBRAS
-
253
INTRODUCTION
-
253
LINEAR
ALGEBRA
PRELIMINARIES
-
253
TYPES
OF
LIE
ALGEBRAS
AND
LEVI
DECOMPOSITION
-
256
SOLVABLE
LIE
ALGEBRAS
-
256
SEMI-SIMPLE
LIE
ALGEBRAS
-
259
LEVI
S
DECOMPOSITION
OF
LIE
ALGEBRAS
-
260
AN
ILLUSTRATIVE
EXAMPLE:
THE
GALILEI
GROUP
-
265
THE
ADJOINT
REPRESENTATION
AND
CARTAN
S
CRITERIA
-
266
CARTAN
S
CRITERIA
-
267
BIBLIOGRAPHICAL
NOTE
-
269
XX
-
CONTENTS
11
11.1
11.2
11.2.1
11.2.2
11.3
11.4
11.4.1
11.4.2
11.5
11.5.1
11.5.2
11.5.3
11.5.4
11.5.5
11.6
ROOT
SYSTEMS
AND
THEIR
CLASSIFICATION
-
270
CARTAN
SUBALGEBRAS
-
270
ROOT
SYSTEMS
-
273
FINAL
FORM
OF
THE
SEMI-SIMPLE
LIE
ALGEBRA
-
276
PROPERTIES
OF
ROOT
SYSTEMS
-
276
SIMPLE
ROOTS,
THE
WEYL
GROUP,
AND
THE
CARTAN
MATRIX
-
280
CLASSIFICATION
OF
THE
IRREDUCIBLE
ROOT
SYSTEMS
-
282
DYNKIN
DIAGRAMS
-
282
THE
CLASSIFICATION
THEOREM
-
284
IDENTIFICATION
OF
THE
CLASSICAL
LIE
ALGEBRAS
-
290
THE
AE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
290
THE
DE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
295
THE
BE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
296
THE
CE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
298
THE
EXCEPTIONAL
UE
ALGEBRAS
-
299
BIBLIOGRAPHICAL
NOTE
-
299
12
12.1
12.1.1
12.2
12.2.1
12.2.2
LIE
ALGEBRA
REPRESENTATION
THEORY
-
301
LINEAR
REPRESENTATIONS
OF
A
LIE
ALGEBRA
-
301
WEIGHTS
OF
A
REPRESENTATION
AND
THE
WEIGHT
LATTICE
-
302
DISCUSSION
OF
TENSOR
PRODUCTS
AND
EXAMPLES
-
311
TENSOR
PRODUCTS
AND
IRREPS
-
311
THE
LIE
ALGEBRA
A
2
,
ITS
WEYL
GROUP,
AND
EXAMPLES
OF
ITS
REPRESENTATIONS
-
315
12.2.3
THE
LIE
ALGEBRA
SP(4,
IR)
~
SO(2,
3),
ITS
FUNDAMENTAL
REPRESENTATION,
AND
ITS
WEYL
GROUP
-
326
12.3
12.4
CONCLUSIONS
FOR
THIS
CHAPTER
-
336
BIBLIOGRAPHICAL
NOTE
-
337
13
13.1
13.2
13.2.1
EXCEPTIONAL
LIE
ALGEBRAS
-
338
THE
EXCEPTIONAL
LIE
ALGEBRA
G2
-
338
THE
LIE
ALGEBRA
F4
AND
ITS
FUNDAMENTAL
REPRESENTATION
-
342
EXPLICIT
CONSTRUCTION
OF
THE
FUNDAMENTAL
AND
ADJOINT
REPRESENTATION
OF
FA
-
346
13.3
13.3.1
13.3.2
13.4
THE
EXCEPTIONAL
LIE
ALGEBRA
EG
-
352
CONSTRUCTION
OF
THE
ADJOINT
REPRESENTATION
-
354
FINAL
COMMENTS
ON
THE
EG
ROOT
SYSTEMS
-
358
BIBLIOGRAPHICAL
NOTE
-
359
14
14.1
14.1.1
IN
DEPTH
STUDY
OF
A
SIMPLE
GROUP
-
360
A
SIMPLE
CRYSTALLOGRAPHIC
POINT
GROUP
IN
SEVEN
DIMENSIONS
-
360
THE
SIMPLE
GROUP
L
168
-
361
CONTENTS
-
XXI
14.1.2
14.1.3
14.1.4
14.1.5
14.1.6
14.1.7
14.2
STRUCTURE
OF
THE
SIMPLE
GROUP
L168
=
PSL(2,
Z7)
-
362
THE
SEVEN-DIMENSIONAL
IRREDUCIBLE
REPRESENTATION
-
363
THE
THREE-DIMENSIONAL
COMPLEX
REPRESENTATIONS
-
366
THE
SIX-DIMENSIONAL
REPRESENTATION
-
367
THE
EIGHT-DIMENSIONAL
REPRESENTATION
-
368
THE
PROPER
SUBGROUPS
OF
L
168
-
368
BIBLIOGRAPHICAL
NOTE
-
374
15
15.1
15.2
15.2.1
15.3
15.4
15.4.1
15.5
A
PRIMARY
ON
THE
THEORY
OF
CONNECTIONS
AND
METRICS
-
375
INTRODUCTION
-
375
CONNECTIONS
ON
PRINCIPAL
BUNDLES:
THE
MATHEMATICAL
DEFINITION
-
376
EHRESMANN
CONNECTIONS
ON
A
PRINCIPAL
FIBER
BUNDLE
-
377
CONNECTIONS
ON
A
VECTOR
BUNDLE
-
385
AN
ILLUSTRATIVE
EXAMPLE
OF
FIBER
BUNDLE
AND
CONNECTION
-
388
THE
MAGNETIC
MONOPOLE
AND
THE
HOPF
FIBRATION
OF
S
3
-
389
RIEMANNIAN
AND
PSEUDO-RIEMANNIAN
METRICS:
THE
MATHEMATICAL
DEFINITION
-
394
15.5.1
15.6
15.6.1
15.6.2
15.7
15.8
SIGNATURES
-
395
THE
LEVI-CIVITA
CONNECTION
-
397
AFFINE
CONNECTIONS
-
397
CURVATURE
AND
TORSION
OF
AN
AFFINE
CONNECTION
-
398
GEODESICS
-
401
GEODESICS
IN
LORENTZIAN
AND
RIEMANNIAN
MANIFOLDS:
TWO
SIMPLE
EXAMPLES
-
403
15.8.1
15.8.2
15.8.3
15.9
THE
LORENTZIAN
EXAMPLE
OF
DS
2
-
403
THE
RIEMANNIAN
EXAMPLE
OF
THE
LOBACHEVSKY-POINCARE
PLANE
-
409
ANOTHER
RIEMANNIAN
EXAMPLE:
THE
CATENOID
-
412
BIBLIOGRAPHICAL
NOTE
-
415
16
16.1
16.1.1
16.1.2
16.1.3
16.2
16.3
16.3.1
16.4
16.5
16.5.1
16.6
ISOMETRIES
AND
THE
GEOMETRY
OF
COSET
MANIFOLDS
-
416
CONCEPTUAL
AND
HISTORICAL
INTRODUCTION
-
416
SYMMETRIC
SPACES
AND
ELIE
CARTAN
-
417
WHERE
AND
HOW
DO
COSET
MANIFOLDS
COME
INTO
PLAY?
-
418
THE
DEEP
INSIGHT
OF
SUPERSYMMETRY
-
419
ISOMETRIES
AND
KILLING
VECTOR
FIELDS
-
420
COSET
MANIFOLDS
-
420
THE
GEOMETRY
OF
COSET
MANIFOLDS
-
425
THE
REAL
SECTIONS
OF
A
COMPLEX
LIE
ALGEBRA
AND
SYMMETRIC
SPACES
-
436
THE
SOLVABLE
GROUP
REPRESENTATION
OF
NON-COMPACT
COSET
MANIFOLDS
-
439
THE
TITS-SATAKE
PROJECTION:
JUST
A
FLASH
-
442
BIBLIOGRAPHICAL
NOTE
-
443
XXII
-
CONTENTS
17
17.1
17.2
17.2.1
17.2.2
FUNCTIONAL
SPACES
AND
NON-COMPACT
LIE
ALGEBRAS
-
444
INTRODUCTION
TO
AN
INTRODUCTION
-
444
THE
IDEA
OF
FUNCTIONAL
SPACES
-
447
LIMITS
OF
SUCCESSIONS
OF
CONTINUOUS
FUNCTIONS
-
449
AVERY
SHORT
INTRODUCTION
TO
MEASURE
THEORY
AND
THE
LEBESGUE
INTEGRAL
-
450
17.2.3
17.2.4
17.2.5
17.2.6
17.2.7
17.3
17.3.1
17.3.2
17.4
SPACE
OF
SQUARE
SUMMABLE
FUNCTIONS
-
452
HILBERT
SPACE
-
453
INFINITE
ORTHONORMAL
BASES
AND
THE
WEIERSTRASS
THEOREM
-
455
CONSEQUENCES
OF
THE
WEIERSTRASS
THEOREM
-
456
THE
SCHMIDT
ORTHOGONALIZATION
ALGORITHM
-
457
ORTHOGONAL
POLYNOMIALS
-
459
THE
CLASSICAL
ORTHOGONAL
POLYNOMIALS
-
459
THE
DIFFERENTIAL
EQUATION
SATISFIED
BY
ORTHOGONAL
POLYNOMIALS
-
461
THE
HEISENBERG
GROUP,
HERMITE
POLYNOMIALS,
AND
THE
HARMONIC
OSCILLATOR
-
463
17.4.1
17.5
THE
HEISENBERG
GROUP
AND
ITS
LIE
ALGEBRA
-
463
SELF-ADJOINT
OPERATORS
-
469
18
18.1
18.1.1
18.1.2
18.2
HARMONIC
ANALYSIS
AND
CONCLUSIVE
REMARKS
-
471
A
FEW
HIGHLIGHTS
ON
HARMONIC
ANALYSIS
-
472
HARMONICS
ON
COSET
SPACES
-
473
DIFFERENTIAL
OPERATORS
ON
H-HARMONICS
-
475
CONCLUSIVE
REMARKS
-
476
A
AVAILABLE
MATHEMATICA
NOTEBOOKS
WRITTEN
BY
THE
AUTHOR
-
479
BIBLIOGRAPHY
-
497
ABOUT
THE
AUTHOR
-
503
INDEX
-
505
|
| adam_txt |
CONTENTS
PREFACE
-
VII
SOME
REMARKS
ABOUT
NOTATIONS
-
XIII
1
1.1
1.2
1.3
1.3.1
GROUPS:
THE
INTUITIVE
NOTION
-
1
EXAMPLES
-
2
GROUPS
AS
TRANSFORMATION
GROUPS
-
3
REPRESENTATIONS
OF
A
GROUP
-
4
VECTOR
SPACES
-
5
2
2.1
2.1.1
2.1.2
2.2
2.3
2.3.1
2.3.2
2.3.3
2.3.4
2.3.5
2.3.6
2.3.7
2.3.8
2.3.9
2.3.10
2.4
2.5
2.6
2.6.1
2.6.2
2.7
2.8
2.8.1
2.8.2
2.8.3
2.8.4
2.8.5
2.9
2.9.1
FUNDAMENTAL
NOTIONS
OF
ALGEBRA
-
8
HISTORICAL
REMARKS
-
8
CAYLEY
AND
SYLVESTER:
A
SHORT
ACCOUNT
OF
THEIR
LIVES
-
8
GALOIS
AND
THE
ADVENT
OF
GROUP
THEORY
-
10
SUMMARY
OF
THE
CONTENT
OF
THIS
CHAPTER
-
12
GROUPS
-
12
SOME
EXAMPLES
-
13
ABELIAN
GROUPS
-
13
THE
GROUP
COMMUTATOR
-
13
CONJUGATE
ELEMENTS
-
14
ORDER
AND
DIMENSION
OF
A
GROUP
-
14
ORDER
OF
AN
ELEMENT
-
14
THE
MULTIPLICATION
TABLE
OF
A
FINITE
GROUP
-
15
HOMOMORPHISMS,
ISOMORPHISMS,
AND
AUTOMORPHISMS
-
15
RANK,
GENERATORS,
AND
RELATIONS
(A
FIRST
BIRD
'
S-EYE
VIEW)
-
16
SUBGROUPS
-
17
RINGS
-
18
FIELDS
-
19
VECTOR
SPACES
-
20
DUAL
VECTOR
SPACES
-
20
INNER
PRODUCTS
-
21
ALGEBRAS
-
22
LIE
ALGEBRAS
-
22
HOMOMORPHISM
-
23
SUBALGEBRAS
AND
IDEALS
-
23
REPRESENTATIONS
-
24
ISOMORPHISM
-
24
ADJOINT
REPRESENTATION
-
24
MODULI
AND
REPRESENTATIONS
-
25
FIELDS,
ALGEBRAIC
CLOSURE,
AND
DIVISION
ALGEBRAS
-
26
XVI
-
CONTENTS
2.10
THERE
ARE
ONLY
TWO
OTHER
DIVISION
ALGEBRAS:
THE
QUATERNIONS
AND
THE
OCTONIONS
-
27
2.10.1
2.10.2
2.11
FROBENIUS
AND
HIS
THEOREM
-
27
GALOIS
AND
FIELD
EXTENSIONS
-
28
BIBLIOGRAPHICAL
NOTE
-
30
3
3.1
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.2.7
3.2.8
3.3
3.3.1
3.4
3.4.1
3.4.2
3.5
3.5.1
3.5.2
3.5.3
3.5.4
3.5.5
3.5.6
3.5.7
3.6
GROUPS:
NOTICEABLE
EXAMPLES
AND
SOME
DEVELOPMENTS
-
31
SURVEY
OF
THE
CONTENTS
OF
THIS
CHAPTER
-
31
GROUPS
OF
MATRICES
-
31
GENERAL
LINEAR
GROUPS
-
31
SPECIAL
LINEAR
GROUPS
-
32
UNITARY
GROUPS
-
32
SPECIAL
UNITARY
GROUPS
-
32
ORTHOGONAL
GROUPS
-
33
SPECIAL
ORTHOGONAL
GROUPS
-
33
SYMPLECTIC
GROUPS
-
33
GROUPS
OF
TRANSFORMATIONS
-
34
SOME
EXAMPLES
OF
FINITE
GROUPS
-
34
THE
PERMUTATION
GROUPS
S
N
-
35
GROUPS
AS
THE
INVARIANCES
OF
A
GIVEN
GEOMETRY
-
36
THE
EUCLIDEAN
GROUPS
-
37
PROJECTIVE
GEOMETRY
-
39
MATRIX
GROUPS
AND
VECTOR
SPACES
-
40
GENERAL
LINEAR
GROUPS
AND
BASIS
CHANGES
IN
VECTOR
SPACES
-
40
TENSOR
PRODUCT
SPACES
-
40
(ANTI)SYMMETRIZED
PRODUCT
SPACES
-
41
EXTERIOR
FORMS
-
42
SPECIAL
LINEAR
GROUPS
AS
VOLUME-PRESERVING
TRANSFORMATIONS
-
44
METRIC-PRESERVING
CHANGES
OF
BASIS
-
44
ISOMETRIES
-
47
BIBLIOGRAPHICAL
NOTE
-
48
4
4.1
4.2
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.2.7
BASIC
ELEMENTS
OF
FINITE
GROUP
THEORY
-
49
INTRODUCTION
-
49
BASIC
NOTIONS
AND
STRUCTURAL
THEOREMS
FOR
FINITE
GROUPS
-
50
CAYLEY
'
S
THEOREM
-
50
LEFT
AND
RIGHT
COSETS
-
51
LAGRANGE
'
S
THEOREM
-
52
CONJUGACY
CLASSES
-
53
CONJUGATE
SUBGROUPS
-
55
CENTER,
CENTRALIZERS,
AND
NORMALIZERS
-
56
THE
DERIVED
GROUP
-
57
CONTENTS
-
XVII
4.2.8
4.2.9
4.2.10
4.2.11
4.2.12
4.2.13
4.3
4.3.1
4.3.2
4.3.3
4.3.4
4.4
SIMPLE,
SEMI-SIMPLE,
AND
SOLVABLE
GROUPS
-
57
EXAMPLES
OF
SIMPLE
GROUPS
-
58
HOMOMORPHISM
THEOREMS
-
58
DIRECT
PRODUCTS
-
59
ACTION
OF
A
GROUP
ON
A
SET
-
60
SEMI-DIRECT
PRODUCTS
-
62
LINEAR
REPRESENTATIONS
OF
FINITE
GROUPS
-
64
SCHUR
'
S
LEMMAS
-
66
CHARACTERS
-
69
DECOMPOSITION
OF
A
REPRESENTATION
INTO
IRREDUCIBLE
REPRESENTATIONS
-
70
THE
REGULAR
REPRESENTATION
-
71
STRATEGY
TO
CONSTRUCT
THE
IRREDUCIBLE
REPRESENTATIONS
OF
A
SOLVABLE
GROUP
-
72
4.4.1
4.4.2
4.5
THE
INDUCTIVE
ALGORITHM
FOR
IRREPS
-
72
THE
OCTAHEDRAL
GROUP
024
~
S
4
AND
ITS
IRREPS
-
75
BIBLIOGRAPHICAL
NOTE
-
80
5
5.1
5.2
5.2.1
5.2.2
5.2.3
5.2.4
5.3
FINITE
SUBGROUPS
OF
SO(3):
THE
ADE
CLASSIFICATION
-
81
INTRODUCTION
-
81
ADE
CLASSIFICATION
OF
THE
FINITE
SUBGROUPS
OF
SU(2)
-
82
THE
ARGUMENT
LEADING
TO
THE
DIOPHANTINE
EQUATION
-
83
CASE
R
=
2:
THE
INFINITE
SERIES
OF
CYCLIC
GROUPS
AN
-
88
CASE
R
=
3
AND
ITS
SOLUTIONS
-
88
SUMMARY
OF
THE
ADE
CLASSIFICATION
OF
FINITE
ROTATION
GROUPS
-
91
BIBLIOGRAPHICAL
NOTE
-
93
6
6.1
6.2
6.2.1
6.2.2
6.2.3
6.3
MANIFOLDS
AND
LIE
GROUPS
-
94
INTRODUCTION
-
94
DIFFERENTIABLE
MANIFOLDS
-
95
HOMEOMORPHISMS
AND
THE
DEFINITION
OF
MANIFOLDS
-
97
FUNCTIONS
ON
MANIFOLDS
-
102
GERMS
OF
SMOOTH
FUNCTIONS
-
104
HOLOMORPHIC
FUNCTIONS
REVISITED
AND
THE
CONFORMAL
AUTOMORPHISM
GROUP
OF
CUM
----
105
6.3.1
6.4
6.4.1
6.4.2
6.5
6.6
6.7
6.8
THE
ANNULUS
AND
ITS
PREIMAGE
ON
THE
RIEMANN
SPHERE
-
105
TANGENT
AND
COTANGENT
SPACES
-
116
TANGENT
VECTORS
AT
A
POINT
P
YY
J
-
117
DIFFERENTIAL
FORMS
AT
A
POINT
P
YY
J
-
121
ABOUT
THE
CONCEPT
OF
FIBER
BUNDLE
-
123
THE
NOTION
OF
LIE
GROUP
-
124
DEVELOPING
THE
NOTION
OF
FIBER
BUNDLE
-
125
TANGENT
AND
COTANGENT
BUNDLES
-
131
XVIII
-
CONTENTS
6.8.1
6.8.2
6.8.3
6.8.4
6.9
6.9.1
6.9.2
6.9.3
6.9.4
6.10
SECTIONS
OF
A
BUNDLE
-
133
THE
LIE
ALGEBRA
OF
VECTOR
FIELDS
-
135
THE
COTANGENT
BUNDLE
AND
DIFFERENTIAL
FORMS
-
136
DIFFERENTIAL
K-FORMS
-
139
HOMOTOPY,
HOMOLOGY,
AND
COHOMOLOGY
-
141
HOMOTOPY
-
143
HOMOLOGY
-
146
HOMOLOGY
AND
COHOMOLOGY
GROUPS:
GENERAL
CONSTRUCTION
-
152
RELATION
BETWEEN
HOMOTOPY
AND
HOMOLOGY
-
154
HOLOMORPHIC
FUNCTIONS
AND
THEIR
INTEGRALS
IN
VIEW
OF
HOMOTOPY,
HOMOLOGY,
AND
COHOMOLOGY
-
155
6.10.1
6.10.2
6.10.3
CURVILINEAR
INTEGRALS
IN
THE
IR
2
PLANE
-
155
THE
PRIMITIVE
OF
A
DIFFERENTIAL
1-FORM
-
158
THE
GREEN-RIEMANN
FORMULA,
AN
INSTANCE
OF
THE
GENERAL
STOKES
THEOREM
-
160
6.10.4
6.10.5
6.10.6
6.10.7
6.10.8
6.11
ILLUSTRATIONS
OF
HOMOTOPY
IN
2
-
162
CONSEQUENCES
OF
A
FUNDAMENTAL
THEOREM
-
166
HOLOMORPHICITY
-
167
HOLOMORPHICITY
CONDITION
-
169
CAUCHY
THEOREM
-
170
BIBLIOGRAPHICAL
NOTE
-
176
7
7.1
7.1.1
7.2
7.3
7.4
7.4.1
7.4.2
7.5
THE
RELATION
BETWEEN
LIE
GROUPS
AND
LIE
ALGEBRAS
-
177
THE
LIE
ALGEBRA
OF
A
LIE
GROUP
-
177
LEFT/RIGHT-INVARIANT
VECTOR
FIELDS
-
179
MAURER-CARTAN FORMS
ON
LIE
GROUP
MANIFOLDS
-
184
MAURER-CARTAN
EQUATIONS
-
187
MATRIX
LIE
GROUPS
-
187
SOME
PROPERTIES
OF
MATRICES
-
187
LINEAR
LIE
GROUPS
-
189
BIBLIOGRAPHICAL
NOTE
-
189
8
8.1
8.1.1
8.1.2
8.1.3
8.1.4
CRYSTALLOGRAPHIC
GROUPS
AND
GROUP
EXTENSIONS
-
191
LATTICES
AND
CRYSTALLOGRAPHIC
GROUPS
-
191
LATTICES
-
191
THE
N-TORUST
"
-
192
CRYSTALLOGRAPHIC
GROUPS
AND
THE
BRAVAIS
LATTICES
FOR
N
=
3
AND
N
=
2
-
193
RIGOROUS
MATHEMATICAL
CLASSIFICATION
OF
THE
POINT
GROUPS
IN
TWO
DIMENSIONS
-
198
8.2
8.2.1
8.2.2
THE
PROPER
POINT
GROUPS:
THE
OCTAHEDRAL
GROUP
024
-
200
THE
CUBIC
LATTICE
AND
THE
OCTAHEDRAL
POINT
GROUP
-
200
IRREDUCIBLE
REPRESENTATIONS
OF
THE
OCTAHEDRAL
GROUP
-
201
CONTENTS
-
XIX
8.3
THE
FULL
TETRAHEDRAL
GROUP
T$2
AND
THE
OCTAHEDRAL
GROUP
024
ARE
ISOMORPHIC
-
203
8.3.1
8.3.2
8.4
8.4.1
8.5
8.5.1
8.5.2
8.5.3
8.5.4
8.5.5
8.6
THE
VIBRATIONS
OF
XY4
MOLECULES
-
203
STRUCTURE
OF
THE
FULL
TETRAHEDRAL
GROUP
-
205
GROUP
EXTENSIONS
AND
SPACE
GROUPS
-
212
GENERAL
THEORY
OF
GROUP
EXTENSIONS
-
212
SPACE
GROUPS
-
215
THE
BASIC
INGREDIENT
OF
THE
CONSTRUCTION:
THE
LIFTING
MAP
-
216
THE
PRODUCT
LAW
AND
THE
CLOSURE
CONDITION
-
217
FINITE
GROUP
COHOMOLOGY
-
219
FROBENIUS
CONGRUENCES
-
220
ANOTHER
EXAMPLE
IN
THE
PLANE:
THE
TETRAGONAL
LATTICE
-
224
BIBLIOGRAPHICAL
NOTE
-
224
9
9.1
9.2
9.3
9.4
9.4.1
9.4.2
MONODROMY
GROUPS
OF
DIFFERENTIAL
EQUATIONS
-
225
THE
ROLE
OF
DIFFERENTIAL
EQUATIONS
IN
ALL
BRANCHES
OF
SCIENCE
-
225
GROUPS
IN
A
NEW
CAPACITY
-
226
ORDINARY
DIFFERENTIAL
EQUATIONS
-
227
SECOND
ORDER
DIFFERENTIAL
EQUATIONS
WITH
SINGULAR
POINTS
-
227
SOLUTIONS
AT
REGULAR
SINGULAR
POINTS
AND
THE
INDICIAL
EQUATION
-
229
FUCHSIAN
EQUATIONS
WITH
THREE
REGULAR
SINGULAR
POINTS
AND
THE
P-SYMBOL
-
232
9.4.3
9.5
THE
HYPERGEOMETRIC
EQUATION
AND
ITS
SOLUTIONS
-
235
AN
EXAMPLE
OF
MONODROMY
GROUP:
DIFFERENTIAL
EQUATIONS
AND
TOPOLOGY
-
241
9.5.1
9.5.2
9.5.3
9.6
9.7
THE
TETRAHEDRAL
GROUP
IN
TWO
DIMENSIONS
AND
THE
TORUS
-
241
AN
ALGEBRAIC
REPRESENTATION
OF
THE
TORUS
BY
MEANS
OF
A
CUBIC
-
244
A
DIFFERENTIAL
EQUATION
ENTERS
THE
PLAY
AND
BRINGS
IN
A
NEW
GROUP
-
245
CONCLUSIVE
REMARKS
ON
THIS
CHAPTER
-
252
BIBLIOGRAPHICAL
NOTE
-
252
10
10.1
10.2
10.3
10.3.1
10.3.2
10.3.3
10.3.4
10.4
10.4.1
10.5
STRUCTURE
OF
LIE
ALGEBRAS
-
253
INTRODUCTION
-
253
LINEAR
ALGEBRA
PRELIMINARIES
-
253
TYPES
OF
LIE
ALGEBRAS
AND
LEVI
DECOMPOSITION
-
256
SOLVABLE
LIE
ALGEBRAS
-
256
SEMI-SIMPLE
LIE
ALGEBRAS
-
259
LEVI
'
S
DECOMPOSITION
OF
LIE
ALGEBRAS
-
260
AN
ILLUSTRATIVE
EXAMPLE:
THE
GALILEI
GROUP
-
265
THE
ADJOINT
REPRESENTATION
AND
CARTAN
'
S
CRITERIA
-
266
CARTAN
'
S
CRITERIA
-
267
BIBLIOGRAPHICAL
NOTE
-
269
XX
-
CONTENTS
11
11.1
11.2
11.2.1
11.2.2
11.3
11.4
11.4.1
11.4.2
11.5
11.5.1
11.5.2
11.5.3
11.5.4
11.5.5
11.6
ROOT
SYSTEMS
AND
THEIR
CLASSIFICATION
-
270
CARTAN
SUBALGEBRAS
-
270
ROOT
SYSTEMS
-
273
FINAL
FORM
OF
THE
SEMI-SIMPLE
LIE
ALGEBRA
-
276
PROPERTIES
OF
ROOT
SYSTEMS
-
276
SIMPLE
ROOTS,
THE
WEYL
GROUP,
AND
THE
CARTAN
MATRIX
-
280
CLASSIFICATION
OF
THE
IRREDUCIBLE
ROOT
SYSTEMS
-
282
DYNKIN
DIAGRAMS
-
282
THE
CLASSIFICATION
THEOREM
-
284
IDENTIFICATION
OF
THE
CLASSICAL
LIE
ALGEBRAS
-
290
THE
AE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
290
THE
DE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
295
THE
BE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
296
THE
CE
ROOT
SYSTEM
AND
THE
CORRESPONDING
LIE
ALGEBRA
-
298
THE
EXCEPTIONAL
UE
ALGEBRAS
-
299
BIBLIOGRAPHICAL
NOTE
-
299
12
12.1
12.1.1
12.2
12.2.1
12.2.2
LIE
ALGEBRA
REPRESENTATION
THEORY
-
301
LINEAR
REPRESENTATIONS
OF
A
LIE
ALGEBRA
-
301
WEIGHTS
OF
A
REPRESENTATION
AND
THE
WEIGHT
LATTICE
-
302
DISCUSSION
OF
TENSOR
PRODUCTS
AND
EXAMPLES
-
311
TENSOR
PRODUCTS
AND
IRREPS
-
311
THE
LIE
ALGEBRA
A
2
,
ITS
WEYL
GROUP,
AND
EXAMPLES
OF
ITS
REPRESENTATIONS
-
315
12.2.3
THE
LIE
ALGEBRA
SP(4,
IR)
~
SO(2,
3),
ITS
FUNDAMENTAL
REPRESENTATION,
AND
ITS
WEYL
GROUP
-
326
12.3
12.4
CONCLUSIONS
FOR
THIS
CHAPTER
-
336
BIBLIOGRAPHICAL
NOTE
-
337
13
13.1
13.2
13.2.1
EXCEPTIONAL
LIE
ALGEBRAS
-
338
THE
EXCEPTIONAL
LIE
ALGEBRA
G2
-
338
THE
LIE
ALGEBRA
F4
AND
ITS
FUNDAMENTAL
REPRESENTATION
-
342
EXPLICIT
CONSTRUCTION
OF
THE
FUNDAMENTAL
AND
ADJOINT
REPRESENTATION
OF
FA
-
346
13.3
13.3.1
13.3.2
13.4
THE
EXCEPTIONAL
LIE
ALGEBRA
EG
-
352
CONSTRUCTION
OF
THE
ADJOINT
REPRESENTATION
-
354
FINAL
COMMENTS
ON
THE
EG
ROOT
SYSTEMS
-
358
BIBLIOGRAPHICAL
NOTE
-
359
14
14.1
14.1.1
IN
DEPTH
STUDY
OF
A
SIMPLE
GROUP
-
360
A
SIMPLE
CRYSTALLOGRAPHIC
POINT
GROUP
IN
SEVEN
DIMENSIONS
-
360
THE
SIMPLE
GROUP
L
168
-
361
CONTENTS
-
XXI
14.1.2
14.1.3
14.1.4
14.1.5
14.1.6
14.1.7
14.2
STRUCTURE
OF
THE
SIMPLE
GROUP
L168
=
PSL(2,
Z7)
-
362
THE
SEVEN-DIMENSIONAL
IRREDUCIBLE
REPRESENTATION
-
363
THE
THREE-DIMENSIONAL
COMPLEX
REPRESENTATIONS
-
366
THE
SIX-DIMENSIONAL
REPRESENTATION
-
367
THE
EIGHT-DIMENSIONAL
REPRESENTATION
-
368
THE
PROPER
SUBGROUPS
OF
L
168
-
368
BIBLIOGRAPHICAL
NOTE
-
374
15
15.1
15.2
15.2.1
15.3
15.4
15.4.1
15.5
A
PRIMARY
ON
THE
THEORY
OF
CONNECTIONS
AND
METRICS
-
375
INTRODUCTION
-
375
CONNECTIONS
ON
PRINCIPAL
BUNDLES:
THE
MATHEMATICAL
DEFINITION
-
376
EHRESMANN
CONNECTIONS
ON
A
PRINCIPAL
FIBER
BUNDLE
-
377
CONNECTIONS
ON
A
VECTOR
BUNDLE
-
385
AN
ILLUSTRATIVE
EXAMPLE
OF
FIBER
BUNDLE
AND
CONNECTION
-
388
THE
MAGNETIC
MONOPOLE
AND
THE
HOPF
FIBRATION
OF
S
3
-
389
RIEMANNIAN
AND
PSEUDO-RIEMANNIAN
METRICS:
THE
MATHEMATICAL
DEFINITION
-
394
15.5.1
15.6
15.6.1
15.6.2
15.7
15.8
SIGNATURES
-
395
THE
LEVI-CIVITA
CONNECTION
-
397
AFFINE
CONNECTIONS
-
397
CURVATURE
AND
TORSION
OF
AN
AFFINE
CONNECTION
-
398
GEODESICS
-
401
GEODESICS
IN
LORENTZIAN
AND
RIEMANNIAN
MANIFOLDS:
TWO
SIMPLE
EXAMPLES
-
403
15.8.1
15.8.2
15.8.3
15.9
THE
LORENTZIAN
EXAMPLE
OF
DS
2
-
403
THE
RIEMANNIAN
EXAMPLE
OF
THE
LOBACHEVSKY-POINCARE
PLANE
-
409
ANOTHER
RIEMANNIAN
EXAMPLE:
THE
CATENOID
-
412
BIBLIOGRAPHICAL
NOTE
-
415
16
16.1
16.1.1
16.1.2
16.1.3
16.2
16.3
16.3.1
16.4
16.5
16.5.1
16.6
ISOMETRIES
AND
THE
GEOMETRY
OF
COSET
MANIFOLDS
-
416
CONCEPTUAL
AND
HISTORICAL
INTRODUCTION
-
416
SYMMETRIC
SPACES
AND
ELIE
CARTAN
-
417
WHERE
AND
HOW
DO
COSET
MANIFOLDS
COME
INTO
PLAY?
-
418
THE
DEEP
INSIGHT
OF
SUPERSYMMETRY
-
419
ISOMETRIES
AND
KILLING
VECTOR
FIELDS
-
420
COSET
MANIFOLDS
-
420
THE
GEOMETRY
OF
COSET
MANIFOLDS
-
425
THE
REAL
SECTIONS
OF
A
COMPLEX
LIE
ALGEBRA
AND
SYMMETRIC
SPACES
-
436
THE
SOLVABLE
GROUP
REPRESENTATION
OF
NON-COMPACT
COSET
MANIFOLDS
-
439
THE
TITS-SATAKE
PROJECTION:
JUST
A
FLASH
-
442
BIBLIOGRAPHICAL
NOTE
-
443
XXII
-
CONTENTS
17
17.1
17.2
17.2.1
17.2.2
FUNCTIONAL
SPACES
AND
NON-COMPACT
LIE
ALGEBRAS
-
444
INTRODUCTION
TO
AN
INTRODUCTION
-
444
THE
IDEA
OF
FUNCTIONAL
SPACES
-
447
LIMITS
OF
SUCCESSIONS
OF
CONTINUOUS
FUNCTIONS
-
449
AVERY
SHORT
INTRODUCTION
TO
MEASURE
THEORY
AND
THE
LEBESGUE
INTEGRAL
-
450
17.2.3
17.2.4
17.2.5
17.2.6
17.2.7
17.3
17.3.1
17.3.2
17.4
SPACE
OF
SQUARE
SUMMABLE
FUNCTIONS
-
452
HILBERT
SPACE
-
453
INFINITE
ORTHONORMAL
BASES
AND
THE
WEIERSTRASS
THEOREM
-
455
CONSEQUENCES
OF
THE
WEIERSTRASS
THEOREM
-
456
THE
SCHMIDT
ORTHOGONALIZATION
ALGORITHM
-
457
ORTHOGONAL
POLYNOMIALS
-
459
THE
CLASSICAL
ORTHOGONAL
POLYNOMIALS
-
459
THE
DIFFERENTIAL
EQUATION
SATISFIED
BY
ORTHOGONAL
POLYNOMIALS
-
461
THE
HEISENBERG
GROUP,
HERMITE
POLYNOMIALS,
AND
THE
HARMONIC
OSCILLATOR
-
463
17.4.1
17.5
THE
HEISENBERG
GROUP
AND
ITS
LIE
ALGEBRA
-
463
SELF-ADJOINT
OPERATORS
-
469
18
18.1
18.1.1
18.1.2
18.2
HARMONIC
ANALYSIS
AND
CONCLUSIVE
REMARKS
-
471
A
FEW
HIGHLIGHTS
ON
HARMONIC
ANALYSIS
-
472
HARMONICS
ON
COSET
SPACES
-
473
DIFFERENTIAL
OPERATORS
ON
H-HARMONICS
-
475
CONCLUSIVE
REMARKS
-
476
A
AVAILABLE
MATHEMATICA
NOTEBOOKS
WRITTEN
BY
THE
AUTHOR
-
479
BIBLIOGRAPHY
-
497
ABOUT
THE
AUTHOR
-
503
INDEX
-
505 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Fré, Pietro 1952- |
| author_GND | (DE-588)1107208297 |
| author_facet | Fré, Pietro 1952- |
| author_role | aut |
| author_sort | Fré, Pietro 1952- |
| author_variant | p f pf |
| building | Verbundindex |
| bvnumber | BV049104105 |
| classification_rvk | SK 260 |
| ctrlnum | (OCoLC)1395894694 (DE-599)DNB1284500225 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV049104105 |
| illustrated | Illustrated |
| index_date | 2024-07-03T22:33:27Z |
| indexdate | 2024-07-10T09:55:26Z |
| institution | BVB |
| institution_GND | (DE-588)10095502-2 |
| isbn | 9783111200750 3111200752 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-034365579 |
| oclc_num | 1395894694 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-29T DE-703 |
| owner_facet | DE-19 DE-BY-UBM DE-29T DE-703 |
| physical | XXII, 508 Seiten Illustrationen, Diagramme 24 cm x 17 cm, 862 g |
| publishDate | 2023 |
| publishDateSearch | 2023 |
| publishDateSort | 2023 |
| publisher | De Gruyter |
| record_format | marc |
| series2 | De Gruyter STEM |
| spelling | Fré, Pietro 1952- Verfasser (DE-588)1107208297 aut Discrete, finite and Lie groups comprehensive group theory in geometry and analysis Pietro Giuseppe Fré Berlin ; Boston De Gruyter [2023] XXII, 508 Seiten Illustrationen, Diagramme 24 cm x 17 cm, 862 g txt rdacontent n rdamedia nc rdacarrier De Gruyter STEM Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Analysis (DE-588)4001865-9 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Homogene Räume Geodäten Dynkin-Diagramme Punkt- und Raumgruppen Weyl Gruppe Homogeneous Spaces Geodesics Dynkin Diagrams Point and Space Groups Weyl Group Homogene Räume; Geodäten; Dynkin-Diagramme; Punkt- und Raumgruppen; Weyl Gruppe Homogeneous Spaces; Geodesics; Dynkin Diagrams; Point and Space Groups; Weyl Group Gruppentheorie (DE-588)4072157-7 s Lie-Algebra (DE-588)4130355-6 s Differentialgeometrie (DE-588)4012248-7 s Analysis (DE-588)4001865-9 s DE-604 Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Online-Ausgabe, PDF 978-3-11-120153-5 Erscheint auch als Online-Ausgabe, EPUB 978-3-11-120277-8 X:MVB https://www.degruyter.com/isbn/9783111200750 B:DE-101 application/pdf https://d-nb.info/1284500225/04 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034365579&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p vlb 20230327 DE-101 https://d-nb.info/provenance/plan#vlb |
| spellingShingle | Fré, Pietro 1952- Discrete, finite and Lie groups comprehensive group theory in geometry and analysis Gruppentheorie (DE-588)4072157-7 gnd Lie-Algebra (DE-588)4130355-6 gnd Analysis (DE-588)4001865-9 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4130355-6 (DE-588)4001865-9 (DE-588)4012248-7 |
| title | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis |
| title_auth | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis |
| title_exact_search | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis |
| title_exact_search_txtP | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis |
| title_full | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis Pietro Giuseppe Fré |
| title_fullStr | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis Pietro Giuseppe Fré |
| title_full_unstemmed | Discrete, finite and Lie groups comprehensive group theory in geometry and analysis Pietro Giuseppe Fré |
| title_short | Discrete, finite and Lie groups |
| title_sort | discrete finite and lie groups comprehensive group theory in geometry and analysis |
| title_sub | comprehensive group theory in geometry and analysis |
| topic | Gruppentheorie (DE-588)4072157-7 gnd Lie-Algebra (DE-588)4130355-6 gnd Analysis (DE-588)4001865-9 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Gruppentheorie Lie-Algebra Analysis Differentialgeometrie |
| url | https://www.degruyter.com/isbn/9783111200750 https://d-nb.info/1284500225/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034365579&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT frepietro discretefiniteandliegroupscomprehensivegrouptheoryingeometryandanalysis AT walterdegruytergmbhcokg discretefiniteandliegroupscomprehensivegrouptheoryingeometryandanalysis |
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