Lie algebras and applications:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Heidelberg [u.a.]
Springer
2015
|
| Edition: | 2. ed. |
| Series: | Lecture notes in physics
891 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 272 S. Ill., graph. Darst. |
| ISBN: | 9783662444931 |
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MARC
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| 100 | 1 | |a Iachello, Francesco |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Lie algebras and applications |c Francesco Iachello |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Heidelberg [u.a.] |b Springer |c 2015 | |
| 300 | |a XVIII, 272 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Lecture notes in physics |v 891 | |
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Physics | |
| 650 | 4 | |a Quantum theory | |
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Record in the Search Index
| _version_ | 1804152698479575040 |
|---|---|
| adam_text | 1 BASIC CONCEPTS 1
1.1 DEFINITIONS 1
1.2 LIE ALGEBRAS 2
1.3 CHANGE OF BASIS 3
1.4 COMPLEX EXTENSIONS 4
1.5 LIE SUBALGEBRAS 5
1.6 ABELIAN ALGEBRAS 6
1.7 DIRECT SUM 6
1.8 IDEALS (INVARIANT SUBALGEBRAS) 8
1.9 SEMISIMPLE ALGEBRAS 8
1.10 SEMIDIRECT SUM 9
1.11 METRIC TENSOR 9
1.12 COMPACT AND NON-COMPACT ALGEBRAS 11
1.13 DERIVED ALGEBRAS 11
1.14 NILPOTENT ALGEBRAS 12
1.15 INVARIANT CASIMIR OPERATORS 13
1.16 INVARIANT OPERATORS FOR NON-SEMISIMPLE ALGEBRAS 14
1.17 CONTRACTIONS OF LIE ALGEBRAS 15
1.18 STRUCTURE OF LIE ALGEBRAS 17
1.18.1 ALGEBRAS WITH ONE ELEMENT 17
1.18.2 ALGEBRAS WITH TWO ELEMENTS 18
1.18.3 ALGEBRAS WITH THREE ELEMENTS 18
2 SEMISIMPLE LIE ALGEBRAS 21
2.1 CARTAN-WEYL FORM OF A (COMPLEX) SEMISIMPLE LIE ALGEBRA 21
2.2 GRAPHICAL REPRESENTATION OF ROOT VECTORS 22
2.3 EXPLICIT CONSTRUCTION OF THE CARTAN-WEYL FORM 24
2.4 DYNKIN DIAGRAMS 25
2.5 CLASSIFICATION OF (COMPLEX) SEMISIMPLE LIE ALGEBRAS 27
IX
HTTP://D-NB.INFO/1053677340
X CONTENTS
2.6 RULES FOR CONSTRUCTING THE ROOT VECTOR DIAGRAMS
OF CLASSICAL LIE ALGEBRAS 27
2.7 RULES FOR CONSTRUCTING ROOT VECTOR DIAGRAMS
OF EXCEPTIONAL LIE ALGEBRAS 29
2.8 DYNKIN DIAGRAMS OF CLASSICAL LIE ALGEBRAS 30
2.9 DYNKIN DIAGRAMS OF EXCEPTIONAL LIE ALGEBRAS 30
2.10 CARTAN MATRICES 30
2.11 CARTAN MATRICES OF CLASSICAL LIE ALGEBRAS 31
2.12 CARTAN MATRICES OF EXCEPTIONAL LIE ALGEBRAS 32
2.13 REAL FORMS OF COMPLEX SEMISIMPLE LIE ALGEBRAS 32
2.14 ISOMORPHISMS OF COMPLEX SEMISIMPLE LIE ALGEBRAS 33
2.15 ISOMORPHISMS OF REAL LIE ALGEBRAS 33
2.16 ENVELOPING ALGEBRA 33
2.17 REALIZATIONS OF LIE ALGEBRAS 34
2.18 OTHER REALIZATIONS OF LIE ALGEBRAS 35
3 LIE GROUPS 37
3.1 GROUPS OF TRANSFORMATIONS 37
3.2 GROUPS OF MATRICES 38
3.3 PROPERTIES OF MATRICES 38
3.4 CONTINUOUS MATRIX GROUPS 39
3.5 EXAMPLES OF GROUPS OF TRANSFORMATIONS 43
3.5.1 THE ROTATION GROUP IN TWO DIMENSIONS, SO(2) 43
3.5.2 THE LORENTZ GROUP IN ONE PLUS ONE
DIMENSION, S0(1,1) 44
3.5.3 THE ROTATION GROUP IN THREE DIMENSIONS, SO(3) 45
3.5.4 THE SPECIAL UNITARY GROUP IN TWO
DIMENSIONS, SU(2) 46
3.5.5 RELATION BETWEEN SO(3) AND SU(2) 47
3.6 OTHER IMPORTANT GROUPS OF TRANSFORMATIONS 48
3.6.1 TRANSLATION GROUP, T(N) 48
3.6.2 AFFINE GROUP, A(N) 49
3.6.3 EUCLIDEAN GROUP, E(N) 49
3.6.4 POINCARE GROUP, P(N) 50
3.6.5 DILATATION GROUP, D(L) 50
3.6.6 SPECIAL CONFORMAL GROUP, C(N) 51
3.6.7 GENERAL CONFORMAL GROUP, GC(M) 51
4 LIE ALGEBRAS AND LIE GROUPS 53
4.! THE EXPONENTIAL MAP 53
4.2 DEFINITION OF EXP 53
4.3 MATRIX EXPONENTIALS 54
4.4 MORE ON EXPONENTIAL MAPS 55
4.5 INFINITESIMAL TRANSFORMATIONS 56
CONTENTS XI
5 HOMOGENEOUS AND SYMMETRIC SPACES (COSET SPACES) 57
5.1 DEFINITIONS 57
5.2 CARTAN CLASSIFICATION 58
5.3 HOW TO CONSTRUCT COSET SPACES 58
6
IRREDUCIBLE BASES (REPRESENTATIONS) 61
6.1 DEFINITIONS 61
6.2 ABSTRACT CHARACTERIZATION 61
6.3 IRREDUCIBLE TENSORS 61
6.3.1 IRREDUCIBLE TENSORS WITH RESPECT TO GL(N) 61
6.3.2 CONSTRUCTION OF IRREDUCIBLE TENSORS
WITH RESPECT TO GL(N). YOUNG METHOD 63
6.3.3 IRREDUCIBLE TENSORS WITH RESPECT TO SU(N) 65
6.3.4 IRREDUCIBLE TENSORS WITH RESPECT TO SO(N) 65
6.4 TENSOR REPRESENTATIONS OF CLASSICAL COMPACT ALGEBRAS 66
6.4.1 UNITARY ALGEBRAS U(N) 66
6.4.2 SPECIAL UNITARY ALGEBRAS SU(N) 66
6.4.3 ORTHOGONAL ALGEBRAS SO(N), N = ODD 67
6.4.4 ORTHOGONAL ALGEBRAS SO(N), N = EVEN 67
6.4.5 SYMPLECTIC ALGEBRAS SP(N), N = EVEN 67
6.5 SPINOR REPRESENTATIONS 68
6.5.1 ORTHOGONAL ALGEBRAS SO(N), = ODD 68
6.5.2 ORTHOGONAL ALGEBRAS SO(N), N = EVEN 69
6.6 FUNDAMENTAL REPRESENTATIONS 69
6.6.1 UNITARY ALGEBRAS 69
6.6.2 SPECIAL UNITARY ALGEBRAS 70
6.6.3 ORTHOGONAL ALGEBRAS, = ODD 70
6.6.4 ORTHOGONAL ALGEBRAS, N = EVEN 70
6.6.5 SYMPLECTIC ALGEBRAS 71
6.7 REALIZATION OF BASES 71
6.8 CHAINS OF ALGEBRAS 72
6.9 CANONICAL CHAINS 72
6.9.1 UNITARY ALGEBRAS 72
6.9.2 ORTHOGONAL ALGEBRAS 74
6.10 ISOMORPHISMS OF SPINOR ALGEBRAS 75
6.11 NOMENCLATURE FOR U(N) 76
6.12 DIMENSIONS OF THE REPRESENTATIONS 77
6.12.1 DIMENSIONS OF THE REPRESENTATIONS OF U(N) 77
6.12.2 DIMENSIONS OF THE REPRESENTATIONS OF SU(N) 78
6.12.3 DIMENSIONS OF THE REPRESENTATIONS
OF A* = SU(N + 1) 78
6.12.4 DIMENSIONS OF THE REPRESENTATIONS
OF B* = SO(2N
+ 1) 79
6.12.5 DIMENSIONS OF THE REPRESENTATIONS OF C* = SP(2RT) 79
6.12.6 DIMENSIONS OF THE REPRESENTATIONS OF D* = SO(2N) 80
XII CONTENTS
6.13 ACTION OF THE ELEMENTS OF G ON THE BASIS B 80
6.14 TENSOR PRODUCTS 83
6.15 NON-CANONICAL CHAINS 87
7 CASIMIR OPERATORS AND THEIR EIGENVALUES 93
7.1 DEFINITIONS 93
7.2 INDEPENDENT CASIMIR OPERATORS 93
7.2.1 CASIMIR OPERATORS OF U(N) 93
7.2.2 CASIMIR OPERATORS OF SU(N) 94
7.2.3 CASIMIR OPERATORS OF SO(N), N = ODD 95
7.2.4 CASIMIR OPERATORS OF SOFA), N = EVEN 95
7.2.5 CASIMIR OPERATORS OF SP(N), N = EVEN 95
7.2.6 CASIMIR OPERATORS OF THE EXCEPTIONAL ALGEBRAS 96
7.3 COMPLETE SET OF COMMUTING OPERATORS 96
7.3.1 THE UNITARY ALGEBRA U(N) 96
7.3.2 THE ORTHOGONAL ALGEBRA SO(N), N = ODD 97
7.3.3 THE ORTHOGONAL ALGEBRA SO(N), N = EVEN 97
7.4 EIGENVALUES OF CASIMIR OPERATORS 97
7.4.1 THE ALGEBRAS U(N) AND SU(N) 97
7.4.2 THE ORTHOGONAL ALGEBRA SO(2N + 1) 100
7.4.3 THE SYMPLECTIC ALGEBRA SP(2N) 102
7.4.4 THE ORTHOGONAL ALGEBRA SO(2N) 103
7.5 EIGENVALUES OF CASIMIR OPERATORS OF ORDER ONE AND TWO 105
8 TENSOR OPERATORS 107
8.1 DEFINITIONS 107
8.2 COUPLING COEFFICIENTS 108
8.3 WIGNER-ECKART THEOREM 109
8.4 NESTED ALGEBRAS: RACAH S FACTORIZATION LEMMA ILL
8.5 ADJOINT OPERATORS 114
8.6 RECOUPL
ING COEFFICIENTS 115
8.7 SYMMETRY PROPERTIES OF COUPLING COEFFICIENTS 117
8.8 HOW TO COMPUTE COUPLING COEFFICIENTS 117
8.9 HOW TO COMPUTE RECOUPLING COEFFICIENTS 118
8.10 PROPERTIES OF RECOUPLING COEFFICIENTS 119
8.11 DOUBLE RECOUPLING COEFFICIENTS 120
8.12 COUPLED TENSOR OPERATORS 121
8.13 REDUCTION FORMULA OF THE FIRST KIND 122
8.14 REDUCTION FORMULA OF THE SECOND KIND 123
9 BOSON REALIZATIONS * 125
9.1 BOSON OPERATORS 125
9.2 THE UNITARY ALGEBRA M(1) 126
9.3 THE ALGEBRAS (2) AND SU(2) ** 128
9.3.1 SUBALGCBRA CHAINS 128
CONTENTS
XIII
9.4 THE ALGEBRAS U(N),N 3 132
9.4.1 RACAHFORM 132
9.4.2 TENSOR COUPLED FORM OF THE COMMUTATORS 133
9.4.3 SUBALGEBRA CHAINS CONTAINING SO(3) 134
9.5 THE ALGEBRAS
U(3)
AND
SU(3)
135
9.5.1 SUBALGEBRA CHAINS 135
9.5.2 LATTICE OF ALGEBRAS 139
9.5.3 BOSON CALCULUS OF
U(
3)
D SO(
3) 139
9.5.4 MATRIX ELEMENTS OF OPERATORS IN
U(3) D SO(3)
141
9.5.5 TENSOR CALCULUS OF
U(3)
D SO
(3)
142
9.5.6 OTHER BOSON CONSTRUCTIONS OF U(3) 143
9.6 THE UNITARY ALGEBRA
U(
4) 145
9.6.1 SUBALGEBRA CHAINS NOT CONTAINING SO(3) 145
9.6.2 SUBALGEBRA CHAINS CONTAINING SO(3) 146
9.7 THE UNITARY ALGEBRA U(6) 152
9.7.1 SUBALGEBRA CHAINS NOT CONTAINING SO(3) 152
9.7.2 SUBALGEBRA CHAINS CONTAINING SO(3) 152
9.8 THE UNITARY ALGEBRA U(7) 162
9.8.1 SUBALGEBRA CHAIN CONTAINING 162
9.8.2 THE TRIPLET CHAINS 164
9.9 CONTRACTIONS OF BOSONIC ALGEBRAS 170
9.9.1 THE HEISENBERG ALGEBRA H (2) 170
9.9.2 THE HEISENBERG ALGEBRA H(N),
N= EVEN 171
10 FERMION REALIZATIONS 175
10.1 FERMION OPERATORS 175
10.2 LIE ALGEBRAS CONSTRUCTED WITH FERMION OPERATORS 176
10.3 RACAHFORM 177
10.4 THE ALGEBRAS U(2J + 1) 178
10.4.1 SUBALGEBRA CHAIN CONTAINING SPIN(3) 178
10.4.2 THE ALGEBRAS
U(
2) AND
SU(
2): SPINORS 179
10.4.3 THE ALGEBRA M(4) 180
10.4.4 THE ALGEBRA U(6) 182
10.4.5 BRANCHINGS OF U(2J + 1) 183
10.5 THE ALGEBRA U (, (2J + 1)) 183
10.6 INTERNAL DEGREES OF FREEDOM (DIFFERENT SPACES) 184
10.6.1 THE ALGEBRAS M(4) AND JM(4) 184
10.6.2 THE ALGEBRAS
U(
6) AND
SU(
6) 185
10.7 INTERNAL DEGREES OF FREEDOM (SAME SPACE) 187
10.7.1 THE ALGEBRA U((2L + L)(2.V + 1)): L-S COUPLING 187
10.7.2 THE ALGEBRA M (2J + 1)): J-J COUPLING 190
10.7.3 THE ALGEBRA U ((,,(21 + 1)) (2S +1)):
MIXED L-S CONFIGURATIONS 191
193
193
193
194
197
197
198
199
201
201
202
205
205
207
208
211
211
212
212
213
214
215
215
216
216
218
220
220
221
221
222
222
224
226
229
229
230
231
231
233
235
235
236
DIFFERENTIAL REALIZATIONS
11.1 DIFFERENTIAL OPERATORS
11.2 UNITARY ALGEBRAS U(N)
11.3 ORTHOGONAL ALGEBRAS SO(N)
11.3.1 CASIMIR OPERATORS: LAPLACE-BELTRAMI FORM....
11.3.2 BASIS FOR THE REPRESENTATIONS
11.4 ORTHOGONAL ALGEBRAS SO(N ,M)
11.5 SYMPLECTIC ALGEBRAS SP(2N)
MATRIX REALIZATIONS
12.1 MATRICES
12.2 UNITARY ALGEBRAS U(N)
12.3 ORTHOGONAL ALGEBRAS SO(N)
12.4 SYMPLECTIC ALGEBRAS SP(2N)
12.5 BASIS FOR THE REPRESENTATION
12.6 CASIMIR OPERATORS
COSET SPACES
13.1 COSET SPACES U(N)/U(N * 1) T/(L)
13.2 COHERENT (OR INTRINSIC) STATES
13.2.1 ALGEBRAIC COHERENT STATES
13.2.2 GROUP COHERENT STATES
13.2.3 PROJECTIVE COHERENT STATES
13.2.4 COSET SPACES
13.3 COHERENT STATES OF U(N), N = EVEN
13.3.1 COHERENT STATES OF U(2)
13.3.2 COHERENT STATES OF U(4)
13.3.3 COHERENT STATES OF U(6)
13.4 COHERENT STATES OF M(H), N = ODD
13.4.1 COHERENT STATES OF (3)
13.5 GENERALIZED COHERENT STATES
13.6 THE GEOMETRY OF ALGEBRAIC MODELS
13.7 COSET SPACES SO(N + M)/SO(N) G
SO(M)
13.7.1 THE COSET SPACE 50(3)/50(2): THE SPHERE S
2
13.7.2 THE COSET SPACE 50(4)/50(3)
13.7.3 THE COSET SPACES SO(N + 2)/SO(N) 8 50(2).
13.8 COSET SPACES SO(N,M)/SO(N) SO(M)
13.8.1 THE COSET SPACE 50(2, L)/50(2):
THE HYPERBOLOID H
2
13.8.2 THE COSET SPACE 50(3,1 )/50(3)
13.9 ACTION OF THE COSET P = G/H
13.9.1 COSETS SO(N + L)/50(N)
13.9.2 COSETS SO(N, 1 )/SO(N)
SPECTRUM GENERATING ALGEBRAS AND DYNAMIC SYMMETRIES.
14.1 SPECTRUM GENERATING ALGEBRAS
14.2 DYNAMIC SYMMETRIES
CONTENTS
XV
14.3 BOSONIC SYSTEMS 236
14.3.1 DYNAMIC SYMMETRIES OF K(4) 237
14.3.2 DYNAMIC SYMMETRIES OF M(6) 240
14.4 FERMIONIC SYSTEMS 242
14.4.1 DYNAMIC SYMMETRY OF U (4) 243
14.4.2 DYNAMIC SYMMETRY OF M(6) 244
15 DEGENERACY ALGEBRAS AND DYNAMICAL ALGEBRAS 247
15.1 DEGENERACY ALGEBRAS 247
15.2 DEGENERACY ALGEBRAS IN V 2 DIMENSIONS 248
15.2.1 THE ISOTROPIC HARMONIC OSCILLATOR 248
15.2.2 THE COULOMB PROBLEM 251
15.3 DEGENERACY ALGEBRA IN V = 1 DIMENSION 256
15.4 DYNAMICAL ALGEBRAS 256
15.5 DYNAMICAL ALGEBRAS IN V 2 DIMENSIONS 257
15.5.1 HARMONIC OSCILLATOR 257
15.5.2 COULOMB PROBLEM 257
15.6 DYNAMICAL ALGEBRA IN V = 1 DIMENSION 258
15.6.1 POSCHL-TELLER POTENTIAL 258
15.6.2 MORSE POTENTIAL 260
15.6.3 LATTICE OF ALGEBRAS 263
REFERENCES 265
INDEX 269
|
| any_adam_object | 1 |
| author | Iachello, Francesco |
| author_facet | Iachello, Francesco |
| author_role | aut |
| author_sort | Iachello, Francesco |
| author_variant | f i fi |
| building | Verbundindex |
| bvnumber | BV042186497 |
| classification_rvk | SK 340 UD 8220 |
| ctrlnum | (OCoLC)897403466 (DE-599)BVBBV042186497 |
| dewey-full | 530.15 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.15 |
| dewey-search | 530.15 |
| dewey-sort | 3530.15 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV042186497 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:14:51Z |
| institution | BVB |
| isbn | 9783662444931 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027625623 |
| oclc_num | 897403466 |
| open_access_boolean | |
| owner | DE-11 DE-384 |
| owner_facet | DE-11 DE-384 |
| physical | XVIII, 272 S. Ill., graph. Darst. |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in physics |
| series2 | Lecture notes in physics |
| spelling | Iachello, Francesco Verfasser aut Lie algebras and applications Francesco Iachello 2. ed. Heidelberg [u.a.] Springer 2015 XVIII, 272 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in physics 891 Mathematische Physik Quantentheorie Physics Quantum theory Mathematical physics Theoretische Physik (DE-588)4117202-4 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Theoretische Physik (DE-588)4117202-4 s Lie-Algebra (DE-588)4130355-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-662-44494-8 Lecture notes in physics 891 (DE-604)BV000003166 891 DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027625623&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Iachello, Francesco Lie algebras and applications Lecture notes in physics Mathematische Physik Quantentheorie Physics Quantum theory Mathematical physics Theoretische Physik (DE-588)4117202-4 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| subject_GND | (DE-588)4117202-4 (DE-588)4130355-6 |
| title | Lie algebras and applications |
| title_auth | Lie algebras and applications |
| title_exact_search | Lie algebras and applications |
| title_full | Lie algebras and applications Francesco Iachello |
| title_fullStr | Lie algebras and applications Francesco Iachello |
| title_full_unstemmed | Lie algebras and applications Francesco Iachello |
| title_short | Lie algebras and applications |
| title_sort | lie algebras and applications |
| topic | Mathematische Physik Quantentheorie Physics Quantum theory Mathematical physics Theoretische Physik (DE-588)4117202-4 gnd Lie-Algebra (DE-588)4130355-6 gnd |
| topic_facet | Mathematische Physik Quantentheorie Physics Quantum theory Mathematical physics Theoretische Physik Lie-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027625623&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003166 |
| work_keys_str_mv | AT iachellofrancesco liealgebrasandapplications |