Group theory: birdtracks, Lie's, and exceptional groups
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Princeton, N.J.
Princeton Univ. Press
2008
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| Subjects: | |
| Online Access: | Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XII, 273 S. graph. Darst. |
| ISBN: | 9780691118369 0691118361 |
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| 020 | |a 9780691118369 |c alk. paper |9 978-0-691-11836-9 | ||
| 020 | |a 0691118361 |c alk. paper |9 0-691-11836-1 | ||
| 035 | |a (OCoLC)213133467 | ||
| 035 | |a (DE-599)BVBBV023423609 | ||
| 040 | |a DE-604 |b ger |e aacr | ||
| 041 | 0 | |a eng | |
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| 100 | 1 | |a Cvitanović, Predrag |d 1946- |e Verfasser |0 (DE-588)1077897871 |4 aut | |
| 245 | 1 | 0 | |a Group theory |b birdtracks, Lie's, and exceptional groups |c Predrag Cvitanović |
| 264 | 1 | |a Princeton, N.J. |b Princeton Univ. Press |c 2008 | |
| 300 | |a XII, 273 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 7 | |a Gruppentheorie |2 swd | |
| 650 | 7 | |a Halbeinfache Lie-Algebra |2 swd | |
| 650 | 7 | |a Lie-Gruppe |2 swd | |
| 650 | 4 | |a Group theory | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0834/2008062101-d.html |3 Publisher description | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016606005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016606005 | ||
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| _version_ | 1804137817519947776 |
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| adam_text | Contents
Acknowledgments
xi
Chapter
1.
Introduction
1
Chapter
2.
A preview
5
2.1
Basic concepts
5
2.2
First example: SU(n)
9
2.3
Second example:
Ев
family
12
Chapter
3.
Invariants and reductibility
14
3.1
Preliminaries
14
18
19
22
24
25
27
27
29
32
32
36
38
38
39
40
Chapter
5.
Recouplings
43
5.1
Couplings and recouplings
43
5.2
Wigner
Зп-ј
coefficients
46
5.3
Wigner-Eckart theorem
47
Chapter
6.
Permutations
50
6.1
Symmetrization
50
6.2
Antisymmetrization
52
6.3
Levi-Civita tensor
54
6.4
Determinants
56
6.5
Characteristic equations
58
3.2
Defining space, tensors, reps
3.3
Invariants
3.4
Invariance
groups
3.5
Projection operators
3.6
Spectral decomposition
Chapter
4.
Diagrammatic notation
4.1
Birdtracks
4.2
Clebsch-Gordan coefficients
4.3
Zero- and one-dimensional subspaces
4.4
Infinitesimal transformations
4.5
Lie algebra
4.6
Other forms of Lie algebra commutators
4.7
Classification of Lie algebras by their primitive invariants
4.8
Irrelevancy of clebsches
4.9
A brief history of birdtracks
6.6
Fully
(
antisymmetric tensors
6.7
Identically vanishing tensors
Chapter
7. Casimir
operators
7.1
Casimirs
and Lie algebra
7.2
Independent
Casimirs
7.3
Adjoint rep
Casimirs
7.4
Casimir
operators
7.5
Dynkin indices
7.6
Quadratic, cubic
Casimirs
7.7
Quartic
Casimirs
7.8
Sundry relations between quartic
Casimirs
7.9
Dynkin labels
Viii
CONTENTS
58
59
61
62
63
65
66
67
70
71
73
76
Chapter
8.
Group integrals
78
8.1
Group integrals for arbitrary reps
79
8.2
Characters
81
8.3
Examples of group integrals
82
Chapter
9.
Unitary groups
84
P. Cvitanovic, H. Elvang, and A. D. Kennedy
9.1
Two-index tensors
84
9.2
Three-index tensors
85
9.3
Young tableaux
86
9.4
Young projection operators
92
9.5
Reduction of tensor products
96
9.6
U{n) recoupling relations
100
9.7
U{n)
Ъп
-j
symbols
101
9.8
SU(n) and the adjoint rep
105
9.9
An application of the negative dimensionality theorem
107
9.10
SU{n) mixed two-index tensors
108
9.11
SU{ri) mixed defining ® adjoint tensors
109
9.12
S U (n)
two-index adjoint tensors
112
9.13 Casimirs
for the fully symmetric reps of SU(n)
117
9.14
SU(n), U(n) equivalence in adjoint rep
118
9.15
Sources
119
Chapter
10.
Orthogonal groups
121
10.1
Two-index tensors
122
10.2
Mixed adjoint ® defining rep tensors
123
10.3
Two-index adjoint tensors
124
10.4
Three-index tensors
128
10.5
Gravity tensors
130
10.6 50(71)
Dynkin labels
133
Chapter
11.
Spinors
135
P. Cvitanovic and A. D. Kennedy
11.1
Spinography
136
11.2
Fierzmg around
139
11.3
Fierz coefficients
143
CONTENTS
11.4
6-j
coefficients
11.5
Exemplary evaluations, continued
11.6
Invariance
of 7-matrices
11.7
Handedness
11.8
Kahane
algorithm
IX
144
146
147
148
149
Chapter
12.
Symplectic groups
152
12.1
Two-index tensors
153
Chapter
13.
Negative dimensions
155
P. Cvitanovic and A. D. Kennedy
13.1
SU(n)=W{-n)
156
13.2
SO(n)
=
Sp(-n)
158
Chapter
14.
Spinors symplectic sisters
160
P. Cvitanovic and A. D. Kennedy
14.1
Spinsters
160
14.2
Racah coefficients
165
14.3 Heisenberg
algebras
166
Chapter
15.
SU(n) family of
invariance
groups
168
15.1
RepsofSí7(2)
168
15.2
SU (S) as invariance
group of a cubic invariant
170
15.3
Levi-Civita tensors and SU(n)
173
15.4
SU(á)-SO(6)
isomorphism
174
Chapter
16.
G2 family of
invariance
groups
176
16.1
Jacobi relation
178
16.2
Aitemativity and reduction of /-contractions
178
16.3
Primitivity
implies aitemativity
181
16.4 Casimirs
for
Ó2
183
16.5
Hurwitz s theorem
184
Chapter
17.
Es family of
invariance
groups
186
17.1
Two-index tensors
187
17.2
Decomposition of Sym3 A
190
17.3
Diophantine conditions
192
17.4
Dynkin labels and Young tableaux for Ek
193
Chapter
18.
E6 family of
invariance
groups
196
18.1
Reduction of two-index tensors
196
18.2
Mixed two-index tensors
198
18.3
Diophantine conditions and the £e family
199
18.4
Three-index tensors
200
18.5
Defining ® adjoint tensors
202
18.6
Two-index adjoint tensors
205
18.7
Dynkin labels and Young tableaux for Ek
209
18.8 Casimirs
for
Efe
210
18.9
Subgroups of
Efe
213
18.10
Springer relation
213
X
CONTENTS
18.11
Springer s construction of E&
214
Chapter
19.
Fa family of
invariance
groups
216
19.1
Two-index tensors
216
19.2
Defining ® adjoint tensors
219
19.3
Jordan algebra and F4(26)
222
19.4
Dynkin labels and Young tableaux for Fa
223
Chapter
20.
Е
-j
family and its negative-dimensional cousins
224
20.1
SO(4) family
225
20.2
Defining ® adjoint tensors
227
20.3
Lie algebra identification
228
20.4
Εη
family
230
20.5
Dynkin labels and Young tableaux for
Εη
233
Chapter
21.
Exceptional magic
235
21.1
Magic Triangle
235
21.2
A brief history of exceptional magic
238
21.3
Extended supergravities and the Magic Triangle
241
Epilogue
242
Appendix A. Recursive decomposition
244
Appendix B. Properties of Young projections
246
H. Elvang and P. Cvitanovic
B.I Uniqueness of Young projection operators
246
B.2 Orthogonality
247
B.3 Normalization and completeness
247
B.4 Dimension formula
248
Bibliography
251
Index
269
|
| adam_txt |
Contents
Acknowledgments
xi
Chapter
1.
Introduction
1
Chapter
2.
A preview
5
2.1
Basic concepts
5
2.2
First example: SU(n)
9
2.3
Second example:
Ев
family
12
Chapter
3.
Invariants and reductibility
14
3.1
Preliminaries
14
18
19
22
24
25
27
27
29
32
32
36
38
38
39
40
Chapter
5.
Recouplings
43
5.1
Couplings and recouplings
43
5.2
Wigner
Зп-ј
coefficients
46
5.3
Wigner-Eckart theorem
47
Chapter
6.
Permutations
50
6.1
Symmetrization
50
6.2
Antisymmetrization
52
6.3
Levi-Civita tensor
54
6.4
Determinants
56
6.5
Characteristic equations
58
3.2
Defining space, tensors, reps
3.3
Invariants
3.4
Invariance
groups
3.5
Projection operators
3.6
Spectral decomposition
Chapter
4.
Diagrammatic notation
4.1
Birdtracks
4.2
Clebsch-Gordan coefficients
4.3
Zero- and one-dimensional subspaces
4.4
Infinitesimal transformations
4.5
Lie algebra
4.6
Other forms of Lie algebra commutators
4.7
Classification of Lie algebras by their primitive invariants
4.8
Irrelevancy of clebsches
4.9
A brief history of birdtracks
6.6
Fully
(
antisymmetric tensors
6.7
Identically vanishing tensors
Chapter
7. Casimir
operators
7.1
Casimirs
and Lie algebra
7.2
Independent
Casimirs
7.3
Adjoint rep
Casimirs
7.4
Casimir
operators
7.5
Dynkin indices
7.6
Quadratic, cubic
Casimirs
7.7
Quartic
Casimirs
7.8
Sundry relations between quartic
Casimirs
7.9
Dynkin labels
Viii
CONTENTS
58
59
61
62
63
65
66
67
70
71
73
76
Chapter
8.
Group integrals
78
8.1
Group integrals for arbitrary reps
79
8.2
Characters
81
8.3
Examples of group integrals
82
Chapter
9.
Unitary groups
84
P. Cvitanovic, H. Elvang, and A. D. Kennedy
9.1
Two-index tensors
84
9.2
Three-index tensors
85
9.3
Young tableaux
86
9.4
Young projection operators
92
9.5
Reduction of tensor products
96
9.6
U{n) recoupling relations
100
9.7
U{n)
Ъп
-j
symbols
101
9.8
SU(n) and the adjoint rep
105
9.9
An application of the negative dimensionality theorem
107
9.10
SU{n) mixed two-index tensors
108
9.11
SU{ri) mixed defining ® adjoint tensors
109
9.12
S U (n)
two-index adjoint tensors
112
9.13 Casimirs
for the fully symmetric reps of SU(n)
117
9.14
SU(n), U(n) equivalence in adjoint rep
118
9.15
Sources
119
Chapter
10.
Orthogonal groups
121
10.1
Two-index tensors
122
10.2
Mixed adjoint ® defining rep tensors
123
10.3
Two-index adjoint tensors
124
10.4
Three-index tensors
128
10.5
Gravity tensors
130
10.6 50(71)
Dynkin labels
133
Chapter
11.
Spinors
135
P. Cvitanovic and A. D. Kennedy
11.1
Spinography
136
11.2
Fierzmg around
139
11.3
Fierz coefficients
143
CONTENTS
11.4
6-j'
coefficients
11.5
Exemplary evaluations, continued
11.6
Invariance
of 7-matrices
11.7
Handedness
11.8
Kahane
algorithm
IX
144
146
147
148
149
Chapter
12.
Symplectic groups
152
12.1
Two-index tensors
153
Chapter
13.
Negative dimensions
155
P. Cvitanovic and A. D. Kennedy
13.1
SU(n)=W{-n)
156
13.2
SO(n)
=
Sp(-n)
158
Chapter
14.
Spinors' symplectic sisters
160
P. Cvitanovic and A. D. Kennedy
14.1
Spinsters
160
14.2
Racah coefficients
165
14.3 Heisenberg
algebras
166
Chapter
15.
SU(n) family of
invariance
groups
168
15.1
RepsofSí7(2)
168
15.2
SU (S) as invariance
group of a cubic invariant
170
15.3
Levi-Civita tensors and SU(n)
173
15.4
SU(á)-SO(6)
isomorphism
174
Chapter
16.
G2 family of
invariance
groups
176
16.1
Jacobi relation
178
16.2
Aitemativity and reduction of /-contractions
178
16.3
Primitivity
implies aitemativity
181
16.4 Casimirs
for
Ó2
183
16.5
Hurwitz's theorem
184
Chapter
17.
Es family of
invariance
groups
186
17.1
Two-index tensors
187
17.2
Decomposition of Sym3 A
190
17.3
Diophantine conditions
192
17.4
Dynkin labels and Young tableaux for Ek
193
Chapter
18.
E6 family of
invariance
groups
196
18.1
Reduction of two-index tensors
196
18.2
Mixed two-index tensors
198
18.3
Diophantine conditions and the £e family
199
18.4
Three-index tensors
200
18.5
Defining ® adjoint tensors
202
18.6
Two-index adjoint tensors
205
18.7
Dynkin labels and Young tableaux for Ek
209
18.8 Casimirs
for
Efe
210
18.9
Subgroups of
Efe
213
18.10
Springer relation
213
X
CONTENTS
18.11
Springer's construction of E&
214
Chapter
19.
Fa family of
invariance
groups
216
19.1
Two-index tensors
216
19.2
Defining ® adjoint tensors
219
19.3
Jordan algebra and F4(26)
222
19.4
Dynkin labels and Young tableaux for Fa
223
Chapter
20.
Е
-j
family and its negative-dimensional cousins
224
20.1
SO(4) family
225
20.2
Defining ® adjoint tensors
227
20.3
Lie algebra identification
228
20.4
Εη
family
230
20.5
Dynkin labels and Young tableaux for
Εη
233
Chapter
21.
Exceptional magic
235
21.1
Magic Triangle
235
21.2
A brief history of exceptional magic
238
21.3
Extended supergravities and the Magic Triangle
241
Epilogue
242
Appendix A. Recursive decomposition
244
Appendix B. Properties of Young projections
246
H. Elvang and P. Cvitanovic
B.I Uniqueness of Young projection operators
246
B.2 Orthogonality
247
B.3 Normalization and completeness
247
B.4 Dimension formula
248
Bibliography
251
Index
269 |
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| author | Cvitanović, Predrag 1946- |
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| dewey-search | 512/.2 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| index_date | 2024-07-02T21:31:53Z |
| indexdate | 2024-07-09T21:18:19Z |
| institution | BVB |
| isbn | 9780691118369 0691118361 |
| language | English |
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| spelling | Cvitanović, Predrag 1946- Verfasser (DE-588)1077897871 aut Group theory birdtracks, Lie's, and exceptional groups Predrag Cvitanović Princeton, N.J. Princeton Univ. Press 2008 XII, 273 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Gruppentheorie swd Halbeinfache Lie-Algebra swd Lie-Gruppe swd Group theory http://www.loc.gov/catdir/enhancements/fy0834/2008062101-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016606005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Cvitanović, Predrag 1946- Group theory birdtracks, Lie's, and exceptional groups Gruppentheorie swd Halbeinfache Lie-Algebra swd Lie-Gruppe swd Group theory |
| title | Group theory birdtracks, Lie's, and exceptional groups |
| title_auth | Group theory birdtracks, Lie's, and exceptional groups |
| title_exact_search | Group theory birdtracks, Lie's, and exceptional groups |
| title_exact_search_txtP | Group theory birdtracks, Lie's, and exceptional groups |
| title_full | Group theory birdtracks, Lie's, and exceptional groups Predrag Cvitanović |
| title_fullStr | Group theory birdtracks, Lie's, and exceptional groups Predrag Cvitanović |
| title_full_unstemmed | Group theory birdtracks, Lie's, and exceptional groups Predrag Cvitanović |
| title_short | Group theory |
| title_sort | group theory birdtracks lie s and exceptional groups |
| title_sub | birdtracks, Lie's, and exceptional groups |
| topic | Gruppentheorie swd Halbeinfache Lie-Algebra swd Lie-Gruppe swd Group theory |
| topic_facet | Gruppentheorie Halbeinfache Lie-Algebra Lie-Gruppe Group theory |
| url | http://www.loc.gov/catdir/enhancements/fy0834/2008062101-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016606005&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT cvitanovicpredrag grouptheorybirdtracksliesandexceptionalgroups |