An introduction to Lie groups and Lie algebras:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2008
|
| Edition: | 1. publ. |
| Series: | Cambridge studies in advanced mathematics
113 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XI, 222 S. Ill. |
| ISBN: | 9780521889698 0521889693 |
Staff View
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| 100 | 1 | |a Kirillov, Alexander |c Jr. |d 1967- |e Verfasser |0 (DE-588)135930243 |4 aut | |
| 245 | 1 | 0 | |a An introduction to Lie groups and Lie algebras |c Alexander Kirillov |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2008 | |
| 300 | |a XI, 222 S. |b Ill. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge studies in advanced mathematics |v 113 | |
| 500 | |a Includes bibliographical references and index | ||
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| 650 | 4 | |a Lie algebras | |
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Record in the Search Index
| _version_ | 1804137695632424960 |
|---|---|
| adam_text | Contents
Preface
page
xi
1
Introduction
1
2
Lie groups: basic definitions
4
2.1.
Reminders from differential geometry
4
2.2.
Lie groups, subgroups, and cosets
5
2.3.
Lie subgroups and homomorphism theorem
10
2.4.
Action of Lie groups on manifolds and
representations
10
2.5.
Orbits and homogeneous spaces
12
2.6.
Left, right, and adjoint action
14
2.7.
Classical groups
16
2.8.
Exercises
21
3
Lie groups and Lie algebras
25
3.1.
Exponential map
25
3.2.
The commutator
28
3.3.
Jacobi identity and the definition of a Lie algebra
30
3.4.
Subalgebras, ideals, and center
32
3.5.
Lie algebra of vector fields
33
3.6.
Stabilizers and the center
36
3.7.
Campbell-Hausdorff formula
38
3.8.
Fundamental theorems of Lie theory
40
3.9.
Complex and real forms
44
3.10.
Example: so(3,
Ж),
su{2), and sl(2, C)
46
3.11.
Exercises
48
vu
viii Contents
4
Representations of Lie groups and Lie algebras
52
4.1.
Basic definitions
52
4.2.
Operations on representations
54
4.3.
Irreducible representations
57
4.4.
Intertwining operators and Schur s lemma
59
4.5.
Complete reducibility of unitary representations:
representations of finite groups
61
4.6. Haar
measure on compact Lie groups
62
4.7.
Orthogonality of characters and Peter-Weyl theorem
65
4.8.
Representations of si(2,C)
70
4.9.
Spherical Laplace operator and the hydrogen atom
75
4.10.
Exercises
80
5
Structure theory of Lie algebras
84
5.1.
Universal enveloping algebra
84
5.2.
Poincare-Birkhoff-Witt theorem
87
5.3.
Ideals and
commutant
90
5.4.
Solvable and
nilpotent
Lie algebras
91
5.5.
Lie s and Engel s theorems
94
5.6.
The radical.
Semisimple
and reductive algebras
96
5.7.
Invariant bilinear forms and semisimplicity of classical Lie
algebras
99
5.8.
Killing form and Cartan s criterion
101
5.9.
Jordan decomposition
104
5.10.
Exercises
106
6
Complex
semisimple
Lie algebras
108
6.1.
Properties of
semisimple
Lie algebras
108
6.2.
Relation with compact groups
110
6.3.
Complete reducibility of representations
112
6.4. Semisimple
elements and
toral subalgebras
116
6.5.
Cartan
subalgebra
119
6.6.
Root decomposition and root systems
120
6.7.
Regular elements and conjugacy of Cartan
subalgebras
126
6.8.
Exercises
130
7
Root systems
132
7.1.
Abstract root systems
132
7.2.
Automorphisms and the Weyl group
134
7.3.
Pairs of roots and rank two root systems
135
Contents ix
7.4. Positive
roots and
simple
roots
137
7.5.
Weight and root lattices
140
7.6.
Weyl chambers
142
7.7.
Simple reflections
146
7.8.
Dynkin diagrams and classification of root systems
149
7.9.
Serre
relations and classification of semisimple
Lie algebras
154
7.10.
Proof of the classification theorem in
simply-laced case
157
7.11.
Exercises
160
8
Representations of semisimple Lie algebras
163
8.1.
Weight decomposition and characters
163
8.2.
Highest weight representations and Verma modules
167
8.3.
Classification of irreducible finite-dimensional
representations
171
8.4.
Bernstein-Gelfand-Gelfand resolution
174
8.5.
Weyl character formula
177
8.6.
Multiplicities
182
8.7.
Representations of sl(n,C)
183
8.8.
Harish-Chandra isomorphism
187
8.9.
Proof of Theorem
8.25 192
8.10.
Exercises
194
Overview of the literature
197
Basic textbooks
197
Monographs
198
Further reading
198
Appendix A Root systems and simple Lie algebras
202
АЛ.
An=sl(n+l,C),n>
1 202
A.2. Bn
=
so(2n
+1,
C), n
> 1 204
A.3. Cn
=
sp(n,
C), n
> 1 206
A.4.
D„=so(2n,C),n>2 207
Appendix
В
Sample syllabus
210
List of notation
213
Bibliography
216
Index
220
|
| adam_txt |
Contents
Preface
page
xi
1
Introduction
1
2
Lie groups: basic definitions
4
2.1.
Reminders from differential geometry
4
2.2.
Lie groups, subgroups, and cosets
5
2.3.
Lie subgroups and homomorphism theorem
10
2.4.
Action of Lie groups on manifolds and
representations
10
2.5.
Orbits and homogeneous spaces
12
2.6.
Left, right, and adjoint action
14
2.7.
Classical groups
16
2.8.
Exercises
21
3
Lie groups and Lie algebras
25
3.1.
Exponential map
25
3.2.
The commutator
28
3.3.
Jacobi identity and the definition of a Lie algebra
30
3.4.
Subalgebras, ideals, and center
32
3.5.
Lie algebra of vector fields
33
3.6.
Stabilizers and the center
36
3.7.
Campbell-Hausdorff formula
38
3.8.
Fundamental theorems of Lie theory
40
3.9.
Complex and real forms
44
3.10.
Example: so(3,
Ж),
su{2), and sl(2, C)
46
3.11.
Exercises
48
vu
viii Contents
4
Representations of Lie groups and Lie algebras
52
4.1.
Basic definitions
52
4.2.
Operations on representations
54
4.3.
Irreducible representations
57
4.4.
Intertwining operators and Schur's lemma
59
4.5.
Complete reducibility of unitary representations:
representations of finite groups
61
4.6. Haar
measure on compact Lie groups
62
4.7.
Orthogonality of characters and Peter-Weyl theorem
65
4.8.
Representations of si(2,C)
70
4.9.
Spherical Laplace operator and the hydrogen atom
75
4.10.
Exercises
80
5
Structure theory of Lie algebras
84
5.1.
Universal enveloping algebra
84
5.2.
Poincare-Birkhoff-Witt theorem
87
5.3.
Ideals and
commutant
90
5.4.
Solvable and
nilpotent
Lie algebras
91
5.5.
Lie's and Engel's theorems
94
5.6.
The radical.
Semisimple
and reductive algebras
96
5.7.
Invariant bilinear forms and semisimplicity of classical Lie
algebras
99
5.8.
Killing form and Cartan's criterion
101
5.9.
Jordan decomposition
104
5.10.
Exercises
106
6
Complex
semisimple
Lie algebras
108
6.1.
Properties of
semisimple
Lie algebras
108
6.2.
Relation with compact groups
110
6.3.
Complete reducibility of representations
112
6.4. Semisimple
elements and
toral subalgebras
116
6.5.
Cartan
subalgebra
119
6.6.
Root decomposition and root systems
120
6.7.
Regular elements and conjugacy of Cartan
subalgebras
126
6.8.
Exercises
130
7
Root systems
132
7.1.
Abstract root systems
132
7.2.
Automorphisms and the Weyl group
134
7.3.
Pairs of roots and rank two root systems
135
Contents ix
7.4. Positive
roots and
simple
roots
137
7.5.
Weight and root lattices
140
7.6.
Weyl chambers
142
7.7.
Simple reflections
146
7.8.
Dynkin diagrams and classification of root systems
149
7.9.
Serre
relations and classification of semisimple
Lie algebras
154
7.10.
Proof of the classification theorem in
simply-laced case
157
7.11.
Exercises
160
8
Representations of semisimple Lie algebras
163
8.1.
Weight decomposition and characters
163
8.2.
Highest weight representations and Verma modules
167
8.3.
Classification of irreducible finite-dimensional
representations
171
8.4.
Bernstein-Gelfand-Gelfand resolution
174
8.5.
Weyl character formula
177
8.6.
Multiplicities
182
8.7.
Representations of sl(n,C)
183
8.8.
Harish-Chandra isomorphism
187
8.9.
Proof of Theorem
8.25 192
8.10.
Exercises
194
Overview of the literature
197
Basic textbooks
197
Monographs
198
Further reading
198
Appendix A Root systems and simple Lie algebras
202
АЛ.
An=sl(n+l,C),n>
1 202
A.2. Bn
=
so(2n
+1,
C), n
> 1 204
A.3. Cn
=
sp(n,
C), n
> 1 206
A.4.
D„=so(2n,C),n>2 207
Appendix
В
Sample syllabus
210
List of notation
213
Bibliography
216
Index
220 |
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| genre_facet | Einführung |
| id | DE-604.BV023342203 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:01:58Z |
| indexdate | 2024-07-09T21:16:23Z |
| institution | BVB |
| isbn | 9780521889698 0521889693 |
| language | English |
| lccn | 2008000267 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016525952 |
| oclc_num | 187568983 |
| open_access_boolean | |
| owner | DE-20 DE-703 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-739 DE-11 DE-91G DE-BY-TUM |
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| physical | XI, 222 S. Ill. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge studies in advanced mathematics |
| series2 | Cambridge studies in advanced mathematics |
| spelling | Kirillov, Alexander Jr. 1967- Verfasser (DE-588)135930243 aut An introduction to Lie groups and Lie algebras Alexander Kirillov 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2008 XI, 222 S. Ill. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 113 Includes bibliographical references and index Lie groups Lie algebras Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Lie-Gruppe (DE-588)4035695-4 s Lie-Algebra (DE-588)4130355-6 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Cambridge studies in advanced mathematics 113 (DE-604)BV000003678 113 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016525952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kirillov, Alexander Jr. 1967- An introduction to Lie groups and Lie algebras Cambridge studies in advanced mathematics Lie groups Lie algebras Lie-Algebra (DE-588)4130355-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4130355-6 (DE-588)4148816-7 (DE-588)4035695-4 (DE-588)4151278-9 |
| title | An introduction to Lie groups and Lie algebras |
| title_auth | An introduction to Lie groups and Lie algebras |
| title_exact_search | An introduction to Lie groups and Lie algebras |
| title_exact_search_txtP | An introduction to Lie groups and Lie algebras |
| title_full | An introduction to Lie groups and Lie algebras Alexander Kirillov |
| title_fullStr | An introduction to Lie groups and Lie algebras Alexander Kirillov |
| title_full_unstemmed | An introduction to Lie groups and Lie algebras Alexander Kirillov |
| title_short | An introduction to Lie groups and Lie algebras |
| title_sort | an introduction to lie groups and lie algebras |
| topic | Lie groups Lie algebras Lie-Algebra (DE-588)4130355-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Lie groups Lie algebras Lie-Algebra Darstellungstheorie Lie-Gruppe Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016525952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT kirillovalexander anintroductiontoliegroupsandliealgebras |