Nilpotent Lie algebras:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
1996
|
| Schriftenreihe: | Mathematics and its applications
361 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 336 S. |
| ISBN: | 0792339320 |
Internformat
MARC
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| 084 | |a SK 340 |0 (DE-625)143232: |2 rvk | ||
| 100 | 1 | |a Goze, Michel |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Nilpotent Lie algebras |c by Michel Goze and Yusupdjan Khakimdjanov |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1996 | |
| 300 | |a XV, 336 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 361 | |
| 650 | 7 | |a Groupes de Lie nilpotents |2 ram | |
| 650 | 7 | |a Lie, Algèbres de |2 ram | |
| 650 | 4 | |a Lie algebras | |
| 650 | 4 | |a Nilpotent Lie groups | |
| 650 | 0 | 7 | |a Nilpotente Lie-Algebra |0 (DE-588)4354815-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nilpotente Lie-Algebra |0 (DE-588)4354815-5 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Hakimǧanov, Yusupǧan |e Verfasser |4 aut | |
| 830 | 0 | |a Mathematics and its applications |v 361 |w (DE-604)BV008163334 |9 361 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007318701&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 940 | 1 | |n oe | |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-007318701 | ||
Datensatz im Suchindex
| _version_ | 1804125430410641408 |
|---|---|
| adam_text | TABLE OF CONTENTS
PREFACE xiii
Chapter 1 : Lie algebras. Generalities 1
§ I Lie algebras. Generalities 1
1.1. Notions of lie algebras 1
1.2. Examples 2
1.3. Lie subalgebras 3
1.4. Classical Lie algebras 3
1.5. Ideals 4
1.6. Quotient lie algebra 5
1.7. Homomorphisms 5
1.8. Direct sum of lie algebras 7
$ II Derivations of Lie algebras 7
11.1. Definition and examples 7
11.2. Inner derivations 8
11.3. Characteristic ideals 9
11.4. Derivations of a direct sum of lie algebras 9
II.5 Semidlrect sum of lie algebras 10
$ III Nilpotent and solvable Lie algebras 11
111.1. Derived sequences, central sequences 11
111.2. Definition of solvable Lie algebras 12
111.3. Definition of nilpotent lie algebras 13
111.4. Engel s Theorem 15
111.5. lie s Theorem 17
111.6. Cartan Criterion for a solvable lie algebra 20
$ IV Semisimple Lie algebra 21
IV. 1. Semisimplicity and radical 21
IV.2. Killing form 22
IV.3 Complete reducibility of representations 23
IV.4. Reductive Lie algebras 24
IV.5. Levi s theorem 25
$ V On the classification of complex semisimple Lie algebras 28
V.I. Regular elements. Cartan subalgebras 28
V.2. Root systems 29
V.3 An order relation on die set of the roots A 30
V.4. Weyl Bases 31 !
V.5 On the classification of complex simple Lie algebras 32
$ VI The nilradical 34
VI. 1. Definition 34
VI.2. On the algebraic Lie algebras 35
$ VII The classical invariants of nilpotent Lie algebras 35
VII.1. The dimension of characteristic ideals and the nilindex 36
VII.2. The characteristic sequence 37
VII.3. The rank of a nilpotent lie algebra 38
VII.4. Other invariants 38
Chapter 2 : Some classes of nilpotent Lie algebras 40
$ I Filiform Lie algebras 40
1.1. Basic notions. Examples 40
1.2. Graded filiform Lie algebras 42
$ II Two step nilpotent Lie algebras 43
ILL Definition and Examples 43
112. On die structure of the two step nilpotent Lie algebras 44
II.3 On die classification of die two step nilpotent Lie
algebras 45
$ III Characteristically nilpotent Lie algebras 46
111.1. On Jacobson s dieorem 46
111.2. Characterization of characteristically nilpotent Lie
algebras 47
111.3. Examples 49
111.4. Direct sum of characteristically nilpotent Lie algebras 49
$ IV Standard Lie algebras 50
IV. 1. Parabolic subalgebras of a semisimple algebra 50
IV.2. Standard subalgebra 51
IV.3. Nilpotent standard algebras 53
IV.4. Structure of die normalizer of a nilpotent standard
subalgebra 53
IV.5. On die nilpotent algebras of maximal rank 56
IV.6. Complete standard lie algebras 56
$V On the classification of nilpotent complex Lie algebras 57
V.I. The classification in dimensions less dian 6 57
V.2. The classification in dimension 7 60
V.3. Odier classifications 76
Chapter 3 : Cohomology of Lie algebras 77
$ I Basic notions 77
1.1. g modules and representations 77
1.2. The space of cochains 78
1.3 The coboundary operator 79
1.4. The cohomology space 80
1.5 Exact sequence 81
$ II Some interpretations of the space W{g,V)for i = 0,l,2£ 84
II. 1. The spaces H°(g,V) 84
11.2. The spaces H^g.V) 84
11.3. The spaces rftg.V) 86
11.4. The spaces H^g.g) 89
$ III Cohomology of filtered and graded Lie algebras 91
III.l. Graded Lie algebras. Filtered lie algebras 91
II.2. Graded and filtered g modules 92
III.3. Graduation and filtration of the cohomology spaces 93
$ IV Spectral sequences 95
IV. 1. Definition 95
IV.2. Some properties 95
IV.3 Exact sequence associated to a filtered complex 97
§ V The Hochschild Serre spectral sequence 99
$ VI. Cohomological calculus and computers 101
VI.1. The MATHEMATICA Program 102
VI.2. How to use this program 110
Chapter 4 : Cohomology of some nilpotent Lie algebras 111
$1 Derivations of some filiform algebras 111
1.1. The algebra of derivations of L,, 112
1.2. The algebra of derivations of Wn 114
13. The algebra of derivations of Qn Cn 2k+l) 115
1.4. The algebra of derivations of Rn 117
1.5. Derivations of the standard nilpotent algebra in
g sl(r+l,C) 118
$ II Cohomology of filiform Lie algebras 121
$ HI Cohomology of the nilradicals of parabolic algebras 123
111.1. A decomposition of the groups Hkn.n), i 1,2 124
111.2. The calculation of the spaces Hkn.g/n) 130
III.3 Particular case: n is the nilradical of a Borel subalgebra 146
111.4. Particular case : n is an Heisenberg algebra 147
111.5. Description of the space H2fond(n,n) 147
111.6. The space H n,n): case of the nilradical of a Borel
subalgebra 154
111.7. The calculus of H2Oi,n) when n is the Heisenberg
algebra 157
§ IV Derivations of some infinite dimensional topologically
nilpotent Lie algebras 159
IV. 1. Infinite dimensional standard algebras 159
IV.2. A topology in a standard nilalgebra 159
IV.3. The weighting derivations of L 161
IV.4. Restriction to the finite dimensional subalgebras 161
IV.5. The structure of the algebra DeKL) 164
$ V Cohomology of the infinite Lie algebra of vector fields
of the real straightline 167
§ VI On the cohomology of nilpotent Lie algebras in small
dimension 169
Chapter 5 : The algebraic variety of the laws of Lie algebras 170
$ I The vector space of tensors Tn2 x 170
1.1. Definition 170
1.2. Action of GLCn.C). Classification of the tensors 171
1.3. Rigid tensors. Open orbits 172
$11 The variety Ln of the Lie algebraic laws 172
II. 1. Lie algebraic laws on Cn 172
11.2. The algebraic variety L« 173
11.3. The scheme L« 174
11.4. The action of GLCn.G). Fibration by orbits 177
II.5 Open orbits. Laws of rigid Lie algebras 179
11.6. The dimension of orbits 180
11.7. The components of Ln 181
$ III Contractions. Deformations 182
111.1. Contractions in Ln 182
111.2. The deformations 183
111.3. Equivalent deformations 186
$ IV Perturbations. Notions of infinitesimal algebra 187
IV. 1. The infinitesimals in Cn 187
IV.2. Theorem of decomposition of a limited vector of Cn 190
IV.3. About the compactness of the Grassmannian manifold 196
IV.4. Some problems of perturbations 199
§ V The tangent geometry to Ln 202
V.I. Cohomological spaces and tangent spaces 202
V.2. Resolution of the equation of deformations 202
V.3. The equation of perturbations 206
$ VI The components of Ln 213
VI.l. The components of Ln for n 7 213
VI.2. Rigid lie algebras 218
$ VII Construction of solvable rigid Lie algebras 224
VII. 1. The decomposability of rigid algebras 225
VII.2. Linear systems of roots associated to a rigid lie algebra 226
VII.3. Rigid Lie algebras whose nilradical is filiform 229
VII.4. Classification when the torus is of dimension 2 231
VII.5. Classification when the torus is 1 dimensional 231
VII.6. On the classification of rigid Lie algebras 232
VII.7. Classification of solvable rigid laws in small dimension 240
% VIII Study of the variety Ln in the neighborhood of the
nilradical of a parabolic subalgebra of a simple complex
Lie algebra 243
Chapter 6 : Variety of nilpotent Lie algebras 250
$ I The tangent space of the variety of nilpotent Lie
algebras 250
$ II On the filiform components of the variety Nn 252
$ III On the reducibility of the variety Nn, n 12 256
$ IV Description of a irreducible component o/Nn+1
containing R,, 259
J V Description of an irreducible component o/Nn+1
containing Wn 263
$ VI Study of the variety 3Snp in a neighborhood of
a nilradical of a parabolic subalgebra 269
VI.l. On the orbit of n 269
VI.2. On the Zariski tangent space at the point n to the
variety Nn 271
VI.3 On the irreducible components of Nnp containing n 272
VI.4. Particular case : n is the nilradical of a Borel subalgebra 274
$ VII. On the components of the variety Nn 275
VII. 1. A nonfiliform component 275
VII.2. An estimation of die number of components 277
Chapter 7 : Characteristically nilpotent Lie algebras 281
$ I Characteristically nilpotent filiform Lie algebras 281
$ II Characteristically nilpotent Lie algebras in the variety Nn 286
$ III On the nonfiliform characteristically nilpotent Lie
algebras 289
111.1. Construction of some families of nonfiliform
characteristically nilpotent lie algebras 289
111.2. The study of special cases 292
111.3. Characteristically nilpotent lie algebras in die variety
NnP 294
Chapter 8 : Applications to differential geometry.
The nilmanifolds 295
$ I A short introduction to the theory of Lie groups 296
1.1. Lie groups 296
1.2. Correspondence between lie groups and lie algebras 296
1.3. The left invariant geometry. Interpretation of g* 298
$ II The nilmanifolds 300
II. 1. Definition of nilmanifolds 300
II.2 Uniform subgroups 300
11.3. Existence of discrete uniform subgroups 301
11.4. Rational nilpotent Lie algebras 301
$ III Contact and symplectic geometry on nilpotent Lie
algebras 302
111.1. The Cartan class of an element of g 302
111.2. Contact lie algebras. A characterization of die
Heisenberg algebra 304
111.3. lie algebras with a symplectic structure 306
$ IV Left invariant metrics on nilpotent Lie groups 313
IV. 1. Left invariant metrics 313
IV.2. Classification of left invariant metrics on the
3 dimensional Heisenberg group 314
IV.3. Some formulas of Riemannian geometry 317
IV.4. Some formulas for left invariant metrics 319
IV.5. Left invariant metrics on nilpotent Lie groups 320
$ V Classification of left invariant metrics on the Heisenberg
group 323
BIBLIOGRAPHY 329
INDEX 335
|
| any_adam_object | 1 |
| author | Goze, Michel Hakimǧanov, Yusupǧan |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV010943625 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:01:26Z |
| institution | BVB |
| isbn | 0792339320 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007318701 |
| oclc_num | 33970919 |
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| owner | DE-12 DE-739 DE-29T DE-11 |
| owner_facet | DE-12 DE-739 DE-29T DE-11 |
| physical | XV, 336 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Goze, Michel Verfasser aut Nilpotent Lie algebras by Michel Goze and Yusupdjan Khakimdjanov Dordrecht [u.a.] Kluwer 1996 XV, 336 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 361 Groupes de Lie nilpotents ram Lie, Algèbres de ram Lie algebras Nilpotent Lie groups Nilpotente Lie-Algebra (DE-588)4354815-5 gnd rswk-swf Nilpotente Lie-Algebra (DE-588)4354815-5 s DE-604 Hakimǧanov, Yusupǧan Verfasser aut Mathematics and its applications 361 (DE-604)BV008163334 361 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007318701&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Goze, Michel Hakimǧanov, Yusupǧan Nilpotent Lie algebras Mathematics and its applications Groupes de Lie nilpotents ram Lie, Algèbres de ram Lie algebras Nilpotent Lie groups Nilpotente Lie-Algebra (DE-588)4354815-5 gnd |
| subject_GND | (DE-588)4354815-5 |
| title | Nilpotent Lie algebras |
| title_auth | Nilpotent Lie algebras |
| title_exact_search | Nilpotent Lie algebras |
| title_full | Nilpotent Lie algebras by Michel Goze and Yusupdjan Khakimdjanov |
| title_fullStr | Nilpotent Lie algebras by Michel Goze and Yusupdjan Khakimdjanov |
| title_full_unstemmed | Nilpotent Lie algebras by Michel Goze and Yusupdjan Khakimdjanov |
| title_short | Nilpotent Lie algebras |
| title_sort | nilpotent lie algebras |
| topic | Groupes de Lie nilpotents ram Lie, Algèbres de ram Lie algebras Nilpotent Lie groups Nilpotente Lie-Algebra (DE-588)4354815-5 gnd |
| topic_facet | Groupes de Lie nilpotents Lie, Algèbres de Lie algebras Nilpotent Lie groups Nilpotente Lie-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007318701&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT gozemichel nilpotentliealgebras AT hakimganovyusupgan nilpotentliealgebras |