This is the 1975 English translation of the first three chapters of Bourbaki’s Groupes et algebres de Lie. The first chapter describes the theory of Lie algebras, their deviations, representations, and enveloping algebras. Chapter two introduces free Lie algebras in order to discuss the exponential, logarithmic and the Hausdorff series. Chapter three deals with the theory of Lie groups over R and C, and over ultrametric fields.
Author(s): Bourbaki, Nicolas
Series: Elements of mathematics
Publisher: Addison-Wesley, Hermann
Year: 1975
Language: English
Commentary: Despite the cover, this is the original translation (which is likely to have more typos than the Springer reprint.)
Pages: 450+xvii
City: Paris; Reading, Mass.
CHAPTER I. Lie Algebras 1
§1. Definition of Lie algebras 1
1. Algebras 1
2. Lie algebras 3
3. Commutative Lie algebras 5
4. Ideals 5
5. Derived series, lower central series 6
6. Upper central series 6
7. Extensions 7
8. Semi—direct products 8
9. Change of base ring 11
§2. Enveloping algebra of a Lie algebra 12
1. Definition of the enveloping algebra 12
2. Enveloping algebra of a product 13
3. Enveloping algebra of a subalgebra. 14
4. Enveloping algebra of the opposite algebra. 15
5. Symmetric algebra of a module. 16
6. Filtration of the enveloping algebra 17
7. The Poincaré-Birkhoff-Witt Theorem 18
8. Extension of derivations 23
9. Extension of the base ring. 25
§3. Representations. 25
1. Representations 25
2. Tensor product of representations 28
3. Representations on homomorphism modules 29
4. Examples 31
5. Invariant elements 32
6. Invariant bilinear forms 33
7. Casimir element 35
8. Extension of the base ring 36
§4. Nilpotent Lie algebras 38
1. Definition of nilpotent Lie algebras 38
2. Engel’s Theorem 39
3. The largest nilpotency ideal of a representation 40
4. The largest nilpotent ideal in a Lie algebra 42
5. Extension of the base field 42
§5. Solvable Lie algebras 43
1. Definition of solvable Lie algebras 43
2. Radical of a Lie algebra 44
3. Nilpotent radical of a Lie algebra 44
4. A criterion for solvability 47
5. Further properties of the radical 48
6. Extension of the base field 49
§6. Semi-simple Lie algebras 50
1. Definition of semi-simple Lie algebras 5O
2. Semi-simplicity of representations 51
3. Semi-simple elements and nilpotent elements in semi-simple Lie algebras 54
4. Reductive Lie algebras 56
5. Application: a criterion for semi-simplicity of representations 58
6. Subalgebras reductive in a Lie algebra 59
7. Examples of semi—simple Lie algebras 60
8. The Levi-Malcev Theorem 62
9. The invariants theorem 66
10. Change of base field 68
§7. Ado’s Theorem 68
1. Coefficients of a representation 69
2. The extension theorem 69
3. Ado’s Theorem 71
Exercises for §1 73
Exercises for §2 83
Exercises for §3 85
Exercises for §4 91
Exercises for §5 99
Exercises for §6 102
Exercises for §7 109
CHAPTER II. FREE LIE ALGEBRAS 111
§1. Enveloping bigebra of a Lie algebra, 111
1. Primitive elements of a cogebra 111
2. Primitive elements of a bigebra 113
3. Filtered bigebras 114
4. Enveloping bigebra of a Lie algebra 115
5. Structure of the cogebra U(g) in characteristic 0 116
6. Structure of filtered bigebras in characteristic 0 119
§2. Free Lie algebras, 122
1. Revision of free algebras 122
2. Construction of the free Lie algebra 122
3. Presentation of a Lie algebra 124
4. Lie polynomials and substitutions 124
5. Functorial properties 125
6. Graduations 126
7. Lower central series 128
8. Derivations of free Lie algebras 129
9. Elimination theorem 130
10. Hall sets in a free magma 132
11. Hall bases of a free Lie algebra 134
§3. Enveloping algebra of the free Lie algebra 136
1. Enveloping algebra of L(X) 136
2. Projector of A+ (X) onto L(X) 138
3. Dimension of the homogeneous components of L(X) 140
§4. Central filtrations 142
1. Real filtrations 142
2. Order function 143
3. Graded algebra associated with a filtered algebra 144
4. Central filtrations on a group 145
5. An example of a central filtration 147
6. Integral central filtrations 148
§5. Magnus algebras 149
1. Magnus algebras 149
2. Magnus group 150
3. Magnus group and free group 151
4. Lower central series of a free group 152
5. p-filtration of free groups 154
§6. The Hausdorff series 155
1. Exponential and logarithm in filtered algebras 155
2. Hausdorff group 157
3. Lie formal power series 158
4. The Hausdorff series. 160
5. Substitutions in the Hausdorff series 171
§7. Convergence of the Hausdorff series (real or complex case) 164
1. Continuous-polynomials with values in g 164
2. Group germ defined by a complete normed Lie algebra 165
3. Exponential in complete normed associative algebras 169
§8. Convergence of the Hausdorff series (ultrametric case) 170
1. p-adic upper bounds of the series exp, log and H 171
2. Normed Lie algebras 172
3. Group defined by a complete normed Lie algebra 172
4. Exponential in complete normed associative algebras 174
Appendix. Mobius function 176
Exercises
CHAPTER III. LIE GROUPS I 209
§1. Lie groups 209
1. Definition of a Lie group 209
2. Morphisms of Lie groups 213
3. Lie subgroups 214
4. Semi-direct products of Lie groups 215
5. Quotient of a manifold by a Lie group 217
6. Homogeneous spaces and quotient groups 219
7. Orbits 222
8. Vector bundles with operators 223
9. Local definition of a Lie group 226
10. Group germs 228
11. Law chunks of operation 231
§2. Group of tangent vectors to a Lie group 233
l. Tangent laws of composition 233
2. Group of tangent vectors to a Lie group 235
3. Case of group germs 237
§3. Passage from a Lie group to its Lie algebra 238
1. Convolution of point distributions on a Lie group 238
2. Functorial properties 241
3. Case of a group operating on a manifold 244
4. Convolution of point distributions and functions 245
5. Fields of point distributions defined by the action of a group on a manifold 248
6. Invariant fields of point distributions on a Lie group 249
7. Lie algebra of a Lie group 251
8. Functorial properties of the Lie algebra 254
9. Lie algebra of the group of invertible elements of an algebra 257
10. Lie algebras of certain linear groups 258
11. Linear representations 259
12. Adjoint representation 264
13. Tensors and invariant forms 268
14. Maurer—Cartan formulae 269
15. Construction of invariant difl‘erential forms 271
16. Haar measure on a Lie group 271
17. Left differential 274
18. Lie algebra of a Lie group germ 276
§4. Passage from Lie algebras to Lie groups 279
1. Passage from Lie algebra morphisms to Lie group morphisms 279
2. Passage from Lie algebras to Lie groups 281
3. Exponential mappings 284
4. Functoriality of exponential mappings 288
5. Structure induced on a sub-group 289
6. Primitives of differential forms with values in a Lie algebra 291
7. Passage from laws of infinitesimal operation to laws of operation 294
§5. Formal calculations in Lie groups. 297
1. The coefficients cum. 297
2. Bracket in the Lie algebra 298
3. Powers 300
4. Exponential. 303
§6. Real and complex Lie groups. 304
l. Passage from Lie algebra morphisms to Lie group morphisms 304
2. Integral subgroups 306
3. Passage from Lie algebras to Lie groups 310
4. Exponential mapping. 311
5. Application to linear representations 215
6. Normal integral subgroups 316
7. Primitives of differential forms with values in a Lie algebra 318
8. Passage from laws of infinitesimal operation to laws of operation 318
9. Exponential mapping in the linear group. 320
10. Complexification of a finite-dimensional real Lie group. 322
§7. Lie groups over an ultrametric field. 326
1. Passage from Lie algebras to Lie groups 327
2. Exponential mappings. 328
3. Standard groups. 328
4. Filtration of standard groups 330
5. Powers in standard groups 331
6. Logarithmic mapping 333
§8. Lie groups over R or Qp 337
1. Continuous morphisms 337
2. Closed subgroups 340
§9. Commutators, centralizers and normalizers in a Lie group 342
1. Commutators in a topological group 342
2. Commutators in a Lie group 343
3. Centralizers 346
4. Normalizers 347
5. Nilpotent Lie groups 347
6. Solvable Lie groups 352
7. Radical of a Lie group 354
8. Semi-simple Lie groups 355
§10. The automorphism group of a Lie group 359
1. Infinitesimal automorphisms 359
2. The automorphism group of a Lie group (real or complex case) 362
3. The automorphism group of a Lie group (ultrametric case) 367
Appendix. Operations on linear representations 368
Exercises
HISTORICAL NOTE (CHAPTERS I TO III) 410
BIBLIOGRAPHY 431
INDEX OF NOTATION 435
INDEX OF TERMINOLOGY 439
Summary of certain properties of infinite-dimensional Lie algebras over a field of characteristic 0 449
