Sagle A. A. — Introduction to Lie groups and Lie algebras :: Электронная библиотека попечительского совета мехмата МГУ

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Sagle A. A. — Introduction to Lie groups and Lie algebras

Sagle A. A. — Introduction to Lie groups and Lie algebras

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Название: Introduction to Lie groups and Lie algebras

Автор: Sagle A. A.

Аннотация:

This text is intended for the beginning graduate student with minimal preparation. However since Lie groups abstract the analytic properties of matrix groups, the student is expected to have some knowledge of senior level algebra, topology, and analysis as given in юте of the references. In Chapter 1 we review some advanced calculus and extend these results to manifolds in Chapter 2. Consequently the reader knowing these results can skip these chapters but should pay attention to the examples on matrix groups. After this the reader probably should follow the order given in the contents noting that the first part of the text is about Lie groups while the algebraic study of Lie algebras begins in Chapter 9. We have not attempted to prove all basic results so the serious student should take the indicated detours to such texts as those by Freudenthal and de Vries, Helgason, Jacobson, or Wolf. In particular the student must develop his own taste in this subject and ours is only one point of view.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1973

Количество страниц: 372

Добавлена в каталог: 31.10.2010

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Предметный указатель

$<\alpha, \beta>$       280
$Aut(\Delta)$       311
$B_{m}$       41
$C^{n}G$ , $C_{n}G$ , $C^{n}g$ , $C_{n}g$ ,      215 218
$C^{p}$ , $C^{p}(U)$       16 19 20
$C^{\infty}$ , $C^{\infty}(U)$ , $C^{\infty}(M)$       20 47
$C^{\infty}$ -alias      42
$C^{\infty}$ -curve      54
$C^{\infty}$ -differeniiable structure      43
$C^{\infty}$ -manifold      43
$C^{\infty}$ -map      47
$C^{\infty}$ -related      49
$C^{\infty}(U, W)$       23
$D_{i}f(p)$       14
$E_{ij}$       87 242 271
$G(\alpha)$       268
$G^{(k)}$ ,       202 204
$g_{R}$ , $\tilde{g}_{R}$       184
$H \lhd G$ , $h \lhd g$       201 204
$Hom_{K}(V, W)= Hom(V, W)$       2
$H^{1}_{\alpha}$       279
$L^{2}(V, W)$       18 20
$L_{X}$       62
$M_{3}{}^{8}$       309
$n(X)=\mid X \mid =\parallel X \parallel$       3
$N(\alpha, \beta)$       286 288 290
$S^{n}$       41
$S_{\alpha}$       282 285
$T^{*}_{p}(M) = T^{*}(M, p) = M^{*}_{p}$       66
$T_{0}$ space      95
$T_{1}$ space      96
$T_{p}(M) = T(M, p) = M_{p}$       62
$V(P)=P\otimes_{K}V$       181
$V\otimes_{K}W$       179
$V_{\lambda}$ , $V(\lambda)$ , V(f)      208
$W^{\bot}$       231
$XY=[XY]_{m}$       165
$X^{(k)}$       23
$X^{T}$       38
$[\alpha]$       168
$[\tilde{X}, \tilde{Y}]$       116
$\dot{\alpha}(t)$       72
$\Gamma^{k}_{ij}$       338
$\mathscr{A}(n, K)$       273 302 313 325 332 334
$\mathscr{C}$ , $\mathscr{C}_{c}$ , $\mathscr{C}<a>$       194 196 200 308
$\mathscr{C}(n, K)$       274 302 313 326 332 334
$\mathscr{D}(n, K)$       274 302 313 324 334
$\mathscr{D}(\mathscr{C})$       199 308
$\mathscr{L}(G)$       116
$\mathscr{L}(P)$       189
$\mathscr{R}(G)$       118
$\nabla_{X}Y$       338
$\partial_{t}=\partial/\partial x_{i}$       63
$\pi_{1}(M)$       169
<X,f>      66
Abelian Lie algebra      130 204
Ad, Ad(a), Ad(G)      156 161 163
Adjoint A* of A      57
Adjoint map ad(X)      39 135 158 191
Adjoint representation      160 161
Admissible chart      43
Ado's Theorem      141 255
Affine connection      338
Affine map      343
Analytic diffeomorphism      31
Analytic function      23
Analytic homomorphism      108
Analytic manifold      43
Analytic parameter      86
Anticommutativity      36
Ascending central series      216 218
Associative algebra $\mathscr{A}(P)$ generated by P      189
Associative bilinear form      230
Atlas      43
Aut(A)      157
Aut(G)      93 176
Automorphism of Dynkin diagram      311
Automorphism of Lie algebra      36
Automorphism of Lie group      176
Automorphism of nonassociauve algebra      157
Basic representation      330
Bounded      5
Bracket multiplication      36 80 116
Broken curve      54
Bump function      27
Bundle homomorphism      76
Bundle isomorphism      77
Burnside's theorem      191
C(U, W), C(U)      5
Campbell — Hausdorff theorem      35 114 131
Canonical chart      124 129
Canonical connection of the first kind      344
Canonical neighborhood      124
Cartan matrix      290
Cartan subalgebra      265
Cartan's criterion      235 236 239 241
Casimir operator      246
Cauchy sequence      4
Cayley algebra      196 271 308 313 323
Cayley — Dickson formulas      196
Center      103 160 163 175 208 336
centralizer      191
Change of coordinates      43
CHARACTER      209
Characteristic function      208
Characteristic roots and values      7 208
Characteristic vector      208
Closed path      168
Closed subgroup      97 144
Commutative Lie algebras and groups      130 153 204
Commutator      36 129
Compact      101
Compact Lie algebra      263
Compact Lie group      263 335
Compact real form      316 335
Compatible      91
Compatible chart      43
Compatible coordinate systems      43
Complete Riemannian manifold      342
Complete vector field      87
Completely reducible module      190 246 250 261
Completely regular      96
Complex manifold      43
Complexification      183
Composition algebra      194
Conjugate Lie algebra      187
Conjugation      185 195 315 333
Connected component      102
Connected space      54
Connection preserving map      343
Continuous function      5
Continuous homomorphism      137
Continuously differentiabit      16
Converges absolutely      4
Coordinate functions      2 5 13 41
Coordinate neighborhood      41
Coordinate system      41
Coordinate vector fields      63
Cotangent bundle      76
Cotangent space      66
Covariant differentiation      338
Covering space      174
Curvature tensor      340
Curve segment      54
D'f, D'f(p)      18 20
Der(A), Der(g)      157 230
DeRham's theorem      342
Derivation      62 157 227 259
Derivative, Df(p)      10
Descending central series      215 218

Determinant function      27
Differentiable functions on manifolds      46
Differentiable manifold      43
Differential of a map, df      66
Differertial equation      85
Difteomorphism      30 48
Dimension of a manifold      41 104
Discrete group      93 99 111 153
Divisible      155
Division ring or algebra      191 197
Djfferertiahle      10
Dual basis      66 246
Dual root system      300
Dual space      66 269
Dynkin diagram      291
D{M)      80
Effective      100
Eigenspace, eigenvalue, eigenvector      208
Embedding      49 70
End(V)      2
Endpoints      168
Engel's theorem      221
Equivalent conjugations      315
Equivalent paths      168
Equivalent representations      327
Euclidean n-space      4
Exceptional Lie algebra      308
Exp(T), exp X      8 122
Exponential      25 122 140
Extending the base field      181
Extension of an endomorphism      182
F(M), F(p)      47 62
F(X, Y)      33 125
f-invariant vector field      82
Faithful representation      160
Fiber      76
Fixed point subgroup      100
Flag      211
frelated      81
Functions of class $C^{\infty}$       20 46
Functions of class, $C^{1}$       16 20
Fundamental group      169
G(p)      100
G-invariant      82 115 211 231 337 344
General linear algebra gl(V)      32
General linear group GL(V), GL(n, R)      9 32 60 87
Geodesic      338
Germ      47 95
Group representation      160 189
H(p), $H^{0}(p)$ , $Hol(\nabla)$       342
h* the dual space of h      269 280
H-space      349
Height of a root      290
Hermuian form      6
Hessian matrix      19
Holonomy irreducible      343
Holonotny group      342
Homogeneous components      13
Homogeneous of degree k      131
Homogeneous space      100 165 341
Homomorphism      116 135 146 170
Homothetic endomorphism      247
Homotopic paths      168
Homotopy      168
Hurwitz's theorem      196
Ideal      146 164 191
Idempotent      193
Identity component      102
idy      13
Im(f)      146
Immersed submanifold      49
Immersion      49 70
Implicit function theorem      29
Implicit function theorem for manifolds      71
Indecomposable root system      285
Inn(A)      156
Inner automorphism      159
Inner derivation      158 199 228 243 260
Inner product      2
Int(A)      154
Integral curve      84
Invariant bilinear form      229 230 346
Invariant connection      344
Invariant flag      211
Invariant metric      95
Invariant subspace      164 191 209
Invariant vector field      82 115 118
Inverse function theorem      21 70
Inverse of a path      168
Involution      195
Irreducible module      190 246
Irreducible representation      246 329 330
Irreducible root system      285 288
Isometry      13
Isomorphic root systems      290
Isomorphism      95 116
Isomorphism theorems      206
Isotropy subgroup      100
Jacobi identity      36
Jacobian matrix      15
Jordan algebras      309 350
Jordan forms      8 224 237
Ker F      146
Kernel      146 154 175
Kill(X, Y)      229
Killing form      229 236
K[A]      236
l(a)      91
L(A), L(X)      56 158
L(V, W)      2
Left invariant metric      95
Left invariant vector field      82 115
Left translation      56 79 91 158
Length of solvable Lie groups and Lie algebras      202 204 342
Levi decomposition      254
Lie algebra      35 36 116 140
Lie algebra homomorphism      116 135 189
Lie algebra of a Lie group      116
Lie algebra representation      160
Lie group      105
Lie module      189
Lie polynomial      131
Lie subalgebra      55 130 140
Lie subgroup      110 139
Lie submodule      189
Lie transformation algebra      158
Lie transformation group      150 349
Lie's theorem      210 212
Linear functionals      66
Linear Lie group      335 345
Linearizing      195
Local embedding      49
Local Group      94
Local homomorphism      120 142
Local inverse      28
Local isomorphism      120 142
Local Lie group      108
Local one-parameter group      86
Locally analytically isomorphic      109
Locally connected      102 174
Locally Euclidean      104
Locally finite      96
Locally invertible functions of class C'      28
Locally isomorphic      95 109 175
Locally paihwise connected      174
Locally simply connected      170
Locally trivial      76
LOG      31 114 124
m-dimensional chart      41
Malcev algebra      349

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