Abelian Lie algebra, 10
Adjoint mapping, 9
Ado’s theorem, 202
Algebra
alternative algebra, 143
associative algebra, 3
associative algebra of linear transformations, 5
constants of multiplication of an algebra, 2
convention on algebra terminology, 15
of differential operators, 175
Jordan algebra, 144
Lie algebra, 3
Lie algebra of linear transformations, 6
non-associative algebra, 2
Almost algebraic Lie algebras, 98
Automorphisms of Al Bl, Cl, Dl, G2 F4, 281–285
Basic irreducible modules, p. 226
Campbell-Hausdorff formula, 175
Canonical generators, 126
Cartan matrix, 121
Cartan subalgebra, 57
conjugacy of Cartan subalgebras, 273
splitting Cartan subalgebra, 108
Casimir operator, 78
Center, 10
Central non-associative algebra, 291
Centralizer, 28
Centroid, 291
Chamber, 263
Characteristic polynomial, 60
primitive character, 259
Cohomology groups, 93–95, 174–185
Compact form, 147
Complete Lie algebra, 11
Complete reducibility, 46, 75–83, 96–103
Composition series, 48
anti-derivation, 179
inner derivation, 9
(s1, S2)-derivation, 177
Derivation algebra, 8
Derived series, 23
Direct sum, 18
Dynkin diagram, 128
Engel’s theorems, 36
Enveloping algebra, 32
Exceptional simple Lie algebras, 142
Extension of base field, 26–28
Extension of a Lie algebra, 18, 88
Exterior algebra, 178
Factor set, 89
Filtered algebra, 164
Fitting decomposition, 37, 39, 57
Free associative algebra, 167
Free Lie algebra, 167
Freudenthal’s formula, 247
Friedrich’s theorem, 170
Goldie-Ore theorem, 165
Graded algebra, 163
Holomorph, 18
Indecomposability (for a set of linear transformations), 46
Invariant form, 69
Irreducibility (for a set of linear transformations), 46
absolute irrepucibility, 223
Irreducible modules for Al, Bl and G2, 226–235
Iwasawa’s theorem, 204
Jacobi identity, 3
Killing form, 69
Kostant’s formula, 261
Levi’s theorem, 91
Lie element, 168
Linear property, 146
Lower central series, 23
Malcev-Harish-Chandra theorem, 92
Module, 14
cyclic module, 212
contragredient module, 22
e-extreme cyclic module, 212
tensor product of modules, 21
Morozov’s lemma, 98
Mostow-Taft theorem, 105
Multiplication algebra, 290
Nil radical, 26
Nilpotent Lie algebra, 25
Orthogonal Lie algebra, 7
Poincaré — Birkhoff — Witt theorem, 159
Polynomial mapping, 266
differential of a polynomial mapping, 268
Qnaternion algebra, 316
Radical
of associative algebra, 26
of Lie algebra, 24
Rank, 58
Real simple Lie algebras, 313–316
Reductive Lie algebra, 104, 105
Regular element, 58
Representation
adjoint representation, 16
for associative algebra, 14
for Lie algebra, 16
regular representation, 16
of split three dimensional simple Lie algebra, 83–86
tensor product of representations, 21
Restricted Lie algebra of characteristic p, 187
Abelian restricted Lie algebra, 192–194
Roots, 64
Schenkman’s derivation tower theorem, 56
Semi-simple and nilpotent components of a linear transformation, 98
Semi-simple associative algebra, 26
Semi-simple Lie algebra, 24
structure theorem for semi-simple Lie algebras, 71
Simple system of roots, 120
indecomposable simple system, 127
Solvable Lie algebra, 24
Specht-Wever theorem, 169
Spin representation, 229
Split Lie algebra of linear transformations, 42, 50
Split semi-simple Lie algebra, 108
existence theorem for, 220
isomorphism theorem for 127, 221
Split simple Lie algebras Al, Bl. Cl, Dl, G2 F4, E6, 135–146; E7 and Es, 223
Split three-dimensional simple Lie algebra, 14
Splittable Lie algebra (in the sense of Malcev), 98
Steinberg’s formula, 262
Subinvariant subalgebra, 29
Symplectic Lie algebra, 7
Trace form, 69
Triality, 287
Triangular Lie algebra, 7
u-algebra of a restricted Lie algebra, 192
Unipotent linear transformation, 285
Universal enveloping algebra, 152
Upper central series, 29
Weakly closed subset of an associative algebra, 31
Weight, 43
Weyl’s character formula, 255
Weyl’s dimensionality formula, 257
Witt’s formula, 194
Witt’s theorem, 168