226 Index
De Rham complex, 187
Defining relations, 16
Deformation tensor, 85
Dehn invariant, 36
Density theorem, 43
Differential, 187
Dimension vector, 163
Direct sum of representations, 5, 11
Double centralizer theorem, 119
Dual group, 62
Dual representation of a Lie
algebra, 37
Dual representation of an
associative algebra, 44
Dynkin diagram, 8, 146
Einstein summation, 32
Enriched category, 179
Equivalence of categories, 182
Exact functor, 191
Exact sequence, 188
Ext functor, 203
Extensions of representations, 53
Exterior algebra, 35
Exterior power, 33
Faithful representation, 18
Filtration, 45
Finite abelian category, 214
Flat module, 202
Formal deformation, 55
Free algebra, 9
Frobenius character formula, 115
Frobenius determinant, 75
Frobenius divisibility theorem, 96
Frobenius formula, 106
Frobenius reciprocity, 107
Frobenius-Schur indicator, 92
Full subcategory, 179
Functor, 179
Functorial morphism, 181
Gabriel’s theorem, 163
General linear group, 13
General linear Lie algebra, 22
Group algebra, 9
Group determinant, 75
Heisenberg group, 81
Highest weight, 130
Hilbert syzygies theorem, 207
Hilbert’s third problem, 36
Homological dimension, 213
Homomorphism of algebras, 9
Homomorphism of Lie algebras, 23
Homomorphism of representations,
11
Hook, 119
Hook length formula, 119
Ideal, 15
Ideal generated by a subset, 15
Idempotent, 67
Indecomposable representation, 6,
11
Induced representation, 105
Intertwining operator, 11
Irreducible representation
(module), 6, 10
Isomorphism of representations, 11
Jordan-H¨ older series, 51
Jordan-H¨ older theorem, 50
Kostka numbers, 114
Koszul resolution, 207
Krull-Schmidt theorem, 51
unneth formula, 190
Left adjoint functor, 184
Left exact functor, 191
Left ideal, 15
Left module, 5, 10
Length of a representation, 51
Lexicographic ordering on
partitions, 112
Lie algebra, 22
Lie group, 24
Lie subalgebra, 22
Lie’s theorem, 39
Linear abelian category (over a
field), 187
Linear algebraic group, 25
Linear functor (over a field), 190
Linked modules, 213
Locally small category, 179
Long exact sequence of
cohomology, 189
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