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

K¨ 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