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

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