Multilinear Algebra

Front Cover......Page 1
Title Page......Page 4
Copyright Information......Page 5
Contents......Page 6
Preface......Page 10
1.1 Multilinear mappings......Page 12
1.2 The tensor notation......Page 15
1.4 Alternating multilinear mappings......Page 17
1.5 Symmetric multilinear mappings......Page 21
1.6 Comments and exercises......Page 24
1.7 Solutions to selected exercises......Page 26
2.1 Basic isomorphisms......Page 30
2.2 Tensor products of homomorphisms......Page 33
2.3 Tensor products and direct sums......Page 36
2.4 Additional structure......Page 39
2.5 Covariant extension......Page 40
2.6 Comments and exercises......Page 42
2.7 Solutions to selected exercises......Page 48
3.1 Basic definitions......Page 53
3.2 Tensor products of algebras......Page 55
3.3 Graded algebras......Page 58
3.4 A modified graded tensor product......Page 62
3.5 Anticommutative algebras......Page 65
3.7 Derivations and skew derivations......Page 67
3.8 Comments and exercises......Page 69
3.9 Solutions to selected exercises......Page 76
4.1 The tensor algebra......Page 80
4.2 Functorial properties......Page 83
4.3 The tensor algebra of a free module......Page 85
4.4 Covariant extension of a tensor algebra......Page 86
4.5 Derivations and skew derivations on a tensor algebra......Page 87
4.6 Comments and exercises......Page 89
4.7 Solutions to selected exercises......Page 91
5.1 The exterior algebra......Page 95
5.2 Functorial properties......Page 98
5.3 The exterior algebra of a free module......Page 100
5.4 The exterior algebra of a direct sum......Page 104
5.5 Covariant extension of an exterior algebra......Page 106
5.6 Skew derivations on an exterior algebra......Page 107
5.7 Pfaffians......Page 111
5.8 Comments and exercises......Page 116
5.9 Solutions to selected exercises......Page 122
General remarks......Page 128
6.1 The symmetric algebra......Page 129
6.2 Functorial properties......Page 131
6.4 The symmetric algebra of a direct sum......Page 132
6.5 Covariant extension of a symmetric algebra......Page 133
6.6 Derivations on a symmetric algebra......Page 134
6.7 Differential operators......Page 135
6.8 Comments and exercises......Page 137
7.1 A fresh look at algebras......Page 141
7.2 Coalgebras......Page 144
7.3 Graded coalgebras......Page 145
7.4 Tensor products of coalgebras......Page 146
7.5 Modified tensor products of coalgebras......Page 154
7.6 Commutative and skew-commutative coalgebras......Page 161
7.7 Linear forms on a coalgebra......Page 162
7.8 Hopf algebras......Page 164
7.9 Tensor products of Hopf algebras......Page 167
7.10 E(M) as a (modified) Hopf algebra......Page 169
7.11 The Grassmann algebra of a module......Page 171
7.12 S(M) as a Hopf algebra......Page 174
7.13 Comments and exercises......Page 177
7.14 Solutions to selected exercises......Page 180
8.1 Modules of linear forms......Page 186
8.2 The graded dual of a graded module......Page 189
8.3 Graded duals of algebras and coalgebras......Page 195
8.4 Graded duals of Hopf algebras......Page 199
8.5 Comments and exercises......Page 202
8.6 Solutions to selected exercises......Page 205
Index......Page 208