Extending Structures Fundamentals and Applications

Cover......Page 1
Half Title......Page 2
Series Page......Page 3
Title Page......Page 4
Copyright Page......Page 5
Contents......Page 6
Introduction......Page 8
Generalities: Basic notions and notation......Page 14
1. Extending structures: The group case......Page 20
1.1 Crossed product and bicrossed product of groups......Page 22
1.2 Group extending structures and unified products......Page 28
1.3 Classifying complements......Page 41
1.4 Examples: Applications to the structure of finite groups......Page 49
2. Leibniz algebras......Page 54
2.1 Unified products for Leibniz algebras......Page 56
2.2 Flag extending structures of Leibniz algebras: Examples......Page 64
2.3 Special cases of unified products for Leibniz algebras......Page 74
2.4 Classifying complements for extensions of Leibniz algebras......Page 81
2.5 Itô's theorem for Leibniz algebras......Page 85
3. Lie algebras......Page 90
3.1 Unified products for Lie algebras......Page 93
3.2 Flag extending structures: Examples......Page 98
3.3 Special cases of unified products for Lie algebras......Page 107
3.4 Matched pair deformations and the factorization index for Lie algebras: The case of perfect Lie algebras......Page 112
3.5 Matched pair deformations and the factorization index for Lie algebras: The case of non-perfect Lie algebras......Page 116
3.6 Application: Galois groups and group actions on Lie algebras......Page 126
4. Associative algebras......Page 144
4.1 Unified products for algebras......Page 147
4.2 Flag and supersolvable algebras: Examples......Page 155
4.3 Special cases of unified products for algebras......Page 168
4.4 The Galois group of algebra extensions......Page 175
4.5 Classifying complements for associative algebras......Page 177
5. Jacobi and Poisson algebras......Page 186
5.1 (Bi)modules, integrals and Frobenius Jacobi algebras......Page 189
5.2 Unified products for Jacobi algebras......Page 198
5.3 Flag Jacobi algebras: Examples......Page 206
5.4 Classifying complements for Poisson algebras......Page 215
Bibliography......Page 224
Index......Page 240