Differential Algebraic Groups (eBook)

Front Cover 1
Differential Algebraic Groups 4
Copyright Page 5
Contents 8
Preface 12
Acknowledgments 16
Conventions 18
Chapter 0. Differential Algebraic Preliminaries 20
1. Compatible homomorphisms 20
2. Numerical polynomials 24
3. Finitely generated extensions 26
4. Differential derivations 27
5. The derivation operators of a differential field 31
6. Varying the set of derivation operators 35
7. A kind of disjointness 38
8. Some groups and rings 43
9. Integrability conditions 45
Chapter I. The Axioms And First Consequences 47
1. Pre differential algebraic sets 48
2. Differential algebraic groups and homogeneous differential algebraic spaces. Differential algebraic sets 51
3. The component theorem and its corollaries 58
Chapter II. Variations in the Data 63
1. Varying the universal differential field 63
2. Extending the basic differential field 67
3. Varying the set of derivation operators 75
Chapter III. Topology 87
1. The differential Zariski topologies 87
2. Differentially closed sets 92
3. Constrained elements 98
Chapter IV. Subgroups and Homomorphisms 106
1. Differential algebraic subgroups 106
2. Differential rational homomorphisms 108
3. Direct products 116
4. Quotients 122
5. Solvable and nilpotent differential algebraic groups 129
Chapter V. Mappings and Functions 132
1. Holomorphicity at a differential specialization 132
2. Differential rational mappings 139
3. Differential rational functions 155
Chapter VI. Other Structures 164
1. Two lemmas 164
2. Some other differential algebraic structures 166
Chapter VII. Cohomology 175
1. The differential algebraic set of a finitely generated constrained extension 176
2. Constrained cohomology 178
3. Differential rational cohomology 189
4. Differential algebraic vector spaces over u 197
5. Principal homogeneous differential algebraic spaces 199
6. Examples 210
Chapter VIII. Lie Theory 213
1. Local differential rings 213
2. Tangent spaces 216
3. Differential desivations 224
4. Invariant differential derivations 227
5. The Lie algebra 229
6. Examples 240
7. Differential algebraic spaces and crossed differential rational homomorphisms 246
8. Logarithmic derivations 254
9. Further properties of logarithmic derivations 259
10. The Lie–Cassidy–Kovacic method 266
11. Images and kernels 268
12. The lie algebras of the connected differential algebraic subgroups of a connected differential algebraic group 270
References 282
Glossary of Notation 284
Index of Definitions 288