Algebraic Groups and Differential Galois TheoryDifferential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book. This book is suitable for a graduate course in differential Galois theory. The last chapter contains several suggestions for further reading encouraging the reader to enter more deeply into different topics of differential Galois theory or related fields. |
Contents
Affine and projective varieties | 3 |
Algebraic varieties | 27 |
Part II Algebraic groups | 53 |
Basic notions | 55 |
Lie algebras and algebraic groups | 75 |
Part III Differential Galois theory | 119 |
PicardVessiot extensions | 121 |
The Galois correspondence | 141 |
Differential equations over ℂ | 165 |
Suggestions for further reading | 215 |
| 219 | |
| 223 | |
Back Cover | 227 |
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Common terms and phrases
affine open affine variety algebraic varieties algebraically closed algorithm assume automorphism basis C-algebra C-vector space closed subgroup closed subset coefficients commutative consider coordinate ring Corollary defined Definition denote derivation differential field differential Galois group differential Galois theory differential ideal differential ring dimension element Example exists field of constants finite dimensional fraction field GL(n GL(V group G hence homogeneous linear differential induced integral irreducible components irreducible variety isomorphism Laurent series Lemma Let G Lie algebra linear algebraic group linear differential equation matrix maximal ideal nilpotent nonzero normal subgroup obtain open set open subset Picard-Vessiot extension Picard-Vessiot theory poles polynomial prevariety prime ideal principal open set projective variety Proof Proposition Prove quotient rational functions resp rotation satisfies semisimple singular point solvable subgroup of G subspace tangent space Theorem topological space unipotent unique Zariski topology zero