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Differential Galois theory of infinite dimension

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Graduate School of Polymathematics, Nagoya University, Chikusa-ku, Nagoya 464-01, Japan, email: [email protected]
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This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 144 , December 1996 , pp. 59 - 135
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1996

References

[D] Drach, J., Essai sur une théorie générale de l’intégation et sur la classification des transcendentes, Annales sci. de l’École normale Sup., (3) 15 (1898), 243–384.Google Scholar
[K] Kolchin, E., Differential algebra and algebraic groups, Accademic Press, New York and London, 1973.Google Scholar
[L] Lang, S., Algebra, Addison-Wesley, Reading, Massachusetts. Menlo Park, California · London · Sydney · Manila, 1971.Google Scholar
[N] Nichols, W. and Weisfeiler, B., Differential formal group of J. F. Ritt, Amer. J. Math., 104 (1982), 943–1003.CrossRefGoogle Scholar
[P] Pommaret, J. F., Differential Galois theory, Gordon and Breach, New York, London, Paris, 1983.Google Scholar
[R] Ritt, J. F., Associative differential operations, Ann. Math., 51 (1950), 756–765.CrossRefGoogle Scholar
[S] Serre, J.-P., Lie algebras and Lie groups, Benjamin, New York, 1965.Google Scholar
[S.G.A.D] Demazure, M. et Grothendieck, A., Schemas en groupes I, Lecture Notes in Math., vol. 151, Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
[U1] Umemura, H., On the irreducibility of the first differential equation of Painlevé, Algebraic geometry and commutative algebra in honor of Masayoshi NAGATA, Kinokuniya, Tokyo, 1987, pp. 771–789.Google Scholar
[U2] Umemura, H., On a class of numbers generated by differential equations related with algebraic groups, Nagoya Math. J., 133 (1994), 1–55.CrossRefGoogle Scholar
[U3] Umemura, H., Galois Theory of algebraic and differential equations, Nagoya Math. J., 144 (1996), 1–58.Google Scholar
[VI] Vessiot, E., Sur la théorie de Galois et ses diverses généralisations, Annales sci. de l’École normale Sup., (3) 21 (1904), 9–85.Google Scholar
[V2] Vessiot, E., Sur une généralisation de la réductibilité des équations et systèmes d’équations finies ou différentielles, Annales sci. de l’Ecole normale Sup., (3) 63 (1946), 1–22.CrossRefGoogle Scholar
[W] Weisfeiler, , On Lie algebras of differential formal groups of Ritt, Bull. Amer. Math. Soc, 84 (1978), 127–130.CrossRefGoogle Scholar