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Zbl 1010.12004
André, Yves
Noncommutative differentials and Galois theory for differential or difference equations. (Différentielles non commutatives et théorie de Galois différentielles ou aux différences.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 34, No. 5, 685-739 (2001). ISSN 0012-9593

There are two classes of nice Galois theories with strong resemblances and analogies: a) The Picard-Vessiot theory of linear differential equations [{\it I. Kaplansky}, An introduction to differential algebra, 2nd ed., Hermann, Paris (1996; Zbl 0954.12500)]. b) The Galois theory of linear difference equations [{\it M. van der Put} and {\it M. F. Singer}, Galois theory of difference equations, Lect. Notes Math. 1666, Springer, Berlin (1997; Zbl 0930.12006); {\it J. Sauloy}, Ann. Inst. Fourier 50, 1021--1071 (2000; Zbl 0957.05012)]. On the other hand, from a dynamical point of view, an essential object in the differential geometry of connections (either for principal or vector bundles) is: c) the holonomy group of the connection. The paper under review is devoted to obtain a unique Galois theory for the three items above: a), b) and c) (in connection with that we remark that recently Malgrange gave a definition of the Galois groupoid of a foliation [{\it B. Malgrange}, Le groupoide de Galois d'un feuilletage, E. Ghys (ed.) et al., Essays on geometry and related topics, Mémoires dédiés à André Haefliger, Vol. 2, Genève: L'Enseignement Mathématique. Monogr. Enseign. Math. 38, 465--501 (2001; Zbl 1033.32020)]. In particular, under some natural assumptions, the author obtains a Galois correspondence theorem which generalizes the theorems on the Galois correspondence for linear differential and difference equations.
[Juan J.Morales-Ruiz (Barcelona)]

MSC 2000:: *12H05 Differential algebra
12H10 Difference algebra
58B34 Noncommutative geometry (a la Connes)
16E45 Differential graded algebras and applications
33D15 Basic hypergeometric functions of one variable

Keywords: differential Galois group; holonomy group; Galois correspondence theorem

Citations: Zbl 0083.03301; Zbl 0954.12500; Zbl 0930.12006; Zbl 0957.05012; Zbl 1033.32020

Cited in: Zbl 1241.33017

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