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Noncommutative differentials and Galois theory for differential or difference equations. (Différentielles non commutatives et théorie de Galois différentielle ou aux différences.) (French) Zbl 1010.12004
There are two classes of nice Galois theories with strong resemblances and analogies:
a) The Picard-Vessiot theory of linear differential equations [I. Kaplansky, An introduction to differential algebra, 2nd ed., Hermann, Paris (1996; Zbl 0954.12500)].
b) The Galois theory of linear difference equations [M. van der Put and M. F. Singer, Galois theory of difference equations, Lect. Notes Math. 1666, Springer, Berlin (1997; Zbl 0930.12006); 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 [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.

MSC:
12H05 Differential algebra
12H10 Difference algebra
58B34 Noncommutative geometry (à la Connes)
16E45 Differential graded algebras and applications (associative algebraic aspects)
33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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