A Galois \(D\)-groupoid for \(q\)-difference equations (Q407794)

\textit{B. Malgrange} [Ghys, Étienne (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)] introduced the notion of \(D\)-groupoid in order to define for non-linear differential equations an object which generalizes the differential Galois group of linear differential equations. In the paper under review the author defines a Galois D-groupoid for \(q\)-difference equations, and then proceeds to compute explicitly the Galois \(D\)-groupoid of a constant linear \(q\)-difference system, showing that it corresponds to the \(q\)-difference Galois group. In addition, the author establishes a conjugation between the Galois \(D\)-groupoids of two equivalent constant linear \(q\)-difference systems, and defines a local Galois \(D\)-groupoid for Fuchsian linear \(q\)-difference systems by giving its realizations.