with Charlotte Hardouin. C. R. Math. Acad. Sci. Paris 348 (2010), no. 17-18, 951--954. 
ABSTRACT:
Combining the results in Hendriks (1996) [6], Di Vizio (2002) [1] and Di Vizio, Hardouin [2], we prove that the generic, algebraic or differential, Galois group of a $q$-difference modules over $\mathbb C\{x\}$ can always be characterized in terms of $v$-curvatures, in the spirit of the work of Katz (1982) [8]. We use this result to prove that the Malgrange–Granier $D$-groupoid of a linear $q$-difference system coincide, in a sense that we specify below, with a sort of Kolchin closure of the dynamics of the linear $q$-difference system and that the group that fixes a transversal, coincide with the differential generic Galois group.
DOI:10.1016/j.crma.2010.08.001
HAL:hal-00629578