Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 055, 35 pages      arXiv:1810.08566      https://doi.org/10.3842/SIGMA.2019.055
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Differential Galois Theory and Isomonodromic Deformations

David Blázquez Sanz a, Guy Casale b and Juan Sebastián Díaz Arboleda a
a) Universidad Nacional de Colombia, Sede Medellín, Facultad de Ciencias, Escuela de Matemáticas, Calle 59A No. 63 - 20, Medellín, Antioquia, Colombia
b) IRMAR, Université de Rennes 1, Campus de Beaulieu, bât. 22-23, 263 avenue du Général Leclerc, CS 74205, 35042 RENNES Cedex, France

Received November 14, 2018, in final form July 29, 2019; Published online August 05, 2019

Abstract
We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group $G$ is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.

Key words: differential Galois theory; isomonodromic deformations; hypergeometric equation.

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