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

SIGMA 15 (2019), 055, 35 pages      arXiv:1810.08566
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

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.

pdf (594 kb)   tex (43 kb)  


  1. Arreche C.E., On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters, J. Symbolic Comput. 75 (2016), 25-55.
  2. Blázquez-Sanz D., Casale G., Parallelisms & Lie connections, SIGMA 13 (2017), 086, 28 pages, arXiv:1603.07915.
  3. Cartier P., Groupoïdes de Lie et leurs algébroïdes, Astérisque 326 (2009), Exp. No. 987, 165-196.
  4. Cassidy P.J., The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, J. Algebra 121 (1989), 169-238.
  5. Cassidy P.J., Singer M.F., Galois theory of parameterized differential equations and linear differential algebraic groups, in Differential Equations and Quantum Groups, IRMA Lect. Math. Theor. Phys., Vol. 9, Eur. Math. Soc., Zürich, 2007, 113-155, arXiv:math.CA/0502396.
  6. Davy D., Spécialisation du pseudo-groupe de Malgrange et irréductibilité, Ph.D. Thesis, Université Rennes 1, 2016.
  7. Demazure M., Grothendieck A., Schémas en groupes. III: Structure des schémas en groupes réductifs, Lecture Notes in Math., Vol. 153, Springer-Verlag, Berlin - New York, 1970.
  8. Dreyfus T., A density theorem in parametrized differential Galois theory, Pacific J. Math. 271 (2014), 87-141, arXiv:1203.2904.
  9. Gillet H., Gorchinskiy S., Ovchinnikov A., Parameterized Picard-Vessiot extensions and Atiyah extensions, Adv. Math. 238 (2013), 322-411, arXiv:1110.3526.
  10. Gómez-Mont X., Integrals for holomorphic foliations with singularities having all leaves compact, Ann. Inst. Fourier (Grenoble) 39 (1989), 451-458.
  11. Gorchinskiy S., Ovchinnikov A., Isomonodromic differential equations and differential categories, J. Math. Pures Appl. 102 (2014), 48-78, arXiv:1202.0927.
  12. Grauert H., On meromorphic equivalence relations, in Contributions to Several Complex Variables, Aspects Math., Vol. E9, Friedr. Vieweg, Braunschweig, 1986, 115-147.
  13. Hardouin C., Minchenko A., Ovchinnikov A., Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence, Math. Ann. 368 (2017), 587-632, arXiv:1505.07068.
  14. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé: a modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  15. Kiso K., Local properties of intransitive infinite Lie algebra sheaves, Japan. J. Math. (N.S.) 5 (1979), 101-155.
  16. Kolchin E.R., Algebraic groups and algebraic dependence, Amer. J. Math. 90 (1968), 1151-1164.
  17. Kolchin E.R., Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York - London, 1973.
  18. Kolchin E.R., Differential algebraic groups, Pure and Applied Mathematics, Vol. 114, Academic Press, Inc., Orlando, FL, 1985.
  19. Landesman P., Generalized differential Galois theory, Trans. Amer. Math. Soc. 360 (2008), 4441-4495, arXiv:0707.3583.
  20. León Sánchez O., Nagloo J., On parameterized differential Galois extensions, J. Pure Appl. Algebra 220 (2016), 2549-2563, arXiv:1507.06338.
  21. Malgrange B., Le groupoïde de Galois d'un feuilletage, in Essays on Geometry and Related Topics, Vols. 1, 2, Monogr. Enseign. Math., Vol. 38, Enseignement Math., Geneva, 2001, 465-501.
  22. Malgrange B., Differential algebraic groups, in Algebraic Approach to Differential Equations, World Sci. Publ., Hackensack, NJ, 2010, 292-312.
  23. Malgrange B., Pseudogroupes de Lie et théorie de Galois différentielle, Preprint IHES/M/10/11, Institut des Hautes études Scientifiques, 2010, available at
  24. Minchenko A., Ovchinnikov A., Singer M.F., Unipotent differential algebraic groups as parameterized differential Galois groups, J. Inst. Math. Jussieu 13 (2014), 671-700, arXiv:1301.0092.
  25. Minchenko A., Ovchinnikov A., Singer M.F., Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations, Int. Math. Res. Not. 2015 (2015), 1733-1793, arXiv:1304.2693.
  26. Pillay A., Algebraic $D$-groups and differential Galois theory, Pacific J. Math. 216 (2004), 343-360.
  27. Wibmer M., Existence of $\partial$-parameterized Picard-Vessiot extensions over fields with algebraically closed constants, J. Algebra 361 (2012), 163-171, arXiv:1104.3514.

Previous article  Next article  Contents of Volume 15 (2019)