, A Hopf-differential ideal of O J(G/B) is a Hopf ideal E of O J(G/B) such that X tot E ? E for every rational vector field X on B

, A differential algebraic subgroup H ? J(G/B) is a Zariski closed subset consisting of the zero locus of some Hopf-differential ideal E of

, As differential-Hopf ideals are radical (Remark A.1) we have that the natural correspondence between differential-Hopf ideals of O J(G/B) and differential algebraic subgroups of J(G/B) is, in fact

, A differential algebraic subgroup H ? J(G/B) is a differential algebraic group of G(U) defined over C(B) in the sense of Kolchin

M. Let, We set m to be the dimension of fibers and n the dimension of S, S be smooth irreducible algebraic affine varieties and ? : M ? S be a bundle

, Definition B.1 Let D M ? End C C[M ] be the ring of operators generated by ? C[M ] acting by multiplication ? The set of derivations Der

, One has two different C[M ]-algebra structures on D M : 1. Left C[M ]-algebra structure given by C[M ] ? C D M ? D M such that f ? P ? f P where (f P )(g) = f P (g)

C. Right and . [m-]-structure-d-m-?-c[m-]-?-d-m-given-by-p-?-f-?-p-?-f-where, P ? f )(g) = P (f g). This action will be

. =-(d-m-?-?c, Then the jet bundle of V over M is the vector bundle whose sheaf of linear functions is given by D M ? ?C

, M ) is the linear sub-bundle L ? J(V /M ) annihilated by a D M -submodule N ? D M ? ?C[M ] ?(V * ). i and this implies Y (p) ? ker v(p) for v(p) ? End

Y. André, Différentielles non commutatives et théorie de galois différentielle ou aux différences, Ann. Sci.École Norm. Sup, vol.34, issue.5, pp.685-739, 2001.

C. E. Arreche, On the computation of the parameterized differential Galois group for a second-order linear differential equation with differential parameters, J. Symbolic Comput, vol.75, pp.25-55, 2016.

D. Bertrand, Book review: Lectures on differential galois theory, by andy r. magid, Bull. Amer. Math. Soc, vol.33, issue.2, pp.289-294, 1996.

D. Blázquez, -. Sanz, and G. Casale, Parallelisms & Lie connections. SIGMA, vol.13, issue.86, pp.1-28, 2017.

P. Cartier, Groupoïdes de Lie et leurs algébroïdes, Séminaire Bourbaki, vol.60, 2007.

P. Cassidy, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, Journal of Algebra, vol.121, issue.1, pp.169-238, 1989.

G. Casale, Sur le groupoïde de Galois d'un feuilletage, 2004.

G. Casale, Une preuve galoisienne de l'irréductibilité au sens de nishiokaumemura de la 1èreéquation de painlevé, Astérisque, vol.323, pp.83-100, 2009.

F. Cano, D. Cerveau, and J. Déserti, Théorieélémentaire des feuilletages holomorphes singuliers, 2013.

T. Crespo and Z. Hajto, Algebraic groups and differential Galois theory, vol.122, 2011.

S. Cantat and F. Loray, Dynamics on character varieties and malgrange irreducibility of painlevé vi equation, Annales de l'institut Fourier, vol.59, pp.2927-2978, 2009.

J. Phyllis, M. Cassidy, and . Singer, Galois theory of parameterized differential equations and linear differential algebraic groups, IRMA Lectures in Mathematics and Theoretical Physics, vol.9, pp.113-157, 2005.

D. Davy, Spécialisation du pseudo-groupe de Malgrange et irréductibilité, 2016.

M. Demazure and A. Grothendieck, Schémas en groupes, Structure des schémas en groupes réductifs: Séminaire du Bois Marie, vol.64, 1962.

J. Drach, Essai sur une théorie générale de l'intégration et sur la classification des transcendantes, Annales scientifiques de l'École Normale Supérieure, vol.15, pp.243-384, 1898.

T. Dreyfus, A density theorem in parametrized differential Galois theory, Pacific J. Math, vol.271, issue.1, pp.87-141, 2014.

H. Gillet, S. Gorchinskiy, and A. Ovchinnikov, Parameterized Picard-Vessiot extensions and Atiyah extensions, Adv. Math, vol.238, pp.322-411, 2013.

X. Gómez-mont, Integrals for holomorphic foliations with singularities having all leaves compact, Ann. Inst. Fourier (Grenoble), vol.39, issue.2, pp.451-458, 1989.

S. Gorchinskiy and A. Ovchinnikov, Isomonodromic differential equations and differential categories, J. Math. Pures Appl, vol.102, issue.1, pp.48-78, 2014.

H. Grauert, On meromorphic equivalence relations, Contributions to several complex variables, pp.115-147, 1986.

C. Hardouin, A. Minchenko, and A. Ovchinnikov, Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence, Math. Ann, vol.368, issue.1-2, pp.587-632, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01927289

K. Iwasaki, H. Kimura, S. Shimemura, and M. Yoshida, From Gauss to Painlevé: a modern theory of special functions, vol.16, 2013.

K. Iwasaki, Finite branch solutions to painlevé vi around a fixed singular point, Advances in Mathematics, vol.217, issue.5, pp.1889-1934, 2008.

N. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France, vol.110, pp.203-239, 1982.

T. Kimura, On riemann's equations which are solvable by quadratures, Funkcial. Ekvac, vol.12, p.1970, 1969.

K. Kiso, Local properties of intransitive infinite Lie algebra sheaves, Japanese journal of mathematics. New series, vol.5, issue.1, pp.101-155, 1979.

E. Kolchin, Algebraic groups and algebraic dependence, Amer. J. Math, vol.90, issue.4, pp.1151-1164, 1968.

E. Kolchin, Differential algebra and algebraic groups, 1973.

E. Kolchin, Differential algebraic groups, 1985.

I. Kolár, J. Slovák, and P. W. Michor, Natural operations in differential geometry, 1999.

P. Landesman, Generalized differential galois theory, Transactions of the American Mathematical Society, vol.360, issue.8, pp.4441-4495, 2008.

O. , L. Sánchez, and J. Nagloo, On parameterized differential Galois extensions, J. Pure Appl. Algebra, vol.220, issue.7, pp.2549-2563, 2016.

O. Lisovyy and Y. Tykhyy, Algebraic solutions of the sixth painlevé equation, Journal of Geometry and Physics, vol.85, pp.124-163, 2014.

C. Kirill and . Mackenzie, General theory of Lie groupoids and Lie algebroids, vol.213, 2005.

A. R. Magid, Lectures on differential Galois theory, vol.7, 1994.

B. Malgrange, Le groupoïde de Galois d'un feuilletage. L'enseignement mathématique, vol.38, pp.465-501, 2001.

B. Malgrange, Differential algebraic groups, Algebraic Approach to Differential Equations (Ed. Lê, D?ng Tráng), pp.292-312, 2010.

B. Malgrange, Pseudogroupes de Lie et théorie de Galois différentielle. IHES, 2010.

I. Moerdijk and J. Mrcun, Introduction to foliations and Lie groupoids, vol.91, 2003.

J. Muñoz and J. Rodríguez, Weil bundles and jet spaces, Czechoslovak Mathematical Journal, vol.50, issue.4, pp.721-748, 2000.

A. Minchenko, A. Ovchinnikov, and M. F. Singer, Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations, Int. Math. Res. Not. IMRN, issue.7, pp.1733-1793, 2015.

A. Minchenko, A. Ovchinnikov, and M. F. Singer, Unipotent differential algebraic groups as parameterized differential Galois groups, J. Inst. Math. Jussieu, vol.13, issue.4, pp.671-700, 2014.

M. Noumi and Y. Yamada, A new lax pair for the sixth painlevé equation associated with, Microlocal analysis and complex Fourier analysis, pp.238-252, 2002.

K. Okamoto, Polynomial hamiltonians associated with painlevé equations, i, Proceedings of the Japan Academy, Series A, Mathematical Sciences, vol.56, issue.6, pp.264-268, 1980.

É. Picard, Sur leséquations différentielles linéaires et les groupes algébriques de transformations, Annales de la Faculté des sciences de Toulouse: Mathématiques, vol.1, pp.1-15, 1887.

A. Pillay, Algebraic D-groups and differential Galois theory, Pacific J. of Math, vol.216, issue.2, pp.343-360, 2004.

. Ha and . Schwartz, Über diejenigen fälle in welchen die gaussische hypergeometrische reihe einer algebraische funktion iheres vierten elementes darstellit, Crelle J, vol.75, pp.292-335, 1873.

J. Serre, Espaces fibrés algébriques. Séminaire Claude Chevalley, vol.3, pp.1-37, 1958.

S. Sternberg, Lectures on differential geometry, vol.316, 1999.

H. Umemura, Differential galois theory of infinite dimension, Nagoya Mathematical Journal, vol.144, pp.59-135, 1996.

M. Van-der-put, F. Michael, and . Singer, Galois theory of linear differential equations, vol.328, 2012.

E. Vessiot, Sur la théorie de galois et ses diverses généralisations, Annales scientifiques de l'École Normale Supérieure, vol.21, pp.9-85, 1904.

E. Vessiot, Sur l'intégration des systèmes différentiels qui admettent des groupes continus de transformations, Acta mathematica, vol.28, issue.1, pp.307-349, 1904.