Projective isomonodromy and Galois groups

Mitschi, Claude; Singer, Michael

doi:10.1090/S0002-9939-2012-11499-6

Projective isomonodromy and Galois groups
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by Claude Mitschi and Michael F. Singer

Proc. Amer. Math. Soc. 141 (2013), 605-617

DOI: https://doi.org/10.1090/S0002-9939-2012-11499-6

Published electronically: June 25, 2012

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Abstract:

In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy-evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic condition for a parameterized linear differential equation to be projectively isomonodromic, in terms of the derived group of its parameterized Picard-Vessiot group.

References

M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, vol. 149, Cambridge University Press, Cambridge, 1991. MR 1149378, DOI 10.1017/CBO9780511623998
D. V. Anosov and A. A. Bolibruch, The Riemann-Hilbert problem, Aspects of Mathematics, E22, Friedr. Vieweg & Sohn, Braunschweig, 1994. MR 1276272, DOI 10.1007/978-3-322-92909-9
A. A. Bolibruch, On isomonodromic deformations of Fuchsian systems, J. Dynam. Control Systems 3 (1997), no. 4, 589–604. MR 1481628, DOI 10.1023/A:1021881809587
A. A. Bolibrukh, On isomonodromic confluences of Fuchsian singularities, Tr. Mat. Inst. Steklova 221 (1998), 127–142 (Russian); English transl., Proc. Steklov Inst. Math. 2(221) (1998), 117–132. MR 1683690
Henri Cartan, Théorie élémentaire des fonctions analytiques d’une ou plusieurs variables complexes, Enseignement des Sciences, Hermann, Paris, 1961 (French). Avec le concours de Reiji Takahashi. MR 0147623
P. J. Cassidy, Differential algebraic groups, Amer. J. Math. 94 (1972), 891–954. MR 360611, DOI 10.2307/2373764
Phyllis Joan Cassidy, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, J. Algebra 121 (1989), no. 1, 169–238. MR 992323, DOI 10.1016/0021-8693(89)90092-6
Phyllis J. Cassidy and Michael F. Singer, Galois theory of parameterized differential equations and linear differential algebraic groups, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., vol. 9, Eur. Math. Soc., Zürich, 2007, pp. 113–155. MR 2322329
S. Chakravarty and M. J. Ablowitz, Integrability, monodromy evolving deformations, and self-dual Bianchi IX systems, Phys. Rev. Lett. 76 (1996), no. 6, 857–860. MR 1372499, DOI 10.1103/PhysRevLett.76.857
M. Boiti, L. Martina, and F. Pempinelli (eds.), Nonlinear evolution equations and dynamical systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1192554
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773, DOI 10.1007/978-1-4684-9443-3
Katsunori Iwasaki, Hironobu Kimura, Shun Shimomura, and Masaaki Yoshida, From Gauss to Painlevé, Aspects of Mathematics, E16, Friedr. Vieweg & Sohn, Braunschweig, 1991. A modern theory of special functions. MR 1118604, DOI 10.1007/978-3-322-90163-7
E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
E. R. Kolchin, Differential algebraic groups, Pure and Applied Mathematics, vol. 114, Academic Press, Inc., Orlando, FL, 1985. MR 776230
Peter Landesman, Generalized differential Galois theory, Trans. Amer. Math. Soc. 360 (2008), no. 8, 4441–4495. MR 2395180, DOI 10.1090/S0002-9947-08-04586-8
C. Mitschi and M. F. Singer, Monodromy groups of parameterized linear differential equations with regular singularities, arXiv:1106.2664v1[math.CA], 2011. To appear in Bulletin of the London Mathematical Society.
Y. Ohyama, Quadratic equations and monodromy evolving deformations, arXiv:0709.4587v1 [math.CA], 2007.
Yousuke Ohyama, Monodromy evolving deformations and Halphen’s equation, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 343–348. MR 2500570, DOI 10.1090/crmp/047/21
Marius van der Put and Michael F. Singer, Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. MR 1960772, DOI 10.1007/978-3-642-55750-7
Yasutaka Sibuya, Linear differential equations in the complex domain: problems of analytic continuation, Translations of Mathematical Monographs, vol. 82, American Mathematical Society, Providence, RI, 1990. Translated from the Japanese by the author. MR 1084379, DOI 10.1090/mmono/082
T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4