## Projective isomonodromy and Galois groups

HTML articles powered by AMS MathViewer

- 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
- PDF | Request permission

## 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

## Bibliographic Information

**Claude Mitschi**- Affiliation: Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg Cedex, France
- Email: mitschi@math.unistra.fr
**Michael F. Singer**- Affiliation: Department of Mathematics, North Carolina State University, Box 8205, Raleigh, North Carolina 27695-8205
- Email: singer@math.ncsu.edu
- Received by editor(s): February 9, 2010
- Received by editor(s) in revised form: July 6, 2011
- Published electronically: June 25, 2012
- Additional Notes: The second author was partially supported by NSF Grants CCF-0634123 and CCF-1017217. He would also like to thank the Institut de Recherche Mathématique Avancée, Université de Strasbourg et C.N.R.S., for its hospitality and support during the preparation of this paper.
- Communicated by: Sergei K. Suslov
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**141**(2013), 605-617 - MSC (2010): Primary 34M56, 12H05, 34M55
- DOI: https://doi.org/10.1090/S0002-9939-2012-11499-6
- MathSciNet review: 2996965