Skip to main content
SpringerLink
Log in

Applications of differential algebra for computing Lie algebras of infinitesimal CR-automorphisms

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We perform detailed computations of Lie algebras of infinitesimal CR-automorphisms associated to three specific model real analytic CR-generic submanifolds in ℂ9 by employing differential algebra computer tools-mostly within the Maple package DifferentialAlgebra — in order to automate the handling of the arising highly complex linear systems of PDE’s. Before treating these new examples which prolong previous works of Beloshapka, of Shananina and of Mamai, we provide general formulas for the explicitation of the concerned PDE systems that are valid in arbitrary codimension k ⩾ 1 and in any CR dimension n ⩾ 1. Also, we show how Ritt’s reduction algorithm can be adapted to the case under interest, where the concerned PDE systems admit so-called complex conjugations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aghasi M, Merker J, Sabzevari M. Effective Cartan-Tanaka connections on C 6-smooth strongly pseudoconvex hypersurfaces M 3 ⊂ ℂ2. Comptes Rendus Mathematique, 2011, 349: 845–848; Arxiv:1104.1509v1

    Article  MATH  MathSciNet  Google Scholar 

  2. Baouendi S, Ebenfelt P, Rothschild L P. Real Submanifolds in Complex Space and Their Mappings. Princeton: Princeton University Press, 1999

    MATH  Google Scholar 

  3. Becker T, Weispfenning V. Gröbner Bases, A Computational Approach to Commutative Algebra. New York: Springer, 1993

    MATH  Google Scholar 

  4. Beloshapka V K. Universal models for real submanifolds. Mathe Notes, 2004, 75: 475–488

    Article  MATH  MathSciNet  Google Scholar 

  5. Beloshapka V K. CR-Varieties of the type (1, 2) as varieties of super-high codimension. Russian J Math Phys, 1997, 5: 399–404

    MATH  MathSciNet  Google Scholar 

  6. Beloshapka V K, Ezhov V, Schmalz G. Canonical Cartan connection and holomorphic invariants on Engel CR manifolds. Russian J Math Phys, 2007, 14: 121–133

    Article  MATH  MathSciNet  Google Scholar 

  7. Beloshapka V K, Kossovskiy I G. Classification of homogeneous CR-manifolds in dimension 4. J Math Anal Appl, 2011, 374: 655–672.

    Article  MATH  MathSciNet  Google Scholar 

  8. Boggess A. CR-Manifolds and the Tangential Cauchy-Riemann Complex. Studies in Advanced Mathematics. New York: CRC Press, 1991

    Google Scholar 

  9. Boulier F, Lazard D, Ollivier F, et al. Computing representations for radicals of finitely generated differential ideals. Appl Algebra Engin Commun Comput, 2009, 20: 73–121

    Article  MATH  MathSciNet  Google Scholar 

  10. Boulier F, Lazard D, Ollivier F, et al. Representation for the radical of a finitely generated differential ideal. ISSAC’95: New York: ACM Press, 1995, 158–166

    Google Scholar 

  11. Chern S S, Moser J. Real hypersurfaces in complex manifolds. Acta Math, 1975, 133: 219–271

    Article  MATH  MathSciNet  Google Scholar 

  12. Chou S C. Mechanical Geometry Theorem Proving. Dordrecht: Springer, 1988

    MATH  Google Scholar 

  13. Ezhov V, McLaughlin B, Schmalz G. From Cartan to Tanaka: Getting real in the complex world. Notices of the AMS, 2011, 58: 20–27

    MATH  MathSciNet  Google Scholar 

  14. Fels G, Kaup W. Classification of Levi degenerate homogeneous CR-manifolds in dimension 5. Acta Math, 2008, 201: 1–82

    Article  MATH  MathSciNet  Google Scholar 

  15. Gallo G. Complexity Issues in Computational Algebra. Ph.D. Thesis. Courant Institute of Mathematical Sciences, New York University, 1992

    Google Scholar 

  16. Gallo G, Mishra B. Efficient algorithms and bounds for Wu-Ritt characteristic sets. In: Progress in Mathematics: Effective Methods in Algebraic Geometry. Boston: Birkhäuser, 1991, 119–142

    Google Scholar 

  17. Gallo G, Mishra B. Wu-Ritt characteristic sets and their complexity. In: Discrete and Computational Geometry: Papers from the DIMACS Special Year, vol. 6. Providence, RI: Amer Math Soc, 1991, 6: 111–136

    Google Scholar 

  18. Gaussier H, Merker J. Nonalgebraizable real analytic tubes in ℂn. Math Z, 2004, 247: 337–383

    Article  MATH  MathSciNet  Google Scholar 

  19. Isaev A. Spherical Tube Hypersurfaces. Lecture Notes in Mathematics. New York: Springer, 2011

    Book  Google Scholar 

  20. Kolchin E R. Differential Algebra and Algebraic Groups. New York: Academic Press, 1973

    MATH  Google Scholar 

  21. Mamai I B. Model CR-manifolds with one-dimensional complex tangent. Russian J Math Phys, 2009, 16: 97–102

    Article  MATH  MathSciNet  Google Scholar 

  22. Mansfield E N, Clarkson P A. Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations. J Symb Comp, 1997, 23: 517–533

    Article  MathSciNet  Google Scholar 

  23. Merker J. Lie symmetries and CR geometry. J Math Sci, 2008, 154: 817–922

    Article  MathSciNet  Google Scholar 

  24. Merker J, Porten E. Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities. Internat Math Res Surveys, 2006, ID 28295, 287 pages

    Google Scholar 

  25. Merker J, Sabzevari M. Cartan equivalence problem for 5-dimensional CR-manifolds in ℂ4. In progress

  26. Merker J, Sabzevari M. Explicit expression of Cartan’s connections for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere. Cent Eur J Math, 2012, 10: 1801–1835

    Article  MATH  MathSciNet  Google Scholar 

  27. Ritt J F. Differential Algebra. New York: Amer Math Soc, 1948

    Google Scholar 

  28. Shananina E N. Models for CR-manifolds of type (1, K) for 3 ⩽ K ⩽ 7 and their automorphisms. Mathe Notes, 2000, 67: 382–388

    Article  MathSciNet  Google Scholar 

  29. Stenström S. Differential Gröbner Bases. Master Thesis. Lulea University of Technology, 2002

    Google Scholar 

  30. Tumanov A E. Finite-dimensionality of the group of CR automorphisms of a standard CR-manifold, and proper holomorphic mappings of Siegel domains. Math USSR Izvestiya, 1989, 32: 655–662

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, University of Shahrekord, 88186-34141, Shahrekord, Iran

    Masoud Sabzevari

  2. Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

    Amir Hashemi

  3. School of Mathematics and Computer Sciences, Damghan University, P.O.Box 3671641167, Damghan, Iran

    Benyamin M.-Alizadeh

  4. Départment de Mathématiques d’Orsay, Bâtiment 425 Faculté des Sciences, Université Paris XI-Orsay, F-91405, Orsay Cedex, France

    Joël Merker

  5. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5746, Tehran, Iran

    Masoud Sabzevari & Amir Hashemi

Authors
  1. Masoud Sabzevari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amir Hashemi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benyamin M.-Alizadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Joël Merker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Masoud Sabzevari.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sabzevari, M., Hashemi, A., M.-Alizadeh, B. et al. Applications of differential algebra for computing Lie algebras of infinitesimal CR-automorphisms. Sci. China Math. 57, 1811–1834 (2014). https://doi.org/10.1007/s11425-013-4751-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-013-4751-5

Keywords

MSC(2010)

Search

Navigation