Skip to main content
SpringerLink
Log in

Algebraic prolongation and rigidity of Carnot groups

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We discuss the known results on rigidity of Carnot groups using Tanaka’s prolongation theory. We also apply Tanaka’s theory to study rigidity of an extended class of H-type groups which we call J-type groups. In particular we obtain a rigidity criterion giving rise to a rigid class of J-type groups which includes the H-type groups, and thus extends the results of H.M. Reimann. We also construct a noncomplex J-type group which is nonrigid and does not satisfy the rank 1 condition over the reals.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bourbaki, N.: Group et algèbre de Lie. Éléments de mathématique Fascicule, XXXIV. Hermann Paris (1968)

  2. Ciatti P.: A new proof of the J 2-condition for real rank one simple Lie alebras and their classification. Proc. Am. Math. Soc. 133(6), 1611–1616 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ciatti P.: A Clifford algebra approach to simple Lie algebras of real rank two. I. The A 2 case. J. Lie Theory 10(1), 53–80 (2000)

    MATH  MathSciNet  Google Scholar 

  4. Cowling M., De Mari F., Korányi A., Reimann H.M.: Contact and conformal mappings in parabolic geometry. I. Geom. Dedicata 111, 65–86 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cowling, M., Reimann, H.M.: Quasiconformal mappings on Carnot groups: three examples. In: Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001). Contemp. Math. Amer. Math. Soc., vol. 320, pp. 111–118. Providence (2003)

  6. Knapp, A.: Lie groups beyond an introduction, 2nd edn. In: Progress in Math, vol. 140. Birkhäuser, Boston (2002)

  7. Kobayashim, S.: Transformation groups in differential geometry. Reprint of the 1972 edition. In: Classics in Mathematics, viii+182 pp. Springer, Berlin (1995)

  8. Kobayashi S., Nomizu K.: Foundations of Differential Geometry Volume II. Wiley, New York (1996)

    Google Scholar 

  9. Korányi A., Reimann H.M.: Quasiconformal mappings on the Heisenberg group. Invent Math. 80(2), 309–338 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kuzmich O.: Graded Nilpotent Lie Algebras in low dimensions. Lobachevskii J. Math. 3, 147–184 (1999)

    MATH  MathSciNet  Google Scholar 

  11. Levstein F., Tiraboschi A.: Classes of 2-step nilpotent Lie algebras. Comm. Algebra 27(5), 2425–2440 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Métivier G.: Hypoellipticité analytique sur des groupes nilpotents de rang 2. Duke Math. J. 47, 195–221 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  13. Müller D., Seeger A.: Singular spherical maximal operators on class of two step nilpotent Lie groups. Israel J. Math. 141, 315–340 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ottazzi A.: A sufficient condition for nonrigidity of Carnot groups. Math. Z 259, 617–629 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ottazzi A.: Multicontact vector fields on Hessenberg manifolds. J. Lie Theory 15(2), 357–377 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Ottazzi, A., Warhurst, B.: Rigidity of Iwasawa nilpotent Lie groups via Tanaka’s theory. Submitted.

  17. Reimann H.M.: Rigidity of H-type groups. Math. Z 237((4), 697–725 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Reimann, H.M., Ricci, F.: The complexified Heisenberg group. In: Proceedings on Analysis and Geometry. International conference in honor of the 70th birthday of Professor Yu. G. Reshetnyak, Novosibirsk, Russia, Aug. 30-Sept. 2, 1999. Izdatelstvo Instituta Matematiki Im. S. L. Soboleva SO RAN, pp. 465-480 (2000)

  19. Saal L.: The automorphism group of a Lie algebra of Heisenberg type. Rend. Sem. Mat. Univ. Politec. Torino 54(2), 101–113 (1996)

    MATH  MathSciNet  Google Scholar 

  20. Spencer D.C.: Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc. 75, 179–239 (1969)

    Article  MATH  Google Scholar 

  21. Sternberg S.: Lectures on Differential Geometry. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  22. Tanaka N.: On generalized graded Lie algebras and geometric structures. I. J. Math. Soc. Jpn. 19, 215–254 (1967)

    Article  MATH  Google Scholar 

  23. Tanaka N.: On differential systems, graded Lie algebras and pseudogroups. J. Math. Kyoto Univ. 10, 1–82 (1970)

    MATH  MathSciNet  Google Scholar 

  24. Warhurst B.: Jet spaces as nonrigid Carnot groups. J. Lie Theory 15(1), 341–356 (2005)

    MATH  MathSciNet  Google Scholar 

  25. Warhurst B.: Tanaka prolongation of free Lie algebras. Geom. Dedicata 130, 59–69 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. In: Progress in differential geometry. Adv. Stud. Pure Math, vol. 22, pp. 413–494. Math. Soc. Japan, Tokyo (1993)

Download references

Author information

Authors and Affiliations

  1. Università di Milano Bicocca, 20126, Milan, Italy

    Alessandro Ottazzi

  2. School of Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    Ben Warhurst

Authors
  1. Alessandro Ottazzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ben Warhurst
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alessandro Ottazzi.

Additional information

Communicated by K. Schmidt.

Ben Warhurst was supported by ARC Discovery grant “Geometry on Nilpotent Groups” and the Institute of Mathematics of the Polish Academy of Sciences.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ottazzi, A., Warhurst, B. Algebraic prolongation and rigidity of Carnot groups. Monatsh Math 162, 179–195 (2011). https://doi.org/10.1007/s00605-009-0170-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-009-0170-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation