Skip to main content
SpringerLink
Log in

INTEGRABILITY OF QUATERNION-KÄHLER SYMMETRIC SPACES

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

We find a necessary condition for the existence of an action of a Lie group G by quaternionic automorphisms on an integrable quaternionic manifold in terms of representations of 𝔤. We check this condition and prove that a Riemannian symmetric space of dimension 4n for n ≥ 2 has an invariant integrable almost quaternionic structure if and only if it is quaternionic vector space, quaternionic hyperbolic space, or quaternionic projective space.

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. M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), no. 2, 85–177.

  2. A. L. Besse, Einstein Manifolds, Classics in Mathematics, Springer, Berlin, 2008.

  3. Bonan, E.: Sur les G-structures de type quaternionien. Cahiers Topologie Géom. Différentielle. 9(4), 389–461 (1967)

    MathSciNet  MATH  Google Scholar 

  4. G. E. Bredon, Introduction to Compact Transformation Groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York, 1972.

  5. R. Cui, The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F), Ph.D. thesis, University of Maryland, College Park, 2016.

  6. W. Fulton, J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics, Vol. 129, Springer, New York, 1991.

  7. Gross, B.H., Wallach, N.R.: On quaternionic discrete series representations, and their continuations. J. Reine Angew. Math. 481, 73–123 (1996)

    MathSciNet  MATH  Google Scholar 

  8. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, Vol. 34, American Mathematical Society, Providence, RI, 2001.

  9. Iwahori, N.: On real irreducible representations of Lie algebras. Nagoya Math. J. 14, 59–83 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  10. Joyce, D.D.: Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts in Mathematics, vol. 12. Oxford University Press, Oxford (2007)

    Google Scholar 

  11. A. W. Knapp, Lie Groups Beyond an Introduction, 2nd Edition, Progress in Mathematics, Vol. 140, Birkhäuser Boston, Boston, MA, 2002.

  12. Kulkarni, R.S.: On the principle of uniformization. J. Differential Geom. 13(1), 109–138 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lee, J.M.: Introduction to Smooth Manifolds, 2nd Edition, Graduate Texts in Mathematics, vol. 218. Springer, New York (2013)

    Google Scholar 

  14. Liu, H., Zhang, G.: Realization of quaternionic discrete series on the unit ball in ℍd. J. Funct. Anal. 262(7), 2979–3005 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Olver, P.J.: Equivalence, Invariants, and Symmetry. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  16. Onishchik, A.L.: Topology of Transitive Transformation Groups. Johann Ambrosius Barth, Leipzig (1994)

    MATH  Google Scholar 

  17. A. L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations, ESI Lectures in Mathematics and Physics, Vol. 1, European Mathematical Society, Zürich, 2004.

  18. S. M. Salamon, Quaternionic manifolds, in: Symposia Mathematica, Vol. XXVI (Rome, 1980), Academic Press, London, 1982, pp. 139–151.

  19. S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31–55.

  20. Schoeneberg, T.: Semisimple Lie algebras and their classification over 𝔭-adic fields. Mém. Soc. Math. Fr. (N.S.). 151, 1–147 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Sternberg, S.: Lectures on Differential Geometry, 2nd edn. Chelsea Publishing Co., New York (1983)

    MATH  Google Scholar 

  22. Terzić, S.: Rational homotopy groups of generalised symmetric spaces. Math. Z. 243(3), 491–523 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. W. P. Thurston, Three-dimensional Geometry and Topology, Vol. 1, Princeton Mathematical Series, Vol. 35, Princeton University Press, Princeton, NJ, 1997.

  24. J. Tits, Classification of algebraic semisimple groups, in: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colorado, 1965), American Mathematical Society, Providence, RI, 1966, pp. 33–62.

  25. J. Tits, Tabellen zu den Einfachen Lie Gruppen und Ihren Darstellungen, Lecture Notes in Mathematics, Vol. 40, Springer, Berlin, 1967.

  26. Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14(6), 1033–1047 (1965)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Technion, 320003, Haifa, Israel

    ANTON HASE

Authors
  1. ANTON HASE
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ANTON HASE.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

HASE, A. INTEGRABILITY OF QUATERNION-KÄHLER SYMMETRIC SPACES. Transformation Groups 28, 803–829 (2023). https://doi.org/10.1007/s00031-022-09724-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-022-09724-w

Search

Navigation