Skip to main content

The spectra of the Laplace operators on connected compact simple Lie groups of rank 3

Abstract

We expose explicit calculations of the spectra of the Laplace operators for smooth real or complex functions on all connected compact simple Lie groups of rank 3 with bi-invariant Riemannian metric and establish the relationship of the obtained formulas with number theory and integer-valued ternary and binary quadratic forms.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    J. F. Adams, Lectures on Lie Groups. Mathematics Lecture Note Series (W. A. Benjamin, New York–Amsterdam, 1969; Nauka, Moscow, 1979).

    Google Scholar 

  2. 2.

    V. N. Berestovskiĭ, “On the spectrum of the Laplace operator for real functions on compact normal homogeneous Riemannian manifolds,” Proc. of the International Conference “Geometry Days in Novosibirsk 2011” dedicated to the 50th anniversary of the Chair of Geometry and Topology of Novosibirsk State University (Novosibirsk State University, Novosibirsk, 2012), 16.

    Google Scholar 

  3. 3.

    V. N. Berestovskiĭ, “Zonal spherical functions on CROSS’s and special functions,” Sib. Mat. Zh. 53, 765 (2012) [Sib.Math. J. 53, 611 (2012)].

    MATH  Google Scholar 

  4. 4.

    V. N. Berestovskiĭand V. M. Svirkin, “The Laplace operator on normal homogeneousRiemannianmanifolds,” Mat. Tr. 12, 3 (2009) [Sib. Adv.Math. 20, 231 (2010)].

    MathSciNet  Google Scholar 

  5. 5.

    V. N. Berestovskiĭand V. M. Svirkin, “The spectrum of the Laplace operator on compact simply connected Lie groups of rank two,” Uch. Zap. Kazan. Gos. Univ., Ser. Fiz.-Mat. Nauki 151 (4), 15 (2009).

    MATH  Google Scholar 

  6. 6.

    N. Bourbaki, Éléments de Mathématique. Groupes et Algèbres de Lie. Chapitres IV, V et VI (Hermann & Cie, Paris, 1968; Mir,Moscow, 1972).

    Google Scholar 

  7. 7.

    A. A. Bukhshtab, Number Theory (Gos. Uch.-Ped. Izdat., Moscow, 1960) [in Russian].

    MATH  Google Scholar 

  8. 8.

    E. Cartan, “Les groupes projectifs qui ne laissent invariante aucune multiplicitéplane,” Bull. Soc. Math. France. 41, 53 (1913).

    MathSciNet  MATH  Google Scholar 

  9. 9.

    J. H. Conway, The Sensual Quadratic Form. The Carus Mathematical Monographs. 26 (The Mathematical Association of America, Washington, DC, 1997).

    Google Scholar 

  10. 10.

    J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups (Springer-Verlag, New York etc., 1988; Mir, Moscow, 1990).

    MATH  Google Scholar 

  11. 11.

    H. Davenport, The Higher Arithmetic. An Introduction to the Theory of Numbers (Hutchinson’s University Library, London, 1952; Nauka, Moscow, 1965).

    MATH  Google Scholar 

  12. 12.

    E. B. Dynkin and A. L. Onishchik, “Compact global Lie groups,” Usp. Mat. Nauk 10 (4), 3 (1955) [Transl., Ser. 2, Am.Math. Soc. 21, 119 (1962)].

    MathSciNet  MATH  Google Scholar 

  13. 13.

    G. P. Matvievskaya, E. P. Ozhigova, N. I. Nevskaya (comp.), and Yu. Kh. Kopelevich Unpublished Papers of L. Euler on Number Theory (Nauka, Sankt-Peterburg, 1997) [in Russian].

    MATH  Google Scholar 

  14. 14.

    A. L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations. ESI Lectures in Mathematics and Physics. (EuropeanMath. Soc. Publishing House, Zurich, 2004).

    Book  MATH  Google Scholar 

  15. 15.

    V. M. Svirkin, “Spectrum of the Laplace operator on connected compact simple Lie groups of rank one or two,” Uch. Zap. Kazan. Gos. Univ., Ser. Fiz.-Mat. Nauki 152 (1), 219–234 (2010).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    B. A. Venkov, Elementary Number Theory (Obied. Nauch.-Tekhn. Izd-vo NKP SSSR, Moscow, 1937; Wolters-Noordhoff Publishing, Groningen, 1970).

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to V. N. Berestovskiĭ.

Additional information

Original Russian Text © V.N. Berestovski˘ı, I.A. Zubareva, and V.M. Svirkin, 2016, published in Matematicheskie Trudy, 2016, Vol. 19, No. 1, pp. 3–45.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Berestovskiĭ, V.N., Zubareva, I.A. & Svirkin, V.M. The spectra of the Laplace operators on connected compact simple Lie groups of rank 3. Sib. Adv. Math. 26, 153–181 (2016). https://doi.org/10.3103/S1055134416030019

Download citation

Keywords

  • Laplace operator
  • spectrum
  • representation of a group
  • Killing form