Algebras and Representation Theory

, Volume 21, Issue 5, pp 1037–1069 | Cite as

Lie Algebras of Slow Growth and Klein-Gordon PDE

  • Dmitry Millionshchikov


We discuss the notion of characteristic Lie algebra of a hyperbolic PDE. The integrability of a hyperbolic PDE is closely related to the properties of the corresponding characteristic Lie algebra χ. We establish two explicit isomorphisms:
  1. 1)
    the first one is between the characteristic Lie algebra \(\chi (\sinh {u})\) of the sinh-Gordon equation \(u_{xy}=\sinh {u}\) and the non-negative part \({\mathcal {L}}({\mathfrak {sl}}(2,{\mathbb {C}}))^{\ge 0}\) of the loop algebra of \({\mathfrak {sl}}(2,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{1}^{(1)}\)
    $$\chi(\sinh{u})\cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(2,{\mathbb{C}}))^{\ge 0}={\mathfrak{s}\mathfrak{l}}(2, {\mathbb{C}}) \otimes {\mathbb{C}}[t]. $$
  2. 2)
    the second isomorphism is for the Tzitzeica equation uxy = eu + e− 2u
    $$\chi(e^{u}{+}e^{-2u}) \cong {\mathcal{L}}({\mathfrak{s}\mathfrak{l}}(3,{\mathbb{C}}), \mu)^{\ge0}=\bigoplus_{j = 0}^{+\infty}{\mathfrak{g}}_{j (\text{mod} \; 2)} \otimes t^{j}, $$
    where \({\mathcal {L}}({\mathfrak {sl}}(3,{\mathbb {C}}), \mu )=\bigoplus _{j \in {\mathbb {Z}}}{\mathfrak {g}}_{j (\text {mod} \; 2)} \otimes t^{j}\) is the twisted loop algebra of the simple Lie algebra \({\mathfrak {sl}}(3,{\mathbb {C}})\) that corresponds to the Kac-Moody algebra \(A_{2}^{(2)}\).

Hence the Lie algebras \(\chi (\sinh {u})\) and χ(eu + e− 2u) are slowly linearly growing Lie algebras with average growth rates \(\frac {3}{2}\) and \(\frac {4}{3}\) respectively.


Characteristic Lie algebra Naturally graded Lie algebra Loop algebra Kac-Moody algebra Hyperbolic PDE Sine-Gordon equation Tzitzeica equation Bell polynomial Gelfand-Kirillov dimension 

Mathematics Subject Classification (2010)

17B80 17B67 35B06 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The author is very grateful to Sergey Smirnov and Victor Buchstaber for valuable comments and remarks.


  1. 1.
    Agrachev, A., Marigo, A.: Rigid Carnot algebras: a classification. J. Dyn. Control. Syst. 11(4), 449–494 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions, Cambridge Mathematical Library (1St Pbk Ed.) Cambridge University Press, UK (1998)Google Scholar
  3. 3.
    Buchstaber V.M.: Polynomial Lie algebras and the Shalev-Zelmanov theorem, Russian Mathematical Surveys, 72(6) (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Crowdy, D.: General solutions to the 2D Liouville equation. Int. J. of Engng Sci. 35(2), 141–149 (1997)CrossRefGoogle Scholar
  5. 5.
    Fialowski, A.: Classification of graded Lie algebras with two generators. Mosc. Univ. Math. Bull. 38(2), 76–79 (1983)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Gelfand, I.M., Kirillov, A.A.: Sur les corps liés aux algèbres enveloppantes des algèbres de Lie. Inst. Hautes Etudes Sci. Publ. Math. 31, 5–19 (1966)CrossRefGoogle Scholar
  7. 7.
    Goursat, E.: Recherches sur quelques equations auux derivées partielles de second ordre. Annales de la Faculté, des Sciences de l’Université de Toulouse 2e serie 1(1), 31–78 (1899)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Boston (1984)Google Scholar
  9. 9.
    Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd ed. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  10. 10.
    Kac, V.G.: Simple graded Lie algebras of finite growth. Math. USSR Izv. 2, 1271–1311 (1968)CrossRefGoogle Scholar
  11. 11.
    Kac, V.G.: Some problems on infinite-dimensional Lie algebras. In: Lie Algebras and related Topics, Lecture Notes in Mathematics 933. Springer (1982)Google Scholar
  12. 12.
    Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. AMS, Providence (2000)zbMATHGoogle Scholar
  13. 13.
    Lepowsky, J., Milne, S.: Lie algebraic approaches to classical partition identities. Adv. Math. 29, 15–59 (1978)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lepowsky, J., Willson, R.L.: Construction of the Affine Lie Algebra \(A_{1}^{(1)}\). Commun Math. Phys. 62, 43–53 (1978)CrossRefGoogle Scholar
  15. 15.
    Leznov, A.N.: On the complete integrability of a nonlinear system of partial differential equations in two-dimensional space. Theoret. Math. Phys. 42(3), 225–229 (1980)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Leznov, A.N., Savel’ev, M.V., Smirnov, V.G.: Theory of group representations and integration of nonlinear dynamical systems. Theoret. Math. Phys. 48(1), 565–571 (1981)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Leznov, A.N., Savel’ev, M.V.: Two-dimensional nonlinear system of differential equations \(x_{\alpha , z z}=\exp {kx}_{\alpha }\). Funct. Anal. Appl. 14(3), 238–240 (1980)CrossRefGoogle Scholar
  18. 18.
    Leznov, A.N., Smirnov, V.G., Shabat, A.B.: The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems. Theoret. Math. Phys. 51(1), 322–330 (1982)CrossRefGoogle Scholar
  19. 19.
    Mathieu, O.: Classification of simple graded Lie algebras of finite growth. Invent. Math. 108, 455–519 (1990)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Millionschikov, D.V.: Naturally graded Lie algebras (Carnot algebras) of slow growth. arXiv:1705.07494
  21. 21.
    Rinehart, G.: Differential forms for general commutative algebras. Trans. Amer.Math. Soc. 108, 195–222 (1963)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sakieva, A.U.: The characteristic Lie ring of the Zhiber-Shabat-Tzitzeica equation. Ufa Math. J. 4(3), 155–160 (2012)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Shalev, A., Zelmanov, E.I.: Narrow algebras and groups. J. of Math. Sci. 93 (6), 951–963 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Shabat, A.B., Yamilov, R.I.: Exponencialnie systemy typa I i matricy Cartana (in russian), Preprint of Bashkir filial of the Soviet Academy of Sciences, Ufa (1981), 1–22 (1981)Google Scholar
  25. 25.
    Tzitzeica, G.: Sur une nouvelle classe de surfaces. Comptes rendus Acad. Sci. 150, 955–956 (1990)zbMATHGoogle Scholar
  26. 26.
    Zhiber, A., Murtazina, R.D.: On the characteristic Lie algebras for equations ”u xy = f(u, u x)”. J. Math. Sci. 151(4), 3112–3122 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zhiber, A., Murtazina, R.D., Habibullin, I.T., Shabat, A.B.: Characteristic Lie rings and integrable models in mathematical physics. Ufa Math. J. 4(3), 17–85 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Zhiber, A.V., Shabat, A.B.: Klein-gordon equations with a nontrivial group. Sov. Phys. Dokl. 24(8), 608–609 (1979)Google Scholar
  29. 29.
    Zhiber, A.V., Shabat, A.B.: Systems of equations u x = p(u, v),v y = q(u, v) that possess symmetries. Soviet Math. Dokl. 30(1), 23–26 (1984)zbMATHGoogle Scholar
  30. 30.
    Zhiber, A.V., Sokolov, V.V.: Exactly integrable hyperbolic equations of Liouville type. Russ. Math. Surv. 56(1), 61–101 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations