Abstract
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:
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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)
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.
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References
Agrachev, A., Marigo, A.: Rigid Carnot algebras: a classification. J. Dyn. Control. Syst. 11(4), 449–494 (2005)
Andrews, G.E.: The Theory of Partitions, Cambridge Mathematical Library (1St Pbk Ed.) Cambridge University Press, UK (1998)
Buchstaber V.M.: Polynomial Lie algebras and the Shalev-Zelmanov theorem, Russian Mathematical Surveys, 72(6) (2017)
Crowdy, D.: General solutions to the 2D Liouville equation. Int. J. of Engng Sci. 35(2), 141–149 (1997)
Fialowski, A.: Classification of graded Lie algebras with two generators. Mosc. Univ. Math. Bull. 38(2), 76–79 (1983)
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)
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)
Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. Reidel, Boston (1984)
Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd ed. Cambridge University Press, Cambridge (1990)
Kac, V.G.: Simple graded Lie algebras of finite growth. Math. USSR Izv. 2, 1271–1311 (1968)
Kac, V.G.: Some problems on infinite-dimensional Lie algebras. In: Lie Algebras and related Topics, Lecture Notes in Mathematics 933. Springer (1982)
Krause, G.R., Lenagan, T.H.: Growth of Algebras and Gelfand-Kirillov Dimension. AMS, Providence (2000)
Lepowsky, J., Milne, S.: Lie algebraic approaches to classical partition identities. Adv. Math. 29, 15–59 (1978)
Lepowsky, J., Willson, R.L.: Construction of the Affine Lie Algebra \(A_{1}^{(1)}\). Commun Math. Phys. 62, 43–53 (1978)
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)
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)
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)
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)
Mathieu, O.: Classification of simple graded Lie algebras of finite growth. Invent. Math. 108, 455–519 (1990)
Millionschikov, D.V.: Naturally graded Lie algebras (Carnot algebras) of slow growth. arXiv:1705.07494
Rinehart, G.: Differential forms for general commutative algebras. Trans. Amer.Math. Soc. 108, 195–222 (1963)
Sakieva, A.U.: The characteristic Lie ring of the Zhiber-Shabat-Tzitzeica equation. Ufa Math. J. 4(3), 155–160 (2012)
Shalev, A., Zelmanov, E.I.: Narrow algebras and groups. J. of Math. Sci. 93 (6), 951–963 (1999)
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)
Tzitzeica, G.: Sur une nouvelle classe de surfaces. Comptes rendus Acad. Sci. 150, 955–956 (1990)
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)
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)
Zhiber, A.V., Shabat, A.B.: Klein-gordon equations with a nontrivial group. Sov. Phys. Dokl. 24(8), 608–609 (1979)
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)
Zhiber, A.V., Sokolov, V.V.: Exactly integrable hyperbolic equations of Liouville type. Russ. Math. Surv. 56(1), 61–101 (2001)
Acknowledgements
The author is very grateful to Sergey Smirnov and Victor Buchstaber for valuable comments and remarks.
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Presented by: Valentin Ovsienko
Dedicated to Alexander Alexandrovich Kirillov with afeeling of gratitude and sincere admiration
This work is supported by RFBR under the grant 17-01-00671
Appendix: The correspondence tables of different gradings for \({\mathfrak {n}}_{1}\) and \({\mathfrak {n}}_{2}\)
Appendix: The correspondence tables of different gradings for \({\mathfrak {n}}_{1}\) and \({\mathfrak {n}}_{2}\)
It follows from the proof of Theorem 2 that the weighted bigrading of \(\text {Diff}{\mathcal F}\) induces the \({\mathbb {Z}}_{\ge 0}{\times }{\mathbb {Z}}_{3}\)-grading on the Lie algebra \(\tilde {\mathfrak {n}}_{1}\). The corresponding bidegrees of its basic elements \(X_{n}^{\prime }\) are presented in the Table 2.
The Lie algebra \(\tilde {\mathfrak {n}}_{2}\) is \({\mathbb {Z}}_{\ge 0}{\times }{\mathbb {Z}}_{5}\)-graded (see the proof of Theorem 3). We list the corresponding bidegrees of its basic elements \(Y_{n}^{\prime }\) in the Table 3.
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Millionshchikov, D. Lie Algebras of Slow Growth and Klein-Gordon PDE. Algebr Represent Theor 21, 1037–1069 (2018). https://doi.org/10.1007/s10468-018-9794-4
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DOI: https://doi.org/10.1007/s10468-018-9794-4
Keywords
- Characteristic Lie algebra
- Naturally graded Lie algebra
- Loop algebra
- Kac-Moody algebra
- Hyperbolic PDE
- Sine-Gordon equation
- Tzitzeica equation
- Bell polynomial
- Gelfand-Kirillov dimension