Characteristic algebras and integrable exponential systems

D. V. Millionshchikov   ; S. V. Smirnov

Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 41-69

Résumé

In the present paper we study characteristic algebras for exponential systems corresponding to degenerate Cartan matrices. These systems generalize hyperbolic sine-Gordon and Tzitzeica equations well-known in the theory of integrable systems. For such systems, corresponding to Cartan matrices of rank $2$, we describe explicitly characteristic algebras in terms of generators and relations and we prove that they have linear growth. We study the relations between the higher symmetries of these systems and the structure of their characteristic algebras. We describe completely the higher symmetries of exponential systems corresponding to the Cartan matrix of affine Lie algebra $A^{(1)}_2$. We also obtain partial results on symmetries of such systems corresponding to other degenerate Cartan matrices of rank $2$. We propose a conjecture on the structure of higher symmetries of arbitrary exponential system corresponding to a degenerate Cartan matrix. We study an interesting combinatorics related to an operator generating a characteristic algebra in the simplest case for a Darboux integrable Liouville equation. The found combinatorial properties can be very useful for proving the aforementioned conjecture on the structure of higher symmetries. Moreover, in the present paper we give a rigorous meaning to the concept of a characteristic algebra of a hyperbolic system used for a long time in the literature. We do this by means of the notion of Lie-Rinehart algebra and at the examples we demonstrate that such formalization is indeed needed.

Détail

Keywords: characteristic algebra, higher symmetry, exponential system.
Mots-clés : Liouville equation

Bibliographie

[1] A. B. Shabat, R. I. Yamilov, Exponential systems of type I and Cartan matrices, Preprint, Bashkir Branch of Academy of Sciences of USSR, 1981 (in Russian)

[2] A. N. Leznov, V. G. Smirnov, A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems”, Theor. Math. Phys., 51:1 (1982), 322–330 | DOI | MR | Zbl

[3] A. N. Leznov, “On the complete integrability of a nonlinear system of partial differential equations in two-dimensional space”, Theor. Math. Phys., 42:3 (1980), 225–229 | DOI | MR | MR | Zbl | Zbl

[4] V. E. Adler, I. T. Habibullin, “Integrable boundary conditions for the Toda lattice”, J. Phys. A: Math. Gen., 28:23 (1995), 67116129 | DOI | MR

[5] B. Gürel, I. Habibullin, “Boundary conditions for two-dimensional integrable chains”, Phys. Lett. A, 233:3–4 (1997), 68–72 | DOI | MR | Zbl

[6] I. T. Habibullin, “Truncations of Toda chains and the reduction problem”, Theor. Math. Phys., 143:1 (2005), 515–528 | DOI | MR | MR | Zbl

[7] R. S. Ward, “Discrete Toda field equations”, Phys. Lett. A, 199:1–2 (1995), 45–48 | DOI | MR | Zbl

[8] A. Doliwa, “Geometric discretisation of the Toda system”, Phys. Lett. A, 234:3 (1997), 187–192 | DOI | MR | Zbl

[9] I. T. Habibullin, “$C$-series discrete chains”, Theor. Math. Phys., 146:2 (2006), 170–182 | DOI | MR | Zbl

[10] S. V. Smirnov, “Semidiscrete Toda lattices”, Theor. Math. Phys., 172:3 (2012), 1217–1231 | DOI | MR | Zbl

[11] I. Habibullin, K. Zheltukhin, M. Yangubaeva, “Cartan matrices and integrable lattice Toda field equations”, J. Phys. A: Math. Theor., 44:46 (2011), 465202 | DOI | MR | Zbl

[12] R. Garifullin, I. Habibullin, M. Yangubaeva, “Affine and finite Lie algebras and integrable Toda field equations on discrete time-space”, SIGMA, 8 (2012), 062 | MR | Zbl

[13] S. V. Smirnov, “Darboux integrability of discrete two-dimensional Toda lattices”, Theor. Math. Phys., 182:2 (2015), 189–210 | DOI | MR | Zbl

[14] D. K. Demskoi, D. T. Tran, “Darboux integrability of determinant and equations for principal minors”, Nonlinearity, 29:7 (2016), 1973–1991 | DOI | MR | Zbl

[15] I. T. Habibullin, A. R. Khakimova, “Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras $A^{(1)}_{N-1}$”, J. Phys. A: Math. Theor., 52:36 (2019), 365202 | DOI | MR

[16] I. Habibullin, A. Khakimova, “Integrable boundary conditions for the Hirota-Miwa equation and Lie algebras”, J. Nonlinear Math. Phys., 27:3 (2020), 393–413 | DOI | MR | Zbl

[17] I. T. Habibullin, M. N. Kuznetsova, A. U. Sakieva, “Integrability conditions for two-dimensional Toda-like equations”, J. Phys. A: Math. Theor., 53:39 (2020), 395203 | DOI | MR

[18] E. Goursat, “Recherches sur quelques équations aux dérivées partielles du second ordre”, Ann. Fac. Sci. Toulouse., 1 (1899), 31–78 ; 439–464 | DOI | MR | Zbl

[19] A. V. Zhiber, N. Kh. Ibragimov, A. B. Shabat, “Equations of Liouville type”, Sov. Math. Dokl., 20 (1979), 1183–1187 | MR | Zbl

[20] I. T. Habibullin, “Characteristic algebras of fully discrete hyperbolic type equations”, SIGMA, 1 (2005), 023 | MR | Zbl

[21] A. V. Zhiber, R. D. Murtazina, “On the characteristic Lie algebras for equations $u_{xy}=f(u,u_x)$”, J. Math. Sci., 151:4 (2008), 3112–3122 | DOI | MR | Zbl

[22] I. T. Habibullin, A. Pekcan, “Characteristic Lie algebra and classification of semidiscrete models”, Theor. Math. Phys., 151:3 (2007), 781–790 | DOI | MR | Zbl

[23] A. V. Zhiber, O. S. Kostrigina, “Characteristic algebras of nonlinear hyperbolic systems of equations”, J. Sib. Fed. Univ. Math. Phys., 3:2 (2010), 173–184 (in Russian) | Zbl

[24] N. A. Zheltukhina, A. U. Sakieva, I. T. Habibullin, “Characteristic Lie algebra and Darboux integrable discrete chains”, Ufimskji Matem. Zhurn., 2:4 (2010), 39–51 (in Russian) | MR | Zbl

[25] A. U. Sakieva, “Characteristic Lie ring of the Zhiber-Shabat-Tzitzeica equation”, Ufimskij Matem. Zhurn., 4:3 (2012), 155–160 (in Russian) | MR | Zbl

[26] M. Gürses, A. V. Zhiber, I. T. Habibullin, “Characteristic Lie rings of differential equations”, Ufimskji Matem. Zhurn., 4:1 (2012), 53–62 (in Russian) | MR

[27] A. V. Zhiber, S. N. Kamaeva, “Construction of exact solution to sine-Gordon equation on the base of its characteristic Lie ring”, Ufa Math. J., 8:3 (2016), 49–57 | DOI | MR | Zbl

[28] I. Habibullin, M. Poptsova, “Classification of a subclass of two-dimensional lattices via characteristic Lie rings”, SIGMA, 13 (2017), 073 | MR | Zbl

[29] M. N. Poptsova, I. T. Habibullin, “Algebraic properties of quasilinear two-dimensional lattices connected with integrability”, Ufa Math. J., 10:3 (2018), 86–105 | DOI | MR | Zbl

[30] I. T. Habibullin, M. N. Kuznetsova, “A classification algorithm for integrable two-dimensional lattices via Lie-Rinehart algebras”, Theor. Math. Phys., 203:1 (2020), 569–581 | DOI | MR | Zbl

[31] A. B. Zhiber, R. D. Murtazina, I. T. Habibullin, A. B. Shabat, Equations of mathematical physics. Nonlinear integrable equations, Yurait, M., 2018 (in Russian)

[32] A. V. Zhiber, A. B. Shabat, “Klein-Gordon equations with a nontrivial group”, Sov. Phys. Dokl., 24:8 (1979), 608–609 | MR

[33] D. V. Millionshchikov, “Characteristic Lie algebras of the sinh-Gordon and Tzitzeica equations”, Russ. Math. Surv., 72:6 (2017), 1174–1176 | DOI | MR | Zbl

[34] D. Millionshchikov, “Lie Algebras of Slow Growth and Klein-Gordon PDE”, Algebras and Representation Theory, 21:5 (2018), 1037–1069 | DOI | MR | Zbl

[35] G. Rinehart, “Differential forms for general commutative algebras”, Trans. Amer. Math. Soc., 108:2 (1963), 195–222 | DOI | MR | Zbl

[36] V. M. Buchstaber, D. V. Leykin, “Polynomial Lie algebras”, Func. Anal. Appl., 36:4 (2002), 267–280 | DOI | MR | Zbl

[37] D. V. Millionshchikov, “Polynomial Lie algebras and growth of their finitely generated Lie subalgebras”, Proc. Steklov Inst. Math., 302 (2018), 298–314 | DOI | MR | Zbl

[38] V. E. Adler, S. Ya. Startsev, “Discrete analogues of the Liouville equation”, Theor. Math. Phys., 121:2 (1999), 1484–1495 | DOI | MR | Zbl

[39] V. G. Kac, “Simple irreducible graded Lie algebras of finite growth”, Math. USSR-Izv., 2:6 (1968), 1271–1311 | DOI | MR

[40] O. Mathieu, “Classification of simple graded Lie algebras of finite growth”, Invent. Math., 108 (1990), 455–519 | DOI | MR

[41] G. R. Krause, T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, AMS, Providence, R.I., 2000 | MR | Zbl

[42] I. M. Gelfand, A. A. Kirillov, “Sur les corps liés aux algèbres enveloppantes des algèbres de Lie”, Inst. Hautes Etudes Sci. Publ. Math., 31 (1966), 5–19 | DOI | MR

[43] J. Riordan, Combinatorial identities, John Wiley and Sons, New York, 1968 | MR | MR | Zbl