# Polynomial Lie Algebras and Growth of Their Finitely Generated Lie Subalgebras

Article

First Online:

## Abstract

The concept of polynomial Lie algebra of finite rank was introduced by V. M. Buchstaber in his studies of new relationships between hyperelliptic functions and the theory of integrable systems. In this paper we prove the following theorem: the Lie subalgebra generated by the frame of a polynomial Lie algebra of finite rank has at most polynomial growth. In addition, important examples of polynomial Lie algebras of countable rank are considered in the paper. Such Lie algebras arise in the study of certain hyperbolic partial differential equations, as well as in the construction of self-similar infinite-dimensional Lie algebras (such as the Fibonacci algebra).

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation,” Proc. Steklov Inst. Math.
**294**, 176–200 (2016) [transl. from Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk**294**, 191–215 (2016)].MathSciNetCrossRefzbMATHGoogle Scholar - 2.V. M. Buchstaber, “Polynomial Lie algebras and the Zelmanov–Shalev theorem,” Russ. Math. Surv.
**72**(6), 1168–1170 (2017) [transl. from Usp. Mat. Nauk**72**(6), 199–200 (2017)].MathSciNetCrossRefzbMATHGoogle Scholar - 3.V. M. Buchstaber and D. V. Leykin, “Polynomial Lie algebras,” Funct. Anal. Appl.
**36**, 267–280 (2002) [transl. from Funkts. Anal. Prilozh.**36**(4), 18–34 (2002)].MathSciNetCrossRefGoogle Scholar - 4.V. G. Kac, “Simple irreducible graded Lie algebras of finite growth,” Math. USSR, Izv.
**2**(6), 1271–1311 (1968) [transl. from Izv. Akad. Nauk SSSR, Ser. Mat.**32**(6), 1323–1367 (1968)].CrossRefzbMATHGoogle Scholar - 5.G. R. Krause and T. H. Lenagan,
*Growth of Algebras and Gelfand–Kirillov Dimension*(Am. Math. Soc., Providence, RI, 2000).zbMATHGoogle Scholar - 6.A. N. Leznov, V. G. Smirnov, and A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems,” Theor. Math. Phys.
**51**(1), 322–330 (1982) [transl. from Teor. Mat. Fiz.**51**(1), 10–21 (1982)].CrossRefzbMATHGoogle Scholar - 7.O. Mathieu, “Classification of simple graded Lie algebras of finite growth,” Invent. Math.
**108**, 455–519 (1992).MathSciNetCrossRefzbMATHGoogle Scholar - 8.D. V. Millionschikov, “Graded filiform Lie algebras and symplectic nilmanifolds,” in Geometry, Topology, and Mathematical Physics: S. P. Novikov’s Seminar 2002–2003 (Am. Math. Soc., rovidence, RI, 2004), AMS Transl., Ser. 2, 212, pp. 259–279.Google Scholar
- 9.D. V. Millionshchikov, “Characteristic Lie algebras of the sinh-Gordon and Tzitzeica equations,” Russ. Math. Surv.
**72**(6), 1174–1176 (2017) [transl. from Usp. Mat. Nauk**72**(6), 203–204 (2017)].CrossRefzbMATHGoogle Scholar - 10.D. Millionshchikov, “Lie algebras of slow growth and Klein–Gordon PDE,” Algebr. Represent. Theory
**21**(5), 1037–1069 (2018).MathSciNetCrossRefzbMATHGoogle Scholar - 11.V. M. Petrogradsky, “Examples of self-iterating Lie algebras,” J. Algebra
**302**(2), 881–886 (2006).MathSciNetCrossRefzbMATHGoogle Scholar - 12.G. S. Rinehart, “Differential forms on general commutative algebras,” Trans. Am. Math. Soc.
**108**, 195–222 (1963).MathSciNetCrossRefzbMATHGoogle Scholar - 13.M. K. Smith, “Universal enveloping algebras with subexponential but not polynomially bounded growth,” Proc. Am. Math. Soc.
**60**, 22–24 (1976).MathSciNetCrossRefzbMATHGoogle Scholar - 14.A. V. Zhiber, R. D. Murtazina, I. T. Habibullin, and A. B. Shabat, “Characteristic Lie rings and integrable models in mathematical physics,” Ufa Math. J.
**4**(3), 17–85 (2012) [transl. from Ufim. Mat. Zh.**4**(3), 17–85 (2012)].MathSciNetzbMATHGoogle Scholar

## Copyright information

© Pleiades Publishing, Ltd. 2018