Lie algebras of slow growth and Klein-Gordon equation

TL;DR

This paper explores the structure of characteristic Lie algebras associated with hyperbolic PDEs, establishing isomorphisms with affine Kac-Moody algebras and analyzing their growth properties related to integrability.

Contribution

It introduces explicit isomorphisms between characteristic Lie algebras of sinh-Gordon and Tzitzeica equations and certain affine Kac-Moody algebras, revealing their slow growth.

Findings

Characteristic Lie algebras are isomorphic to pro-solvable subalgebras of affine Kac-Moody algebras.

Both Lie algebras exhibit slow linear growth with specific rates.

The integrability of hyperbolic PDEs is linked to properties of their characteristic Lie algebras.

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 $\chi$. We establish two explicit isomorphisms between characteristic Lie algebras of sinh-Gordon and Tzitzeica equations and pro-solvable Lie subalgebras of affine Kac-Moody algebras $A_1^{(1)}$ and $A_2^{(2)}$ respectively. Hence both characteristic Lie algebras are slowly linearly growing Lie algebras with average growth rates $\frac{3}{2}$ and $\frac{4}{3}$ respectively.