Lax representations via twisted extensions of infinite-dimensional Lie algebras: some new results

TL;DR

This paper introduces new 3D integrable PDEs derived from twisted extensions of infinite-dimensional Lie algebras, expanding the understanding of algebraic structures underlying integrable systems.

Contribution

It applies twisted extension techniques to infinite-dimensional Lie algebras to discover new integrable partial differential equations in three dimensions.

Findings

Derived new 3D integrable PDEs from algebraic deformations

Connected algebraic structures with integrable system classifications

Extended the framework of Lie algebra deformations for PDEs

Abstract

We apply the technique of twisted extensions of infinite-dimensional Lie algebras to find new 3D integrable {\sc pde}s related to the deformations of Lie algebra $\mathbb{R}_N[s]\otimes \mathfrak{w}$ with $N=1, 2$ as well as to the Lie algebra $\mathfrak{h} \oplus \mathfrak{w}$, where $\mathbb{R}_N[s]$ is the algebra of truncated polynomials of degree $N$, $\mathfrak{w}$ is the Lie algebra of polynomial vector fields on $\mathbb{R}$ and $\mathfrak{h}$ is the Lie algebra of polynomial Hamiltonian vector fields on $\mathbb{R}^2$.