Integrable triples in semisimple Lie algebras

TL;DR

This paper classifies all integrable triples in simple Lie algebras, linking each to a hierarchy of bi-Hamiltonian PDEs, including well-known examples like KdV and Drinfeld-Sokolov hierarchies.

Contribution

It provides a complete classification of integrable triples in simple Lie algebras, establishing their correspondence with integrable hierarchies of PDEs.

Findings

Classification of all integrable triples in simple Lie algebras

Connection of triples to known integrable hierarchies

Explicit examples including KdV and Drinfeld-Sokolov hierarchies

Abstract

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple $(f,0,e)$ in $\mathfrak{sl}_2$ corresponds to the KdV hierarchy, and the triple $(f,0,e_\theta)$, where $f$ is the sum of negative simple root vectors and $e_\theta$ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld-Sokolov hierarchy.