Integrability of classical affine W-algebras

TL;DR

This paper proves that most classical affine W-algebras associated with simple Lie algebras admit integrable bi-Hamiltonian PDE hierarchies, with a few specific exceptions in certain exceptional Lie groups.

Contribution

It establishes the integrability of classical affine W-algebras for all simple Lie algebras and nilpotent elements, identifying specific exceptions.

Findings

Most classical affine W-algebras admit integrable hierarchies.

Exceptions are identified in specific nilpotent classes of G_2, F_4, and E_8.

The result advances understanding of the structure and integrability of W-algebras.

Abstract

We prove that all classical affine W-algebras W(g,f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G_2, one in F_4, and five in E_8.