W-Algebras from Soliton Equations and Heisenberg Subalgebras

TL;DR

This paper establishes conditions under which certain Hamiltonian structures of generalized KdV hierarchies correspond to classical W-algebras derived via Drinfeld-Sokolov reduction, linking integrable systems with Lie algebra embeddings.

Contribution

It identifies specific conditions and classes of W-algebras associated with Heisenberg subalgebras of affine Kac-Moody algebras, expanding the understanding of their algebraic structure.

Findings

Connection between Hamiltonian structures and W-algebras is clarified.

Class of W-algebras related to specific Lie algebra embeddings is characterized.

Results recover known Gel'fand-Dickey algebras for principal cases.

Abstract

We derive sufficient conditions under which the ``second'' Hamiltonian structure of a class of generalized KdV-hierarchies defines one of the classical $\cal W$-algebras obtained through Drinfel'd-Sokolov Hamiltonian reduction. These integrable hierarchies are associated to the Heisenberg subalgebras of an untwisted affine Kac-Moody algebra. When the principal Heisenberg subalgebra is chosen, the well known connection between the Hamiltonian structure of the generalized Drinfel'd-Sokolov hierarchies - the Gel'fand-Dickey algebras - and the $\cal W$-algebras associated to the Casimir invariants of a Lie algebra is recovered. After carefully discussing the relations between the embeddings of $A_1=sl(2,{\Bbb C})$ into a simple Lie algebra $g$ and the elements of the Heisenberg subalgebras of $g^{(1)}$, we identify the class of $\cal W$-algebras that can be defined in this way. For $A_n$, this class only includes those associated to the embeddings labelled by partitions of the form $n+1= k(m) + q(1)$ and $n+1= k(m+1) + k(m) + q(1)$.