On the classical $W_N^{(l)}$ algebras

TL;DR

This paper investigates the structure of W_N^{(l)} algebras, providing explicit derivations, supporting the conjecture of their realization as Hamiltonian structures, and exploring their twisted and embedded variants.

Contribution

It offers explicit derivations of W_4^3 algebra, confirms the conjectured realization as Hamiltonian structures, and explores twisted and embedded algebra variants.

Findings

Explicit derivation of W_4^3 algebra

Support for the conjecture of Hamiltonian realization

Identification of twisted and embedded algebra variants

Abstract

We analyze the W_N^l algebras according to their conjectured realization as the second Hamiltonian structure of the integrable hierarchy resulting from the interchange of x and t in the l^{th} flow of the sl(N) KdV hierarchy. The W_4^3 algebra is derived explicitly along these lines, thus providing further support for the conjecture. This algebra is found to be equivalent to that obtained by the method of Hamiltonian reduction. Furthermore, its twisted version reproduces the algebra associated to a certain non-principal embedding of sl(2) into sl(4), or equivalently, the u(2) quasi-superconformal algebra. The general aspects of the W_N^l algebras are also presented.