Generalized Drinfeld-Sokolov Hierarchies II: The Hamiltonian Structures

TL;DR

This paper investigates the bi-Hamiltonian structures of generalized KdV hierarchies, showing they are Kirillov brackets on Kac-Moody algebras, and constructs related extended conformal algebras including $W_n^l$ algebras.

Contribution

It demonstrates the Hamiltonian structures as Kirillov brackets and constructs new extended conformal algebras from these structures.

Findings

Hamiltonian structures are Kirillov brackets on Kac-Moody algebra.

Classical extended conformal algebras derived from the second Poisson bracket.

Construction of $W_n^l$ algebras including the case $n=3$, $l=2$.

Abstract

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct the $W_n^l$ algebras, first discussed for the case $n=3$ and $l=2$ by A. Polyakov and M. Bershadsky.