The Algebra of Differential Operators on the Circle and $W_{KP}^{(q)}$

TL;DR

This paper extends a Lie algebra homomorphism to a deformation of the $W_{KP}$ algebra, providing insights into the symmetry structure of the KP hierarchy and its deformations.

Contribution

It introduces an extension of Radul's map to the deformed algebra $W_{KP}^{(q)}$, linking differential operators on the circle to this new algebra.

Findings

$W_ ext{infinity}$ is the symmetry algebra of KP's additional symmetries

Extended the algebraic framework to include $W_{KP}^{(q)}$

Provided a short proof of the symmetry structure of KP

Abstract

Radul has recently introduced a map from the Lie algebra of differential operators on the circle to $W_n$. In this note we extend this map to $W_{KP}^{(q)}$, a recently introduced one-parameter deformation of $W_{KP}$---the second hamiltonian structure of the KP hierarchy. We use this to give a short proof that $W_\infty$ is the symmetry algebra of additional symmetries of the KP equation.