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

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.