The $W_{1+\infty}(gl_s)$--symmetries of the $S$--component KP hierarchy

TL;DR

This paper extends the understanding of symmetries in integrable hierarchies by generalizing the $W_{1+ abla}(gl_s)$ algebra actions from the KP and Toda hierarchies to the s-component KP hierarchy, revealing a richer algebraic structure.

Contribution

It introduces the $W_{1+ abla}(gl_s)$ algebra as the symmetry algebra for the s-component KP hierarchy, generalizing previous results for KP and Toda hierarchies.

Findings

The vertex operators generate the $W_{1+ abla}(gl_s)$ algebra.

The results reveal a richer symmetry structure in the 2-component KP hierarchy.

The algebraic framework extends the known symmetries of integrable hierarchies.

Abstract

Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau--functions to an action of a vertex operator of pseudo-differential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchyto the $W_{1+\infty}$--, respectively $W_{1+\infty}\times W_{1+\infty}$--algebra symmeties. In this paper we generalize the results to the $s$--component KP hierarchy. The vertex operators generate the algebra $W_{1+\infty}(gl_s)$, the matrix version of $W_{1+\infty}$. Since the Toda lattice hierarchy is equivalent to the $2$--component KP hierarchy, the results of this paper uncover in that particular case a much richer structure than the one obtained by Adler, Shiota and van Moerbeke.