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

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 $W_{1+\infty }(g{l}_s)$, 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.

20 pages of amstex, no figures

References in corpus (3)

• Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
• On additional symmetries of the KP hierarchy and Sato's Bäcklund transformation
• KdV type hierarchies, the string equation and $W_{1+\infty }$ constraints

Cited by in corpus (5)

• The $n$-component KP hierarchy and representation theory
• The geometry of spinors and the multicomponent BKP and DKP hierarchies
• W-Symmetry of the Adelic Grassmannian
• Etingof Trace, Path Hypergeometric Functions and Integrable Systems
• Representations of Centrally-Extended Lie Algebras over Differential Operators and Vertex Algebras