KdV type hierarchies, the string equation and $W_{1+\infty }$ constraints

Abstract

To every partition $n=n_1+n_2+\cdots +n_s$ one can associate a vertex operator realization of the Lie algebras $a_{\infty }$ and ${\widehat{gl}}_n$. Using this construction we make reductions of the $s$--component KP hierarchy, reductions which are related to these partitions. In this way we obtain matrix KdV type equations. Now assuming that (1) $\tau$ is a $\tau$--function of the $[n_1,n_2,\dots ,n_s]$--th reduced KP hierarchy and (2) $\tau$ satisfies a `natural' string equation, we prove that $\tau$ also satisfies the vacuum constraints of the $W_{1+\infty }$ algebra.

29 pages of amstex, Correction of some errors and 1 reference added

References in corpus (4)

• Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
• The $n$-component KP hierarchy and representation theory
• Additional symmetries of KP, Grassmannian, and the string equation
• Additional symmetries of KP, Grassmannian, and the string equation II

Cited by in corpus (4)

• Virasoro Symmetry of Constrained KP Hierarchies
• The $W_{1+\infty }(g{l}_s)$--symmetries of the $S$--component KP hierarchy
• Algebraic Solutions of the Multicomponent KP Hierarchy
• Virasoro symmetries of Multi-Component Gelfand-Dickey systems