The W1+∞(gl 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 pseudodifferential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchy to the W1+∞−, respectively, W1+∞×W1+∞− algebra symmetries. In this paper we generalize the results to the s-component KP hierarchy. The vertex operators generate the algebra W1+∞(gls), the matrix version of W1+∞. Since the Toda lattice hierarchy is formally equivalent to the 2-component KP hierarchy, the results of this article uncover in that particular case a much richer structure than the one obtained by Adler, Shiota, and van Moerbeke.