To every partition n = n1 + n2 + + ns one can associate a vertex operator realization of the Lie algebras α∞ and . 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) τ is a τ-function of the [n1, n2, …, ns]th reduced KP hierarchy and (2) τ satisfies a ‘natural’ string equation, we prove that τ also satisfies the vacuum constraints of the W1+∞ algebra.
- KdV hierarchies
van de Leur, J. W., & van de Leur, J. (1995). KdV type hierarchies, The string equation and W1+infinity constraints. Journal of geometry and physics, 17(17), 95-124. https://doi.org/10.1016/0393-0440(94)00039-7