Higher-spin and W ∞( J) algebras in Virasoro-constrained KP and N-KdV hierarchies

Abstract

Virasoro constraints on the KP hierarchy, arising in matrix models, are studied by reexpressing them in terms of dressing operators of the hierarchy. There exists a one-parameter family of Virasoro representations on the KP hierarchy (depending on a number J which can be identified as the conformal weight of an abstract bc system). The respective full invariance algebra is the “Borel” subalgebra of W ∞( J), which we describe as an extension of the “wedge”, or higher spin, algebra B λ = J−J 2 by the L 2 Virasoro generator. Reductions of these structures of the N-KdV hierarchies are performed explicitly.