Algebra of pseudodifferential operators and symmetries of equations in the Kadomtsev–Petviashvili hierarchy

Abstract

Point symmetries are obtained for all equations in the KP hierarchy. The Lie algebra for each equation is infinite dimensional and involves several arbitrary functions of the corresponding time tN. The symmetry algebra is a semidirect sum of a Virasoro algebra and a Kac–Moody one. The “positive” part of this algebra is embedded into the known W∞ algebra of KP symmetries and into the free fermion algebra ĝl(∞). The corresponding action on the tau-function is presented. The negative part of the point symmetries does not fit into the free fermion algebra, but is embedded into a P∞ algebra, based on the algebra of pseudodifferential operators.