$P_{\infty }$ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators

Abstract

The symmetry algebra $P_{\infty }=W_{\infty }\oplus H\oplus I_{\infty }$ of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one (``positive'') half of the point symmetries belongs to the $W_{\infty }$ symmetries while the other (``negative'') part belongs to the $I_{\infty }$ ones. The corresponing action on the tau-function is obtained for the positive part of the symmetries. The negative part can not be obtained from the free fermion algebra. A new embedding of the Virasoro algebra into $gl(\infty )$n describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.

21 pages, Latex, no figures (some references added and misprints are corrected)