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

摘要

The symmetry algebra $P_\infty = W_\infty \oplus H \oplus I_\infty$ ofintegrable systems is defined. As an example the classical Sophus Lie pointsymmetries 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 partof the symmetries. The negative part can not be obtained from the free fermionalgebra. A new embedding of the Virasoro algebra into $gl(\infty )$n describesconformal transformations of the KP time variables. A free fermion algebracocycle is described as a PDO Lie algebra cocycle.