P∞ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators

The symmetry algebra P-infinity = W-infinity + H + I-infinity of integrable systems is defined. As an example the classical Sophus Lie point symmetries of all higher KP equations are obtained. ft is shown that one ("positive") half of the point symmetries belongs to the W-infinity symmetries while the other ("negative") part belongs to the I-infinity ones. The corresponding action on the tau-function is obtained. A new embedding of the Virasoro algebra into gl(infinity) describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.