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

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 corresponding 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 $\operatorname{gl}(\infty)$ describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.