P∞ algebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operatorsalgebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operators

The symmetry algebraP∞=W∞ ⊕P ⊕I∞ of integrable systems is defined. As an example, the classical Lie point symmetries of all higher Kadomtsev-Petviashvili equations are obtained. It is shown that of the point symmetries, the (positive) ones belong to theW∞ symmetries, while the other (negative) ones belong toI∞ symmetries. The corresponding action on the τ-function is obtained for the positive symmetries. The negative symmetries cannot be obtained from the free fermion algebra. A new embedding of the Virasoro algebra intogl(∞) describes the conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.