P∞ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators
Abstract
The symmetry algebra P∞ = W∞ H I∞ 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∞ symmetries while the other (``negative'') part belongs to the I∞ 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(∞ )n describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.
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