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

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

1997-01-13

arxiv.org/abs/solv-int/9701008v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

21 pages, Latex, no figures (some references added and misprints are corrected)

Scientific paper

10.1142/S0217979297001532

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 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 $gl(\infty )$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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