Poisson-Lie group of pseudodifferential symbols and fractional KP-KdV hierarchies

Abstract

The Lie algebra of pseudodifferential symbols on the circle has a nontrivial central extension (by the ``logarithmic'' 2-cocycle) generalizing the Virasoro algebra. The corresponding extended subalgebra of integral operators generates the Lie group of classical symbols of all real (or complex) degrees. It turns out that this group has a natural Poisson-Lie structure whose restriction to differential operators of an arbitrary integer order coincides with the second Adler-Gelfand-Dickey structure. Moreover, for any real (or complex) αthere exists a hierarchy of completely integrable equations on the degree αpseudodifferential symbols, and this hierarchy for α=1 coincides with the KP one, and for an integer α=n>1$ and purely differential symbol gives the n-KdV-hierarchy.