Poisson-Lie group of pseudodifferential symbols

TL;DR

This paper constructs a Poisson-Lie group structure on pseudodifferential symbols, unifying various integrable systems' Poisson structures and providing a framework for Hamiltonian functions and geometric interpretations.

Contribution

It introduces a new Poisson-Lie structure on the dual group of pseudodifferential symbols, linking multiple integrable systems and their Hamiltonian structures within a universal geometric framework.

Findings

Realizes classical Poisson structures as restrictions of the new Poisson-Lie structure.

Provides a geometric interpretation of W_infinity as a limit of Poisson algebras.

Defines an infinite set of involutive functions serving as Hamiltonians.

Abstract

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes to 0.