Dressing Symmetries

TL;DR

This paper explores the role of non-Abelian Hamiltonians in Lie-Poisson actions on symplectic manifolds, revealing their connection to dressing transformations and quantum group symmetries in soliton theories.

Contribution

It demonstrates that dressing transformations are generated by non-Abelian Hamiltonians, specifically the monodromy matrix, providing a new proof of their Lie-Poisson structure.

Findings

Dressing transformations are generated by the monodromy matrix.

They serve as classical precursors to quantum group symmetries.

The approach is exemplified in Toda and Heisenberg models.

Abstract

We study Lie-Poisson actions on symplectic manifolds. We show that they are generated by non-Abelian Hamiltonians. We apply this result to the group of dressing transformations in soliton theories; we find that the non-Abelian Hamiltonian is just the monodromy matrix. This provides a new proof of their Lie-Poisson property. We show that the dressing transformations are the classical precursors of the non-local and quantum group symmetries of these theories. We treat in detail the examples of the Toda field theories and the Heisenberg model.