Coadjoint Orbits of the Generalised Sl(2) Sl(3) Kdv Hierarchies

TL;DR

This paper develops coadjoint orbit constructions for the phase spaces of generalized Sl(2) and Sl(3) KdV hierarchies, involving group actions, Hamiltonian reduction, and gauge fixing procedures.

Contribution

It introduces new coadjoint orbit frameworks for these hierarchies using Yang-Baxter operators and Hamiltonian reduction methods.

Findings

Constructed two group actions via Yang-Baxter operators.

Reproduced Poisson brackets using Kirillov's method.

Provided a natural gauge fixing procedure for the hierarchies.

Abstract

In this paper we develop two coadjoint orbit constructions for the phase spaces of the generalised $Sl(2)$ and $Sl(3)$ KdV hierachies. This involves the construction of two group actions in terms of Yang Baxter operators, and an Hamiltonian reduction of the coadjoint orbits. The Poisson brackets are reproduced by the Kirillov construction. From this construction we obtain a `natural' gauge fixing proceedure for the generalised hierarchies.