Poisson-Lie structures on the four-dimensional oscillator lie groups

Abstract

This paper studies local normal forms of Poisson–Lie structures on four-dimensional Lie groups, which play a fundamental role in mechanics and physics. A central problem is to determine local coordinates under diffeomorphisms that yield the simplest possible expressions of these structures. We compute all Poisson–Lie structures via their correspondence with Lie bialgebra structures, using the fact that any connected Poisson–Lie group arises from a Lie bialgebra. We then prove their local linearizability near the identity by constructing explicit normal forms, obtained through solving an associated system of differential equations.