Hamiltonian dynamics on matched pairs

TL;DR

This paper explores the structure of Hamiltonian dynamics on matched pair Lie groups, deriving symplectic and Poisson structures, and applying these to Lie-Poisson equations on specific Lie algebras.

Contribution

It demonstrates that the cotangent bundle of a matched pair Lie group forms a matched pair Lie group and explicitly derives symplectic, Poisson, and Lie-Poisson structures.

Findings

Cotangent bundle of a matched pair Lie group is itself a matched pair Lie group.

Explicit formulas for symplectic form and Poisson bracket on the trivialized space.

Derivation of Lie-Poisson equations on rsl(2,\,C).

Abstract

It is shown that the cotangent bundle of a matched pair Lie group is itself a matched pair Lie group. The trivialization of the cotangent bundle of a matched pair Lie group are presented. On the trivialized space, the canonical symplectic two-form and canonical Poisson bracket are explicitly written. Various symplectic and Poisson reductions are perfomed. The Lie-Poisson bracket is derived. As an example, Lie-Poisson equations on $\mathfrak{sl}(2,\mathbb{C})^\ast$ are obtained.