A Simple Model of Double Dynamics on Lie Groups

TL;DR

This paper introduces a unified model for double dynamics on Lie groups, specifically SU(2) and its dual SB(2,C), using a parent action on the Drinfeldd double that captures their coupled behavior.

Contribution

It presents a novel parent action framework on the Drinfeldd double that describes coupled dynamics of a Lie group and its dual, recovering individual models via gauging.

Findings

Unified description of SU(2) and SB(2,C) dynamics

Recovery of individual models through gauging

Extension to systems with doubled degrees of freedom

Abstract

We study the dynamics of the rigid rotator on the group manifold of SU(2) as an instance of dynamics on Lie groups together with a dual model whose carrier space is the Borel group SB(2,C), the Lie Poisson dual of SU(2). We thus introduce a parent action on the Drinfeldd double of the above mentioned groups, which describes the dynamics of a system with twice as many degrees of freedom as the two starting partners. Through a gauging procedure of its global symmetries both the rigid rotor and the dual model are recovered.