Dynamical Aspects of Lie--Poisson Structures

TL;DR

This paper explores the dynamical properties of Lie--Poisson structures, focusing on specific systems related to quantum groups and their classical counterparts, with applications to Hamiltonian and dissipative dynamics.

Contribution

It develops an understanding of Lie--Poisson structures by analyzing dynamical systems on $SU(2)$ and $SU(1,1)$, including Hamiltonian and dissipative systems that persist through quantization.

Findings

Analyzed equations of motion for Lie-Poisson systems.

Identified dissipative systems preserving the Poisson bracket.

Connected classical structures to quantum group quantization.

Abstract

Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at $SU(2)$ and $SU(1,1)$, as submanifolds of a 4--dimensional phase space with constraints, and deal with two classes of problems. In the first set of examples we consider some hamiltonian systems associated with Lie-Poisson structures and we investigate the equations of the motion. In the second set of examples we consider systems which preserve the chosen bracket, but are dissipative. However in this approach, they survive the quantization procedure.