The SU(2) Lie-Poisson Algebra and its Descendants

TL;DR

This paper introduces a new discrete algebra combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation, modeling a discrete string in R3 and generating an infinite hierarchy of Poisson structures.

Contribution

It presents a novel discrete algebra derived from the SU(2) Lie-Poisson bracket and discrete Frenet equations, including the first descendant structure using spinor representation.

Findings

Defines a new Poisson structure involving four vertices

Extends to an infinite hierarchy of Poisson structures

Employs spinor representation for computational simplicity

Abstract

In this paper, a novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R3. The starting point of our derivation is the discrete Frenet frame assigned at each vertix of the string. Then the link vector that connect the neighbouring vertices assigns the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighbouring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial and novel Poisson structure that engages four neighbouring vertices.