SU(2) LiePoisson algebra and its descendants
Abstract
In this paper, a novel discrete algebra is presented which follows by combining the SU(2) LiePoisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R^{3}. The starting point of our derivation is the discrete Frenet frame assigned at each vertex of the string. Then the link vector that connects the neighboring vertices is assigned the SU(2) LiePoisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighboring frames along the string. The procedure extends in a selfsimilar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) LiePoisson 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 neighboring vertices.
 Publication:

Physical Review D
 Pub Date:
 September 2022
 DOI:
 10.1103/PhysRevD.106.054514
 arXiv:
 arXiv:2205.01424
 Bibcode:
 2022PhRvD.106e4514D
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 Extented abstract and extra references, To appear in Physical Review D