Isomorphisms of $\Spin\left( \frac{1}{2}\right) $ to $\SU(1,1)-\mbox{Boson}$: Universal Enveloping and Kangni-type Transformation

TL;DR

This paper explores the mathematical relationship between spin-1/2 groups and SU(1,1)-quasi boson structures, revealing their properties, decompositions, and Fourier transforms, with implications for quantum algebra and harmonic analysis.

Contribution

It establishes an isomorphism between Spin(1/2) and SU(1,1)-quasi bosons, detailing their Haar measure, spherical Fourier transform, and Kangni-type transformation at Planck's constant.

Findings

SU(1,1)-quasi boson has a left invariant Haar measure

The spherical Fourier transform of the structure is explicitly characterized

The Fourier transform reduces to a Kangni-type transform when =1

Abstract

In this study we investigate the nexus between the $\Spin (\frac12)$ and the $\SU(1,1)$-quasi boson Lie structure and reveal related properties as well as some decomposition of spin particles. We show that the $\SU(1,1)$-quasi boson has a left invariant Haar measure and we ascertain its spherical Fourier transformation. We finally show that this spherical Fourier transformation of type delta is a Kangni-type transform when the Planck's constant, $\hbar=1$.