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

Francis Atta Howard, Kinvi Kangni

TL;DR

This work proves an isomorphism between $Spin(\tfrac{1}{2})$ and the $SU(1,1)$-quasi-boson, and develops the corresponding universal enveloping and Hopf-algebraic structure. It derives a concrete Cartan/Iwasawa-type decomposition and explicit Haar measure for the quasi-boson, linking the spin description to a harmonic-analytic setting. Central to the results is that the spherical Fourier transform of type Delta becomes a Kangni-type transform when $\hbar=1$, with Abel-transform machinery enabling spin-particle applications. Overall, the paper provides a unified algebraic and harmonic-analytic framework for spin-1/2 systems, connecting Clifford-algebraic spin structures to $SU(1,1)$-quasi-boson representations and quantum group formalisms with potential implications for representation theory and quantum symmetries.

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, $\hbar=1$.

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

TL;DR

This work proves an isomorphism between and the -quasi-boson, and develops the corresponding universal enveloping and Hopf-algebraic structure. It derives a concrete Cartan/Iwasawa-type decomposition and explicit Haar measure for the quasi-boson, linking the spin description to a harmonic-analytic setting. Central to the results is that the spherical Fourier transform of type Delta becomes a Kangni-type transform when , with Abel-transform machinery enabling spin-particle applications. Overall, the paper provides a unified algebraic and harmonic-analytic framework for spin-1/2 systems, connecting Clifford-algebraic spin structures to -quasi-boson representations and quantum group formalisms with potential implications for representation theory and quantum symmetries.

Abstract

In this study we investigate the nexus between the and the -quasi boson Lie structure and reveal related properties as well as some decomposition of spin particles. We show that the -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, .

Paper Structure

This paper contains 8 sections, 12 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier Transform of Type Delta
  4. Kangni-Type Transform [KTT]
  5. Universal Enveloping Algebra $\mathcal{U}\left({spin}\left(\frac{1}{2}\right)\right)$
  6. Isomorphism from $\mathbf{Spin} (\frac{1}{2})$ to $\mathrm{SU}(1,1)$-quasi boson
  7. Application of Kangni Spherical Fourier Transform of the Type Delta to Spin particles
  8. Concluding Remarks

Key Result

Theorem 2.1

kangni2001transformation Let $E$ be a vector space with finite dimension on $\mathbb{C}$, $\phi$ a quasi-bounded function and $K$ central class of $C^{\infty}$ function. Assume there exist an irreducible representation $u_{\phi}$ of $C$ in $E$ such that: for all $D\in \mathcal{C}$. Thus, there exist $\delta\in \hat{K}$ such that $\phi$ is spherical of type delta.

Theorems & Definitions (27)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 4.1
  • ...and 17 more