Group-algebraic characterization of spin particles: semi-simplicty, SO(2N) structure and Iwasawa decomposition

TL;DR

This paper characterizes the Lie algebra and group structures of quantum spin particles, demonstrating semi-simplicity, SO(2N) structure, and Iwasawa decomposition, with applications to angular momentum coupling.

Contribution

It introduces a method to construct Lie group structures for spin particles and extends semi-simplicity and Iwasawa decomposition results to both algebraic and group levels.

Findings

Proves semi-simplicity of quantum spin particle Lie algebras

Establishes SO(2N) structure for these Lie algebras

Performs Iwasawa decomposition at algebra and group levels

Abstract

In this paper, we focus on the characterization of Lie algebras of fermionic, bosonic and parastatistic operators of spin particles. We provide a method to construct a Lie group structure for the quantum spin particles. We show the semi-simplicity of a quantum spin particle Lie algebra, and extend the results to the Lie group level. Besides, we perform the Iwasawa decomposition of spin particles at both the Lie algebra and Lie group levels. Finally, we investigate the coupling of angular momenta of spin half particles, and give a general construction for such a study.