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

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.