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 the Lie algebra for a quantum spin particle, and extend the results to the Lie group level. Besides, we perform the Iwasawa decomposition for spin particles at both the Lie algebra and the Lie group levels. Then, we give a general decomposition for spin particles. Finally, we investigate the coupling of angular momenta of spin half particles, and give a general construction for such a study.

中文翻译：

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自旋粒子的群代数表征：半简单性、$$\mathbf {\mathbf {SO}(2N)}$$ SO ( 2 N ) 结构和 Iwasawa 分解

在本文中，我们专注于自旋粒子的费米子、玻色子和寄生算子的李代数的表征。我们提供了一种构建量子自旋粒子李群结构的方法。我们展示了量子自旋粒子的李代数的半简单性，并将结果扩展到李群水平。此外，我们在李代数和李群水平上对自旋粒子进行岩泽分解。然后，我们给出了自旋粒子的一般分解。最后，我们研究了自旋半粒子角动量的耦合，并给出了这种研究的一般结构。