类四元数的矩阵表示

Matrix Representations of Quaternion-Like Algebras

摘要: 类四元数在数学与物理中有着重要的应用。类四元数的非交换性和零因子的存在性使得对这些代数系统的研究非常困难。本文将类四元数看作环，基于类四元数的运算法则，建立了其与实数，复数或双曲复数上的矩阵环的同构关系，并由此得到其在相应数域上的矩阵表示。本文的方法可推广到其它的代数系统，将为这些代数系统提供重要的研究工具。

Abstract: Quaternion-like algbras have significant applications in mathematics and physics. The non-commutativity and existence of zero divisors in quaternion-like algebra make their study particularly challenging. This paper treats quaternion-like numbers as rings and establishes an isomorphism between these rings and matrix rings over real num- bers, complex numbers, or hypercomplex numbers based on the operational rules of quaternion-like numbers. Consequently, matrix representations of quaternion-like numbers in corresponding number fields are obtained. The methods used in this paper can be extended to other algebraic systems, providing important tools for their study.