源自3D几何代数的所有Hurwitz代数
All Hurwitz Algebras from 3D Geometric Algebras
AI总结：
本文受相关研究启发，提出通过研究偶子代数、伪标量生成代数及引入两种乘积的方式，将全部7种Hurwitz代数嵌入3D几何代数，并明确了各类共轭运算的对应关系，丰富了几何代数的应用场景。
摘要：
Hurwitz代数是有单位元的合成代数，因其广泛的应用价值在代数学和数学物理领域广为人知。受Lesenby与Hitzer研究工作的启发，本文阐明了如何将全部7种Hurwitz代数（可除与分裂型）嵌入到3D几何代数，即满足p+q=3的$\u2006\mathcal{G}\left(p,q\right)$中。该嵌入通过研究偶子代数（恒为四元数或分裂四元数型）以及由伪标量生成的代数（恒为复或分裂复型）实现。反转、逆运算与Clifford共轭分别对应双四元数共轭、复共轭与四元数共轭。八元数代数即八元数与分裂八元数，可通过引入两种不同乘积、复刻Cayley-Dickson过程的一种变体以两种方式得到。八元数共轭可在几何代数体系中通过全阶逆定义的对合实现。
英文摘要：
Hurwitz algebras are unital composition algebras widely known in algebra and mathematical physics for their useful applications. In this paper, inspired by works of Lesenby and Hitzer, we show how to embed all seven Hurwitz algebras (division and split) in 3D geometric algebras, i.e. $\mathcal{G}\left(p,q\right)$ with $p+q=3$. This is achieved studying the even subalgebra, which is always of quaternionic or split-quaternionic type, and the algebra generated by the pseudoscalar which is always of complex or split-complex type. Reversion, inversion and Clifford conjugation correspond to biquaternionic, complex and quaternionic conjugation respectively. Octonionic algebras, i.e., octonions and split-octonions, are obtained in two ways introducing two different products, replicating a variation of the Cayley-Dickson process. Octonionic conjugation is achieved in the geometric algebra formalism through the involution defined by the full grade inversion.
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