实Clifford代数中-1的平方根
Square Roots of -1 in Real Clifford Algebras
摘要：
众所周知，Clifford（几何）代数以平方为负1的叶片的形式提供了对-1的平方根的几何解释。这扩展到四元数的几何解释为单位立方体的侧面双向量。系统的研究已经在-1的双四元数根上完成[32]，放弃了对叶片的限制。双四元数同构于$\mathbb{R}^3$的Clifford（几何）代数$Cl(3,0)$。对一般代数$Cl(p,q)$的进一步研究，明确地导出了$p+q \leq 4$的-1的几何根[17]。目前的研究放弃了这种维数限制，并使用Clifford代数矩阵代数同构，以代数特征的连续流形的平方根的-1中发现的不同类型的Clifford代数，根据类型的相关环（$\mathbb{R}$，$\mathbb{H}$，$\mathbb{R}^2$，$\mathbb{H}^2$，或$\mathbb{C}$）。文末给出了所有具有$n=5,7$和$s=3 \, (mod 4)$及其环$\mathbb{C}$的Clifford代数的-1的代表平方根的计算机生成表.这包括，例如，$Cl(0,5)$在Clifford分析中起着重要的作用，而$Cl(4,1)$在应用上是共形几何代数的基础。所有这些-1的根在构造新型的几何克利福德傅立叶变换时都是非常有用的。
英文摘要：
It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Systematic research has been done [32] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$. Further research on general algebras $Cl(p,q)$ has explicitly derived the geometric roots of -1 for $p+q \leq 4$ [17]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of -1 found in the different types of Clifford algebras, depending on the type of associated ring ($\mathbb{R}$, $\mathbb{H}$, $\mathbb{R}^2$, $\mathbb{H}^2$, or $\mathbb{C}$). At the end of the paper explicit computer generated tables of representative square roots of -1 are given for all Clifford algebras with $n=5,7$, and $s=3 \, (mod 4)$ with the associated ring $\mathbb{C}$. This includes, e.g., $Cl(0,5)$ important in Clifford analysis, and $Cl(4,1)$ which in applications is at the foundation of conformal geometric algebra. All these roots of -1 are immediately useful in the construction of new types of geometric Clifford Fourier transformations.
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