Clifford代数$\G_{p,q}$与$p+q \leq 4$的-1几何根
Geometric Roots of -1 in Clifford Algebras $\G_{p,q}$ with $p+q \leq 4$
摘要：
众所周知，克利福德（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。研究已经完成[S。张文，张文，等。16（1），pp. 63-68，2006年。在-1的双四元数根上，放弃对叶片的限制。双四元数同构于$\R^3$的Clifford（几何）代数$Cl_{3}$。所有这些根的-1找到直接应用在建设新类型的几何克利福德傅立叶变换。现在我们将这一研究推广到一般代数$Cl_{p,q}$。利用$p+q \leq 4$完全导出了Clifford（几何）代数的-1的几何根.
英文摘要：
It is 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. Research has been done [S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of -1, Adv. Appl. Cliford Alg. 16(1), pp. 63-68, 2006.] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl_{3}$ of $\R^3$. All these roots of -1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations. We now extend this research to general algebras $Cl_{p,q}$. We fully derive the geometric roots of -1 for the Clifford (geometric) algebras with $p+q \leq 4$.
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