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arxiv: 0905.3019 · v1 · submitted 2009-05-19 · 🧮 math.RA · math.CV

Geometric Roots of -1 in Clifford Algebras G_(p,q) with p+q leq 4

classification 🧮 math.RA math.CV
keywords geometricrootscliffordalgebrasalgebrabiquaternionbladesinterpretation
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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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