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Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

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Abstract

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1 in the form of blades that square to –1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [1] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ 3 of \({{\mathbb R^3}}\) . 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 ≤ 4.

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Authors and Affiliations

  1. Department of Applied Physics, University of Fukui, Bunkyo 3-9-1, 910-8507, Fukui, Japan

    Eckhard Hitzer

  2. Department of Mathematics, Tennessee Technological University, Box 5054, Cookeville, TN, 38505, USA

    Rafał Abłamowicz

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  1. Eckhard Hitzer
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  2. Rafał Abłamowicz
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Correspondence to Eckhard Hitzer.

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Hitzer, E., Abłamowicz, R. Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4. Adv. Appl. Clifford Algebras 21, 121–144 (2011). https://doi.org/10.1007/s00006-010-0240-x

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