E.S.M. Hitzer, R. Ablamowicz
Geometric Roots of -1 in Clifford Algebras Cl(p,q) with p+q <= 4
Adv. In Appl. Cliff. Algebras, Vol. 21(1) pp. 121-144, (2011), DOI 10.1007/s00006-010-0240-x. Preprint version: Technical Report 2009-3, Department of Mathematics, Tennessee Technological University, Tennessee, USA, May 2009. Also available as: arXiv:0905.3019v1 [math.RA]

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 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 C(3,0) 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 C(p,q). We fully derive the geometric roots of -1 for the Clifford (geometric) algebras with p+q <= 4.

Mathematics Subject Classification (2000). Primary 15A66; Secondary 11E88, 42A38, 30G35.

Keywords. Keywords. Roots of -1, Clifford (geometric) algebra, Fourier transformation, pseudo scalar.

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