Geometric Roots of −1 in Clifford Algebras, Cl(p,q) with p + q <= 4
by Eckhard Hitzer, Department of Applied Physics, University of Fukui, Japan,
Rafal Ablamowicz, Department of Mathematics, Tennessee Technological University (TTU), USA
TTU, Dep. of Math., Tech. Rep. No. 2009-3.
Submitted to Advances in Applied Clifford Algebras, May 2009.
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. Clifford 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 <= 4.
Mathematics Subject Classification (2000). Primary 15A66; Secondary 11E88, 42A38, 30G35.
Keywords. Roots of −1, Clifford (geometric) algebra, Fourier transformation, pseudo scalar.
Source: Eckhard Hitzer (15 May 2009)