**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.

**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. 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.

**Download **(296 kB, PDF):

http://math.tntech.edu/techreports/TR_2009_3.pdf

arXiv:0905.3019v1 [math.RA]

*Source: *Eckhard Hitzer (15 May 2009)

### Like this:

Like Loading...

*Related*