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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