Square Roots of –1 in Real Clifford Algebras

Abstract

It is well known that Clifford (geometric) algebra offers a geometric interpretation for square roots of –1in 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. Systematic research has been done [33] on the biquaternion roots of –1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cl 3,0 of ℝ3. Further research on general algebras Cl p,q has explicitly derived the geometric roots of –1for p + q≤4 [20]. The current research abandons this dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in order to algebraically characterize the continuous manifolds of square roots of –1f ound in the different types of Clifford algebras, depending on the type of associated ring (ℝ,ℍ,ℝ2,ℍ2, or ℂ). At the end of the chapter explicit computer generated tables of representative square roots of –1 are given for all Clifford algebras with n = 5,7, and s = 3 (mod 4) with the associated ring ℂ. This includes, e.g., Cl 0,5 important in Clifford analysis, and Cl 4,1 which in applications is at the foundation of conformal geometric algebra. All these roots of –1 are immediately useful in the construction of new types of geometric Clifford–Fourier transformations.