# Hitzer, Helmstetter and Ablamowicz: Square Roots of −1 in Real Clifford Algebras

Eckhard Hitzer, Jacques Helmstetter and Rafal Ablamowicz, Square Roots of −1 in Real Clifford Algebras, submitted to E. Hitzer, S. Sangwine (eds.), “Quaternion and Clifford Fourier transforms and wavelets”, Trends in Mathematics, Birkhauser, Basel, 2013.

Abstract It is well 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. Systematic research has been done [S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of −1. Adv. Appl. Clifford Algebras 16(1) (2006), 63–68.] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$. Further research on general algebras $Cl(p,q)$ has explicitly derived the geometric roots of -1 for $p+q \leq 4$ [E. Hitzer, R. Ab lamowicz, Geometric Roots of −1 in Clifford Algebras Cℓ(p, q) with p+q ≤ 4. Adv. Appl. Clifford Algebras, Online First, 13 July 2010. DOI: 10.1007/s00006-010-0240-x.]. 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 -1 found in the different types of Clifford algebras, depending on the type of associated ring ($\mathbb{R}$, $\mathbb{H}$, $\mathbb{R}^2$, $\mathbb{H}^2$, or $\mathbb{C}$). At the end of the paper 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 $\mathbb{C}$. 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.