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.

Download: http://arxiv.org/abs/1204.4576, http://www.tntech.edu/files/math/reports/TR_2012_3.pdf

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.

Source: http://arxiv.org/abs/1204.4576, http://www.tntech.edu/files/math/reports/TR_2012_3.pdf

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