Geometric Roots of –1 in Clifford Algebras Cℓ p,q with p + q ≤ 4

Abstract

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 [1] on the biquaternion roots of −1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra Cℓ3 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 Cℓp,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.

A General Geometric Fourier Transform Convolution Theorem

“…The Fourier transform (FT) is a very important tool for mathematics, physics, computer science and engineering. Since geometric algebras [1] usually contain continuous submanifolds of geometric square roots of minus one [2,3] there are infinitely many ways to construct new geometric Fourier transforms by replacing the imaginary unit in the classical definition of the FT. Every multivector comes with a natural geometric interpretation so the generalization is very useful. It helps to interpret the transform and apply it in a target oriented way to the specific underlying problem.…”

Real clifford windowed Fourier transform

“…(1) For a fixed b, we have (2) The energy density is defined as the modulus square given by 18) which shows that the CWFT is a bounded linear operator on L 2 (R n ; G n ).…”

Two-Sided Clifford Fourier Transform with Two Square Roots of −1 in Cl(p, q)

“…It is well-known that there are elements other than blades, squaring to −1. Motivated by their special relevance for new types of CFTs, they have recently been studied thoroughly [19,21,27]. arXiv:1306.2092v1 [math.RA] 10 Jun 2013…”