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Square roots of complexified quaternions

Adolfas Dargys, Arturas Acus

TL;DR

The paper addresses square roots of complexified quaternions by exploiting isomorphisms with the real Clifford algebra $Cl_{3,0}$, enabling a transfer of root calculations from Clifford multivectors to complex quaternions. It demonstrates explicit isomorphisms for $\,\boldsymbol{H}^{\mathbb{C}}$ and its variants with $Cl_{3,0}$, and provides concrete root constructions for the Hamilton, coquaternion, conectorine, and nectorine, showing that roots can be discrete, continuous, or absent. An accompanying root algorithm for $Cl_{3,0}$ is described, allowing systematic enumeration of all roots and revealing free-parameter families in some cases. The results facilitate transforming quaternionic equations into Clifford-algebra form and underscore the rich structure of complexified quaternionic roots with potential applications in geometry and physics.

Abstract

Square roots of complexified (complex) quaternions, namely, the Hamilton quaternion, coquaternion, nectorine, and conectorine are investigated. The isomorphisms between the complex quaternions and 3-dimensional multivectors of Clifford algebras is employed for this purpose. Root examples for all named quaternions are presented from which follows that the complex quaternionic roots may assume discrete or continuous form, or there may be no roots at all.

Square roots of complexified quaternions

TL;DR

The paper addresses square roots of complexified quaternions by exploiting isomorphisms with the real Clifford algebra , enabling a transfer of root calculations from Clifford multivectors to complex quaternions. It demonstrates explicit isomorphisms for and its variants with , and provides concrete root constructions for the Hamilton, coquaternion, conectorine, and nectorine, showing that roots can be discrete, continuous, or absent. An accompanying root algorithm for is described, allowing systematic enumeration of all roots and revealing free-parameter families in some cases. The results facilitate transforming quaternionic equations into Clifford-algebra form and underscore the rich structure of complexified quaternionic roots with potential applications in geometry and physics.

Abstract

Square roots of complexified (complex) quaternions, namely, the Hamilton quaternion, coquaternion, nectorine, and conectorine are investigated. The isomorphisms between the complex quaternions and 3-dimensional multivectors of Clifford algebras is employed for this purpose. Root examples for all named quaternions are presented from which follows that the complex quaternionic roots may assume discrete or continuous form, or there may be no roots at all.
Paper Structure (11 sections, 19 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 11 sections, 19 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Relations between quaternions and 2-dimensional Clifford algebras
  3. Complex quaternions and isomorphisms
  4. Complexified Hamilton quaternion
  5. Isomorphisms between $\mathit{Cl}_{3,0}$ and complex quaternions
  6. Examples of square roots of complex quaternions
  7. Square roots of a complexified Hamilton quaternion $\mathbb{H}^{\mathbb{C}}$
  8. Square roots of complex coquaternion, $\mathbb{H}^{\mathbb{C}}_{\text{coq}}$
  9. Square roots of complex conectorine, $\mathbb{H}^{\mathbb{C}}_{\text{con}}$
  10. Square roots of complex nectorine, $\mathbb{H}^{\mathbb{C}}_{\text{neq}}$
  11. Square root algorithm for $\mathit{Cl}_{3,0}$

Figures (1)

  • Figure 1: Algorithm for isolated roots of MV in real $\mathit{Cl}_{3,0}$