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Determinant Factorization for Left Multiplication in the Sedenions

Shoot Koebisu

Abstract

We study zero-divisors in the $16$-dimensional sedenion algebra from the viewpoint of the determinant of left multiplication. We show that this determinant admits a canonical factorization into the square of a quartic polynomial, obtained via a $G_2$-invariant reduction to a quaternionic normal form and an explicit block computation. The quartic factor recovers the classical characterization of left zero-divisors in terms of the imaginary components. After normalization, the resulting zero-divisor manifold is identified with the Stiefel manifold $V_2(\mathbb{R}^7)$. We also analyze a $3$-dimensional purely imaginary slice, on which the quartic reduces to a simple quadratic form. This yields a concrete geometric model of the zero-divisor locus as a quadratic cone in the slice.

Determinant Factorization for Left Multiplication in the Sedenions

Abstract

We study zero-divisors in the -dimensional sedenion algebra from the viewpoint of the determinant of left multiplication. We show that this determinant admits a canonical factorization into the square of a quartic polynomial, obtained via a -invariant reduction to a quaternionic normal form and an explicit block computation. The quartic factor recovers the classical characterization of left zero-divisors in terms of the imaginary components. After normalization, the resulting zero-divisor manifold is identified with the Stiefel manifold . We also analyze a -dimensional purely imaginary slice, on which the quartic reduces to a simple quadratic form. This yields a concrete geometric model of the zero-divisor locus as a quadratic cone in the slice.

Paper Structure

This paper contains 13 sections, 11 theorems, 121 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cayley--Dickson construction and the sedenions
  4. Left multiplication operators and the determinant
  5. Determinant Factorization
  6. Definition of the fundamental factors D1 and D2
  7. Block representation of the left multiplication matrix
  8. Main theorem on determinant factorization
  9. Component expressions for D1 and D2
  10. The Normalized Left Zero--Divisor Manifold
  11. A Purely Imaginary Cyclic Slice
  12. Conclusion
  13. The complex determinant in Theorem \ref{['thm:factor']}

Key Result

Lemma 3.2

For $v=v_{1}+v_{2}e_{8}$ we have

Figures (2)

  • Figure 1: Schematic view of the $G_{2}$ reduction used in Lemma \ref{['lem:g2normal']}. An arbitrary pair $(u,w)\in \operatorname{Im}(\mathbb{O})^{2}$ is carried to the normal form $(r e_{1},\, p e_{1}+q e_{2})$, preserving the norm and inner product.
  • Figure 2: Visualizations of the quartic factor on the purely imaginary cyclic slice.

Theorems & Definitions (27)

  • Definition 2.1: Determinant of the left multiplication matrix
  • Definition 3.1: Fundamental factors $D_{1},D_{2}$
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Block matrix representation
  • proof
  • Lemma 3.5: Determinants of left and right multiplication on the octonions
  • proof
  • ...and 17 more