Determinant Factorization for Left Multiplication in the Sedenions

TL;DR

This paper investigates zero-divisors in the 16-dimensional sedenion algebra by factorizing the determinant of left multiplication, revealing geometric and algebraic structures related to classical zero-divisor characterization.

Contribution

It introduces a canonical quartic factorization of the determinant of left multiplication in sedenions, linking algebraic properties to geometric manifolds and simplifying the zero-divisor analysis.

Findings

Determinant admits a canonical quartic factorization

Zero-divisors characterized by imaginary components

Zero-divisor manifold identified with Stiefel manifold V_2(R^7)

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.