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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.13002 |
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| _version_ | 1866910074733068288 |
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| author | Koebisu, Shoot |
| author_facet | Koebisu, Shoot |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13002 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determinant Factorization for Left Multiplication in the Sedenions Koebisu, Shoot Differential Geometry 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. |
| title | Determinant Factorization for Left Multiplication in the Sedenions |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.13002 |