Complex slices, inverses, and deformations of a three-dimensional real algebra
Abstract
We study the real three-dimensional unital algebra with basis {1,i,j} and products i2=j2=−1, ij=0, ji=i+j. Its quadratic and power-associative elements form the union of exactly three complex subalgebras; their nonzero elements are precisely those with a two-sided inverse, which is unique. Every nonzero subalgebra contains the unit, and every element outside these three planes generates the whole algebra. We classify one-sided inverses, including the singular cases, and show that two-sided invertibility does not imply cancellation. The multiplication algebra is M3(ℝ), so the algebra is absolutely central simple. Its derivations vanish, its automorphism group is trivial, and its commutator Lie algebra is ℝ⊕aff(1). Its unique antiautomorphism is involutive, and its complexification admits no complex-linear structurable involution. Finally, replacing ji=i+j by ji=t(i+j) gives real algebras 𝒜t with 𝒜s≅𝒜t exactly when s2=t2. We describe the exceptional parameters and thereby exhibit nontrivial deformations of the original algebra.
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BibTeX
@misc{valdes2026complex,
title = {{Complex slices, inverses, and deformations of a three-dimensional real algebra}},
author = {Alejandro Jesús Bermejo Valdés},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12483},
eprint = {2610.12483},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12483},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complex slices, inverses, and deformations of a three-dimensional real algebra
AU - Alejandro Jesús Bermejo Valdés
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.12483
UR - https://arxiv.org/abs/2610.12483
AB - We study the real three-dimensional unital algebra with basis {1,i,j} and products i^2=j^2=−1, ij=0, ji=i+j. Its quadratic and power-associative elements form the union of exactly three complex subalgebras; their nonzero elements are precisely those with a two-sided inverse, which is unique. Every nonzero subalgebra contains the unit, and every element outside these three planes generates the whole algebra. We classify one-sided inverses, including the singular cases, and show that two-sided invertibility does not imply cancellation. The multiplication algebra is M_3(ℝ), so the algebra is absolutely central simple. Its derivations vanish, its automorphism group is trivial, and its commutator Lie algebra is ℝ⊕aff(1). Its unique antiautomorphism is involutive, and its complexification admits no complex-linear structurable involution. Finally, replacing ji=i+j by ji=t(i+j) gives real algebras 𝒜_t with 𝒜_s≅𝒜_t exactly when s^2=t^2. We describe the exceptional parameters and thereby exhibit nontrivial deformations of the original algebra.
ER -