From Quaternions to Pentarnions I: Algebra, geometry, and a spin model
Abstract
We introduce a family of five-dimensional non-flexible quadratic real algebras {𝒜γ}0≤γ≤1 with canonical basis 1,i,j,k,ℓ, in which γ changes the multiplication but not their norm, and study how this change affects geometry and dynamics. The algebras are pairwise non-isomorphic, 𝒜0 contains Hamilton's quaternion algebra, and 𝒜1 is the Pentarnion algebra. The alternating part of the trilinear form ⟨ u×γv,w⟩ induced by the multiplication of 𝒜γ defines a three-dimensional subspace Hγ⊂spanℝ{i,j,k,ℓ}, tilted by the angle arctan(γ/√3), whose rotation group Gγ≅SO(3) contains the automorphism group of 𝒜γ. For γ1≠γ2 the groups Gγ1 and Gγ2 generate SO(V)≅SO(4). For the six ways of ordering and bracketing the product of i,p,p in 𝒜γ, two map the unit circle in span{j,k} to a point and, for γ>0, four map it to ellipses. As an application, we derive a Hamiltonian model of a classical spin on Hγ whose multiplication-induced quadratic energy is isotropic for γ=0 and anisotropic for γ>0. It conserves spin length and, for fixed γ, energy, and satisfies an exact work identity when γ varies in time. Numerical experiments with periodically varying γ indicate both regular and irregular spin dynamics.
Related
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BibTeX
@misc{poshtkohi2026from,
title = {{From Quaternions to Pentarnions I: Algebra, geometry, and a spin model}},
author = {Alireza Poshtkohi and Susanne Pumpluen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31485},
eprint = {2610.31485},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31485},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Quaternions to Pentarnions I: Algebra, geometry, and a spin model
AU - Alireza Poshtkohi
AU - Susanne Pumpluen
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31485
UR - https://arxiv.org/abs/2610.31485
AB - We introduce a family of five-dimensional non-flexible quadratic real algebras {𝒜_γ}_0≤γ≤1 with canonical basis 1,i,j,k,ℓ, in which γ changes the multiplication but not their norm, and study how this change affects geometry and dynamics. The algebras are pairwise non-isomorphic, 𝒜_0 contains Hamilton's quaternion algebra, and 𝒜_1 is the Pentarnion algebra. The alternating part of the trilinear form ⟨ u×_γv,w⟩ induced by the multiplication of 𝒜_γ defines a three-dimensional subspace H_γ⊂span_ℝ{i,j,k,ℓ}, tilted by the angle arctan(γ/√3), whose rotation group G_γ≅SO(3) contains the automorphism group of 𝒜_γ. For γ_1≠γ_2 the groups G_γ_1 and G_γ_2 generate SO(V)≅SO(4). For the six ways of ordering and bracketing the product of i,p,p in 𝒜_γ, two map the unit circle in span{j,k} to a point and, for γ>0, four map it to ellipses. As an application, we derive a Hamiltonian model of a classical spin on H_γ whose multiplication-induced quadratic energy is isotropic for γ=0 and anisotropic for γ>0. It conserves spin length and, for fixed γ, energy, and satisfies an exact work identity when γ varies in time. Numerical experiments with periodically varying γ indicate both regular and irregular spin dynamics.
ER -