From Quaternions to Pentarnions I: Algebra, geometry, and a spin model

Abstract

We introduce a family of five-dimensional non-flexible quadratic real algebras $\{\mathcal A_γ\}_{0\leqγ\leq1}$ with canonical basis $1,i,j,k,\ell$, 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, $\mathcal A_0$ contains Hamilton's quaternion algebra, and $\mathcal A_1$ is the Pentarnion algebra. The alternating part of the trilinear form $\langle u\times_γv,w\rangle$ induced by the multiplication of $\mathcal{A}_γ$ defines a three-dimensional subspace $H_γ\subset\operatorname{span}_{\mathbb R}\{i,j,k,\ell\}$, tilted by the angle $\arctan(γ/\sqrt3)$, whose rotation group $G_γ\cong\mathrm{SO}(3)$ contains the automorphism group of $\mathcal A_γ$. For $γ_1\neγ_2$ the groups $G_{γ_1}$ and $G_{γ_2}$ generate $\mathrm{SO}(V)\cong\mathrm{SO}(4)$. For the six ways of ordering and bracketing the product of $i,p,p$ in $\mathcal A_γ$, two map the unit circle in ${\rm span}\{j,k\}$ 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.