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

Abstract

We introduce a family of five-dimensional non-flexible quadratic real algebras $\{{\mathcal{A}}_{\gamma }{\}}_{0\le \gamma \le 1}$ with canonical basis $1,i,j,k,\ell$, in which $\gamma$ 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 $⟨u{\times }_{\gamma }v,w⟩$ induced by the multiplication of ${\mathcal{A}}_{\gamma }$ defines a three-dimensional subspace $H_{\gamma }\subset {\mathrm{span}}_{\mathbb{R}}\{i,j,k,\ell \}$, tilted by the angle $\arctan (\gamma /\sqrt{3})$, whose rotation group $G_{\gamma }\cong \mathrm{SO}(3)$ contains the automorphism group of ${\mathcal{A}}_{\gamma }$. For ${\gamma }_1\ne {\gamma }_2$ the groups $G_{{\gamma }_1}$ and $G_{{\gamma }_2}$ generate $\mathrm{SO}(V)\cong \mathrm{SO}(4)$. For the six ways of ordering and bracketing the product of $i,p,p$ in ${\mathcal{A}}_{\gamma }$, two map the unit circle in $\mathrm{span}\{j,k\}$ to a point and, for $\gamma >0$, four map it to ellipses. As an application, we derive a Hamiltonian model of a classical spin on $H_{\gamma }$ whose multiplication-induced quadratic energy is isotropic for $\gamma =0$ and anisotropic for $\gamma >0$. It conserves spin length and, for fixed $\gamma$, energy, and satisfies an exact work identity when $\gamma$ varies in time. Numerical experiments with periodically varying $\gamma$ indicate both regular and irregular spin dynamics.