On linear equations over split-octonions
Artem Lopatin, Alexandr N. Zubkov
TL;DR
This work determines the structure of solution sets for linear equations over split-octonions $oxed{\mathbf{O}}$ on algebraically closed fields, focusing on $ax=c$, $(ax)b=c$, and $a(bx)=c$, and extends to general linear monomial equations. By reducing to canonical non-invertible octonion pairs using ${\rm Aut}(\mathbf{O})={\rm G}_2$ action and leveraging known orbit classifications, the authors show that solution sets are either empty, a singleton, or affine flats $X\in\Omega_r$ with precise allowable dimensions: $r=4$ for $ax=c$, $r\in\{4,5,7\}$ for $(ax)b=c$, and $r\in\{4,6,8\}$ for $a(bx)=c$; for general monomials, $4\le r\le7$ or $X=\mathbf{O}$. They also prove that any linear monomial equation with nonzero constant term that has at least two solutions must admit an invertible solution, and they analyze how these solution sets degenerate under specializations via ${\rm G}_2$-orbit closures. These results provide a detailed, dimension-aware picture of linear equations in the nonassociative, split-octonion setting, with potential implications for related algebraic and physical frameworks that utilize ${\rm G}_2$ symmetries and octonionic structures.
Abstract
Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations $a(xb)=c$ and $a(bx)=c$ over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial equations in the split-octonions. Moreover, we show that if a linear monomial equation over the split-octonions with nonzero constant term has at least two solutions, then it necessarily possesses an invertible solution.