十六元数代数的关系图
Relation graphs of the sedenion algebra
浏览论文内容
中文总结 AI 辅助
该研究探讨十六元数代数的正交图与交换性图，刻画正交图连通分支顶点集与直径、建立其与八元数虚部直线的双射，还得出交换性图相关连通分支的直径范围。
中文摘要 AI 辅助
令$\u211d$表示十六元数代数，$Γ_O(\u211d)$表示其正交图。我们观察到$\u211d$中的任意一对零因子都会在$Γ_O(\u211d)$中生成一个双六边形。双六边形的顶点集可扩展为$\u211d$的一组基，该基具有便于使用的乘法表。我们明确刻画了$Γ_O(\u211d)$任意连通分支的顶点集，并求出了其直径。随后我们建立了$Γ_O(\u211d)$的连通分支与八元数虚部中的直线之间的双射关系。最后，我们研究了十六元数的交换性图，发现所有虚部为零因子的元素都属于同一个连通分支，且其直径在3到4之间。
英文摘要
Let $\mathbb{S}$ denote the algebra of the sedenions, and $Γ_O(\mathbb{S})$ denote its orthogonality graph. We observe that any pair of zero divisors in $\mathbb{S}$ produces a double hexagon in $Γ_O(\mathbb{S})$. The set of vertices of a double hexagon can be extended to a basis of $\mathbb{S}$ which has a convenient multiplication table. We describe explicitly the set of vertices of an arbitrary connected component of $Γ_O(\mathbb{S})$ and find its diameter. We then establish the bijection between the connected components of $Γ_O(\mathbb{S})$ and lines in the imaginary part of the octonions. Finally, we consider the commutativity graph of the sedenions and discover that all elements whose imaginary part is a zero divisor belong to the same connected component, and its diameter lies between $3$ and $4$.
