AI 中文总结
本文研究了四维除环上二维向量空间的子空间重合问题，通过分析Knuth的两种代数系统，给出了显式解法。
AI 中文摘要
设A是一个非结合除环上的域F上的代数。让A通过左乘作用在空间A²上。对于非零元素v, v'∈A²，我们询问何时子空间Av和Av'重合。本文在A为四维且F为阶2的域的情况下给出了答案。在这种情况下，A已知是同构于Knuth的两种代数系统V和W之一。对于这两种代数中的每一种，我们给出了方程Av=Av'的显式解。结果适用于任何四维定义在任意基域F上的代数A，并配备了系统V或W的乘法规则。
英文摘要
Let $A$ be a non-associative division algebra over a field $F$. Let $A$ act on the space $A^2$ by left multiplication. For nonzero elements $v, v'$ of $A^2$ we ask when the subspaces $Av$ and $Av'$ coincide. The paper gives an answer in the case where $A$ is four-dimensional and $F$ is the field of order 2. In this case $A$ is known to be isotopic to either of two algebras, system $V$ and system $W$ of Knuth. For each of these two algebras we give an explicit solution of the equation $Av=Av'$. The result is stated for any four-dimensional algebra $A$ defined over an arbitrary base field $F$ and equipped with the multiplication rule of system $V$ or system $W$.
Comments32 pages
