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arXiv · 2411.18996

Intersection of subspaces in $A^2$ for a three-dimensional division algebra $A$ over a finite field

Daisuke Tambara
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Abstract

Let $A$ be a three-dimensional nonassociative division algebra over a finite field. Let $A$ act on the space $A^2$ by left multiplication. For a nonzero vector $v$ in $A^2$ we have a three-dimensional subspace $Av$ in $A^2$. This paper concerns about possible dimension of the intersection of $Av$ and $Av'$ for $v, v'$ in $A^2$. One of our results is that there exists a two-dimensional intersection if and only if $A$ is isotopic to a commutative algebra. We use a classical theorem that A is a twisted field of Albert.

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Daisuke Tambara. 2024-11-28. Intersection of subspaces in $A^2$ for a three-dimensional division algebra $A$ over a finite field. https://arxiv.org/abs/2411.18996

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