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

TL;DR

This paper investigates the intersection dimensions of subspaces generated by elements in a three-dimensional division algebra over a finite field, revealing conditions for two-dimensional intersections related to algebra commutativity.

Contribution

It establishes a characterization of when the intersection of subspaces has dimension two, linking it to the algebra being isotopic to a commutative algebra, using Albert's theorem.

Findings

Two-dimensional intersection occurs iff A is isotopic to a commutative algebra.

Uses Albert's theorem on twisted fields.

Provides criteria for subspace intersections in nonassociative division algebras.

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.