Affine subspaces of units in simple algebras

TL;DR

This paper determines the maximum dimension of affine subspaces within the units of simple algebras over a field, characterizing their structure and connections to quadratic and Hermitian forms.

Contribution

It provides a classification of maximal affine subspaces of units in simple algebras, linking algebraic structures to quadratic and Hermitian form theory.

Findings

Maximum dimension of affine subspaces in units determined

Classification of spaces with maximal dimension provided

Connections established with quadratic and Hermitian forms

Abstract

Let $A$ be a simple algebra over a field $F$. Under a mild cardinality assumption on $F$, we determine the greatest possible dimension for an $F$-affine subspace of $A$ that is included in the group of units $A^\times$, and we describe the spaces that have the greatest possible dimension. This is equivalent to the problem of determining the greatest possible dimension for an $F$-linear subspace $S$ of $A$ in which $x-1_A$ is a unit for all $x \in S$, and we elucidate the structure of these linear subspaces up to conjugation when their dimension reaches the greatest possible one. These classifications involve the associative composition algebras over $F$. Over fields of characteristic other than $2$, the first problem is essentially reduced to the classification of nonisotropic quadratic forms over $F$ and of nonisotropic Hermitian forms over quadratic and quaternionic extensions of $F$. These results are intimately connected with the problem of intransitive operator spaces between finite-dimensional vector spaces over division rings, which we study in depth: in particular, we generalize a dual version of Atkinson's theorem on primitive spaces of bounded rank matrices.