Affine subspaces of units in simple algebras

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.