Affine subspaces of units in simple algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910193031315456 |
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| author | Pazzis, Clément de Seguins |
| author_facet | Pazzis, Clément de Seguins |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06934 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine subspaces of units in simple algebras Pazzis, Clément de Seguins Rings and Algebras 15A30, 16K20 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. |
| title | Affine subspaces of units in simple algebras |
| topic | Rings and Algebras 15A30, 16K20 |
| url | https://arxiv.org/abs/2508.06934 |