Two-Sided Fundamental Theorem of Affine Geometry

Q3 Mathematics

Arnold Mathematical Journal Pub Date : 2022-03-24 DOI:10.1007/s40598-022-00201-6

Alexey Gorinov

引用次数: 0

Abstract

The fundamental theorem of affine geometry says that if a self-bijection f of an affine space of dimenion n over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then f of the expected type, namely f is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type.

Abstract Image

查看原文本刊更多论文

仿射几何的双侧基本定理

仿射几何的基本定理说，如果二元n的仿射空间在可能偏斜的域上的自双射f取左仿射子空间到相同维度的左仿射子区域，则期望类型的f，即f是仿射映射和域的自同构的组合。我们证明了这一点的双边相似性：即，我们考虑如上所述的自双射，其将仿射子空间带到仿射子空间，但允许将左子空间带到右子空间，反之亦然。我们证明，在某些条件下，这些映射再次是预期类型的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Arnold Mathematical Journal Mathematics-Mathematics (all)

CiteScore

1.50

自引率

0.00%

发文量

期刊介绍： The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.