Two-sided fundamental theorem of affine geometry

2017/02/24 by A. G. Gorinov, Gorinov, A. G.
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1702.07701

openalex publication_date 2017/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The fundamental theorem of affine geometry says that a self-bijection f of a finite-dimensional affine space 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 affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that these maps again are of the expected type.

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