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Two-Sided Fundamental Theorem of Affine Geometry

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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.

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References

  1. Arnold, V. I.: Arnold’s Problems. Springer, Berlin (2005). Translated and revised edition of the Russian 2000 original

  2. Artin, E.: Geometric Algebra. Interscience Publishers Inc., New York (1957)

    MATH  Google Scholar 

  3. Chubarev, A., Pinelis, I.: Fundamental theorem of geometry without the 1-to-1 assumption. Proc. AMS 127(9):2735-2744 (1999)

  4. Gorinov, A.G.: Pseudocomplex and Pseudo(bi)quaternion mappings. Funktsional. Anal. i Prilozhen. 38(2), 84–85 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Gorinov, A.G., Howard, R., Johnson, V., McNulty, G.F.: Maps that must be affine or conjugate affine: a problem of Vladimir Arnold. Arnold Math. J. 6, 213–229 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lam, T.Y.: Algebraic Theory of Quadratic Forms. Revised Second Printing, Benjamin/Cummings Publishing Co. Inc., Reading (1980)

    MATH  Google Scholar 

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Acknowledgements

I am grateful to the referees for their careful reading of the paper, their comments and finding a mistake in the statement of the main theorem in the first version of the paper.

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  1. Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russia

    Alexey Gorinov

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  1. Alexey Gorinov
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Correspondence to Alexey Gorinov.

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Gorinov, A. Two-Sided Fundamental Theorem of Affine Geometry. Arnold Math J. 8, 469–480 (2022). https://doi.org/10.1007/s40598-022-00201-6

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  • DOI: https://doi.org/10.1007/s40598-022-00201-6

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