Real division algebras with a~nontrivial reflection
Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2022) no. 2, pp. 23-35
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In this note, we consider four-dimensional unital real division algebras $\mathcal A$ with $\operatorname{Aut}(\mathcal A)$ containing a nontrivial reflection $\varphi$ (i.e., an automorphism of order two). If such an algebra $\mathcal A$ is a $\mathbb C$-bimodule, then we describe its multiplication table and state division conditions in terms of certain polynomials. Finally, we suggest a new method (different from the duplication process) that can be used to construct families of four-dimensional division algebras $\mathcal A$ with $\mathfrak{Der} (\mathcal A) =\{0\}$, which are generally not third power-associative or quadratic. Under some restrictions on algebra coefficients, we have listed all possible types of their automorphism groups.
@article{FPM_2022_24_2_a1,
author = {D. Gokal and E. Napedenina and M. Tvalavadze},
title = {Real division algebras with a~nontrivial reflection},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {23--35},
publisher = {mathdoc},
volume = {24},
number = {2},
year = {2022},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2022_24_2_a1/}
}
TY - JOUR AU - D. Gokal AU - E. Napedenina AU - M. Tvalavadze TI - Real division algebras with a~nontrivial reflection JO - Fundamentalʹnaâ i prikladnaâ matematika PY - 2022 SP - 23 EP - 35 VL - 24 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/FPM_2022_24_2_a1/ LA - ru ID - FPM_2022_24_2_a1 ER -
D. Gokal; E. Napedenina; M. Tvalavadze. Real division algebras with a~nontrivial reflection. Fundamentalʹnaâ i prikladnaâ matematika, Tome 24 (2022) no. 2, pp. 23-35. http://geodesic.mathdoc.fr/item/FPM_2022_24_2_a1/