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Real Division Algebras with a Nontrivial Reflection

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In this note, we consider four-dimensional unital real division algebras \(\mathcal{A}\) with Aut \(\left(\mathcal{A}\right)\) containing a nontrivial reflection φ (i.e., an automorphism of order two). If such an algebra \(\mathcal{A}\) is a 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 \(\mathfrak{D}\mathfrak{e}\mathfrak{r}\left(\mathcal{A}\right)=\left\{0\right\},\) 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.

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Authors and Affiliations

  1. Department of Mathematical and Computational Sciences, University of Toronto Mississauga, 3359 Mississauga Road N, Mississauga, On, L5L 1C6, Canada

    D. Gokal

  2. Plekhanov Russian University of Economics, Stremyannyy Pereulok, 36, Moscow, 115093, Russia

    E. Napedenina

  3. Department of Mathematical and Computational Sciences, University of Toronto Mississauga, 3359 Mississauga Road N, Mississauga, On, L5L 1C6, Canada

    M. Tvalavadze

Authors
  1. D. Gokal
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  2. E. Napedenina
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  3. M. Tvalavadze
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Correspondence to D. Gokal.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 24, No. 2, pp. 23–35, 2022.

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Gokal, D., Napedenina, E. & Tvalavadze, M. Real Division Algebras with a Nontrivial Reflection. J Math Sci 275, 393–402 (2023). https://doi.org/10.1007/s10958-023-06690-w

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