Abstract
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all two-dimensional algebras with respect to these identities is given.
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REFERENCES
H. Ahmed, U. Bekbaev, and I. Rakhimov, ‘‘Subalgebras, idempotents, ideals and quasi-units of two-dimensional algebras,’’ Int. J. Algebra Comput. 30 (5), 903–929 (2020). https://doi.org/10.1142/S0218196720500253
H. Ahmed, U. Bekbaev and I. Rakhimov, ‘‘Two-dimensional left (right) unital algebras over algebraically closed fields and \(\mathbb{R}\),’’ J. Phys.: Conf. Ser.1489, 012001 (2020). https://doi.org/10.1088/1742-6596/1489/1/012002
H. Ahmed, U. Bekbaev, and I. Rakhimov, ‘‘On two-dimensional power associative algebras over algebraically closed fields and \(\mathbb{R}\),’’ Lobachevskii J. Math.40 (1), 1–13 (2019).
H. Ahmed, U. Bekbaev, and I. Rakhimov, ‘‘Classification of \(2\)-dimensional evolution algebras, their groups of automorphisms and derivation algebras,’’ J. Phys.: Conf. Ser. 1489, 012001 (2020).https://doi.org/10.1088/1742-6596/1489/1/012001
H. Ahmed, U. Bekbaev, and I. Rakhimov, ‘‘Classification of two-dimensional Jordan algebras over \(\mathbb{R}\),’’ Malays. J. Math. Sci. 12, 287–303 (2018).
H. Ahmed, U. Bekbaev, and I. Rakhimov, ‘‘Complete classification of two-dimensional algebras,’’ AIP Conf. Proc. 1830, 070016 (2017). https://doi.org/10.1063/1.4980965
H. Ahmed, U. Bekbaev, and I. Rakhimov, ‘‘Classification of two-dimensional Jordan algebras,’’ AIP Conf. Proc. 1905, 030003-1–030003-8 (2017). https://doi.org/10.1063/1.5012149
R. Durán Díaz, J. M. Masqé, and A. P. Domínguez, ‘‘Classifying quadratic maps from plane to plane,’’ Linear Algebra Appl. 364, 1–12 (2003).
M. Goze and E. Remm, ‘‘2-dimensional algebras,’’ Afr. J. Math. Phys. 10, 81–91 (2011).
H. P. Petersson, ‘‘The classification of two-dimensional nonassicative algebras,’’ Result. Math.3, 120–154 (2000).
Y. C. Casado, ‘‘Evolution algebras,’’ PhD Thesis (EDITA, Univ. Málaga, 2016). http://orchid.org/0000-0003-4299-4392.
J. M. Casas, M. Ladra, B. A. Omirov, and U. A. Rozikov, ‘‘On evolution algebras,’’ Algebra Colloq.21, 331–342 (2014).
I. Kaygorodov and Yu. Volkov, ‘‘The variety of 2-dimensional algebras over an algebraically closed field,’’ Canad. J. Math. 71, 819–842 (2019).
A. Giambruno, S. Mishchenko, and M. Zaicev, ‘‘Codimension growth of two-dimensional non-associative algebras,’’ Proc. Am. Math. Soc. 135, 3405–3415 (2007).
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The authors are grateful to the referee for the careful reading of the manuscript.
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Ahmed, H., Bekbaev, U. & Rakhimov, I. Identities on Two-Dimensional Algebras. Lobachevskii J Math 41, 1615–1629 (2020). https://doi.org/10.1134/S1995080220090024
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DOI: https://doi.org/10.1134/S1995080220090024