Classification of four dimensional baric algebras satisfying polynomial identity of degree six
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Abstract
In this paper, we proceeded to the classification of four dimensional baric algebras strictly satisfying a polynomial identity of degree six. After some results on the structure of such algebras, we show that the type of an algebra of the studied class is an invariant under change of idempotent in the Peirce decomposition. This last result plays a major role in our classification.
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References
[1] J.Bayara, A. Conseibo, Ouattara, M. and A.Micali, Train algebras of degree 2 and exponent 3, Discret and continous dynamical systems series, 4 (6), (2011), 1371–1386. https://doi.org 10.3934/dcdss.2011.4.1371 Google Scholar
[2] P.Beremwidougou, and A.Conseibo, Algebras satisfying identity of almost Bernstein algebras, Far East J. Math. Sci. (FJMS) 131 (2) (2021), 131–152. https://doi.org/10.17654/ms131020131 Google Scholar
[3] S. N.Bernstein, D ́emonstration math ́ematique de la loi d’h ́er ́edit ́e de Mendel, C. R. Acad. Sci. Paris, 177, (1923), 528–531. Google Scholar
[4] S. N.Bernstein, Principe de stationnarit ́e et g ́en ́eralisation de la loi de Mendel, C. R. Acad. Sci. Paris, 177, (1923), 581–584. Google Scholar
[5] I. M. H. Etherington, Genetic algebras, Proceedings of the Royal Society of Edinburgh, 59 (1939), 242–258. Google Scholar
[6] B. L. M.Ferreira, H.Guzzo, J. C. M.Ferreira, The Wedderburn b-decomposition for a class of almost alternative baric algebras, Asian-European Journal of Mathematics, 08 (2015), p. 1550006. https://doi.org/10.1142/S1793557115500060 Google Scholar
[7] B. L. M.Ferreira, R. N.Ferreira, The Wedderburn b-decomposition for Alternative Baric Algebras, Mitt. Math. Ges. Hamburg, 37 (2017), 13–25. https://doi.org/10.48550/arXiv.1410.7078 Google Scholar
[8] B. L. M.Ferreira, The b-radical of generalized alternative balgebras II, PROYECCIONES JOURNAL OF MATHEMATICS, 38 (2019), 969–979. https://doi.org/10.22199/issn.0717-6279-2019-05-0062 Google Scholar
[9] Ph.Holgate, Genetic algebras satisfying Bernstein’s stationarity principle, J. London Math. Soc., 2 (9) (1975), 613–623. https://doi.org/10.1112/jlms/s2-9.4.613 Google Scholar
[10] D. Kabr ́e and A. Conseibo, Structure of baric algebras satisfying a polynomial identity of degree six, JP Journal of Algebra Number Theory and Applications, 61 (1), (2023), 37–52. http://dx.doi.org/10.17654/0972555523010 Google Scholar
[11] D. Kabr ́e and A. Conseibo, Algebras satisfying a polynomial identity of degree six that are principal train, European Journal of Pure and Applied Mathematics, 16 (3), (2023), 1480–1490. https://doi.org/10.29020/nybg.ejpam.v16i3.4787 Google Scholar
[12] C. Mallol and al., On the train algebras of degree four: structures and classifications, Commun. Algebra 37(2), (2009) 532–547. http://dx.doi.org/10.1080/00927870802251146 Google Scholar
[13] M. Nourigat and R. Varro, Study of commutative ω-PI algebras of degree 4. III: Barycentric invariant algebras by gametization, Commun. Algebra 41 (8), (2013), 2825–2851. https://doi.org/10.1080/00927872.2012.665532 Google Scholar
[14] R. D.Schafer, Structure of genetic algebras, Amer. J. Math., 71 (1949), 121–135. https://doi.org/10.2307/2372100 Google Scholar
[15] R.Varro, Multilinear identities of degree 4 for Bernstein algebra and noncommutative mutation algebras, Commun. Algebra 40 (7), (2012), 2426–2448. https://doi.org/10.1080/00927872.2011.578605 Google Scholar
[16] A. W ̈orz-Busekros, Algebras in Genetics, Lecture Notes in Biomathematics, 36, Springer-Verlag, Berlin-New York, 1980. Google Scholar