Abstract:
The main result of the paper is the classification of all (nonassociative) algebras of level two, i.e., such algebras that maximal chains of nontrivial degenerations starting at them have length two. During this classification we obtain an estimation of the level of an algebra via its generation type, i.e., the maximal dimension of its one generated subalgebra. Also we describe all degenerations and levels of algebras of the generation type 1 with a square zero ideal of codimension 1.
Key words and phrases:
Level of algebra, orbit closure, degeneration.
\Bibitem{KayVol19}
\by Ivan~Kaygorodov, Yuru~Volkov
\paper Complete classification of algebras of level two
\jour Mosc. Math.~J.
\yr 2019
\vol 19
\issue 3
\pages 485--521
\mathnet{http://mi.mathnet.ru/mmj744}
\crossref{https://doi.org/10.17323/1609-4514-2019-19-3-485-521}
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https://www.mathnet.ru/eng/mmj744
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This publication is cited in the following 22 articles:
Sin-Ei Takahasi, Kiyoshi Shirayanagi, Makoto Tsukada, “A classification of endo-commutative curled algebras of dimension 2 over a nontrivial field”, Asian-European J. Math., 16:10 (2023)
Yury Volkov, “Anticommutative algebras of the third level”, Linear Algebra and its Applications, 662 (2023), 18
Hani Abdelwahab, Elisabete Barreiro, Antonio J. Calderón, Amir Fernández Ouaridi, “The algebraic and geometric classification of nilpotent Lie triple systems up to dimension four”, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 117:1 (2023)
Ivan Kaygorodov, Mykola Khrypchenko, Samuel A. Lopes, “The geometric classification of nilpotent algebras”, Journal of Algebra, 633 (2023), 857
Sin-Ei Takahasi, Kiyoshi Shirayanagi, Makoto Tsukada, “A Classification of 2-Dimensional Endo-Commutative Straight Algebras of Rank 1 over a non-Trivial Field”, MathPann, 29_NS3:2 (2023), 258
Ouaridi A.F., Kaygorodov I., Khrypchenko M., Volkov Yu., “Degenerations of Nilpotent Algebras”, J. Pure Appl. Algebr., 226:3 (2022), 106850
Ismailov N., Kaygorodov I., Mustafa M., “The Algebraic and Geometric Classification of Nilpotent Right Alternative Algebras”, Period. Math. Hung., 84:1 (2022), 18–30
Yury Volkov, “Anticommutative Engel algebras of the first five levels”, Linear and Multilinear Algebra, 70:1 (2022), 148
Yury Volkov, “n-ary Algebras of the First Level”, Mediterr. J. Math., 19:1 (2022)
D. Jumaniyozov, I. Kaygorodov, A. Khudoyberdiyev, “The algebraic and geometric classification of nilpotent noncommutative Jordan algebras”, J. Algebra. Appl., 20:11 (2021), 2150202
I. Kaygorodov, M. Khrypchenko, “The geometric classification of nilpotent $\mathfrak{CD}$-algebras”, J. Algebra. Appl., 20:11 (2021), 2150198
Ignatyev M., Kaygorodov I., Popov Yu., “The Geometric Classification of 2-Step Nilpotent Algebras and Applications”, Rev. Mat. Complut., 2021
I. Gorshkov, I. Kaygorodov, Yu. Popov, “Degenerations of Jordan algebras and “marginal” algebras”, Algebr. Colloq., 28:02 (2021), 281–294
I. Kaygorodov, M. Khrypchenko, Yu. Popov, “The algebraic and geometric classification of nilpotent terminal algebras”, J. Pure Appl. Algebr., 225:6 (2021), 106625
Nurlan Ismailov, Ivan Kaygorodov, Farukh Mashurov, “The Algebraic and Geometric Classification of Nilpotent Assosymmetric Algebras”, Algebr Represent Theor, 24:1 (2021), 135
I. Kaygorodov, S. A. Lopes, Yu. Popov, “Degenerations of nilpotent associative commutative algebras”, Commun. Algebr., 48:4 (2020), 1632–1639
I. Gorshkov, I. Kaygorodov, M. Khrypchenko, “The geometric classification of nilpotent tortkara algebras”, Commun. Algebr., 48:1 (2020), 204–209
Ivan Kaygorodov, Abror Khudoyberdiyev, Aloberdi Sattarov, “One-generated nilpotent terminal algebras”, Communications in Algebra, 48:10 (2020), 4355
Ivan Kaygorodov, Yury Volkov, “Degenerations of Filippov algebras”, Journal of Mathematical Physics, 61:2 (2020)
Luisa M. Camacho, Ivan Kaygorodov, Viktor Lopatkin, Mohamed A. Salim, “The variety of dual mock-Lie algebras”, Communications in Mathematics, 28:2 (2020), 161
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