The variety of nilpotent Tortkara algebras
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 2, pp. 173-184.

Voir la notice de l'article provenant de la source Math-Net.Ru

We give algebraic and geometric classification of all $5$-dimensional nilpotent Tortkara algebras over complex field.
Keywords: Tortkara algebra, central extension, geometric classification, degenerations.
Mots-clés : algeraic classification
@article{JSFU_2019_12_2_a3,
     author = {Ilya B. Gorshkov and Ivan Kaygorodov and Alexey A. Kytmanov and Mohamed A. Salim},
     title = {The variety of nilpotent {Tortkara} algebras},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {173--184},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2019_12_2_a3/}
}
TY  - JOUR
AU  - Ilya B. Gorshkov
AU  - Ivan Kaygorodov
AU  - Alexey A. Kytmanov
AU  - Mohamed A. Salim
TI  - The variety of nilpotent Tortkara algebras
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2019
SP  - 173
EP  - 184
VL  - 12
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2019_12_2_a3/
LA  - en
ID  - JSFU_2019_12_2_a3
ER  - 
%0 Journal Article
%A Ilya B. Gorshkov
%A Ivan Kaygorodov
%A Alexey A. Kytmanov
%A Mohamed A. Salim
%T The variety of nilpotent Tortkara algebras
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2019
%P 173-184
%V 12
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2019_12_2_a3/
%G en
%F JSFU_2019_12_2_a3
Ilya B. Gorshkov; Ivan Kaygorodov; Alexey A. Kytmanov; Mohamed A. Salim. The variety of nilpotent Tortkara algebras. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 2, pp. 173-184. http://geodesic.mathdoc.fr/item/JSFU_2019_12_2_a3/

[1] J. Adashev, L. Camacho, S. Gomez-Vidal, I. Karimjanov, “Naturally graded Zinbiel algebras with nilindex $n-3$”, Linear Algebra Appl., 443 (2014), 86–104 | DOI | MR | Zbl

[2] J. Adashev, L. Camacho, B. Omirov, “Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras,”, J. Algebra, 479 (2017), 461–486 | DOI | MR | Zbl

[3] M.A. Alvarez, “On rigid $2$-step nilpotent Lie algebras”, Algebra Colloquium, 25:2 (2018), 349–360 | DOI | MR | Zbl

[4] M.A. Alvarez, “The variety of $7$-dimensional $2$-step nilpotent Lie algebras”, Symmetry, 10:1 (2018), 26 | DOI | MR | Zbl

[5] M.A. Alvarez, I. Hernández, “On degenerations of Lie superalgebras”, Linear and Multilinear Algebra | DOI

[6] M.A. Alvarez, I. Hernández, I. Kaygorodov, “Degenerations of Jordan superalgebras,”, Bulletin of the Malaysian Mathematical Sciences Society, 2018 | DOI

[7] G.G.A. Bauerle, E.A. de Kerf, A.P.E. ten Kroode, Lie Algebras, v. 2, Studies in Mathematical Physics, 7, Finite and Infinite Dimensional Lie Algebras and Applications in Physics, edited and with a preface by E.M. de Jager, North-Holland Publishing Co., Amsterdam, 1997 | MR | Zbl

[8] T. Benes, D. Burde, “Degenerations of pre-Lie algebras”, Journal of Mathematical Physics, 50:11 (2009), 112102 | DOI | MR | Zbl

[9] T. Benes, D. Burde, “Classification of orbit closures in the variety of three dimensional Novikov algebras”, Journal of Algebra and Its Applications, 13:2 (2014), 1350081 | DOI | MR | Zbl

[10] M. Bremner, “On Tortkara triple systems”, Communications in Algebra, 46:6 (2018), 2396–2404 | DOI | MR | Zbl

[11] D. Burde, “Degenerations of nilpotent Lie algebras”, Journal of Lie Theory, 9:1 (1999), 193–202 | MR | Zbl

[12] D. Burde, C. Steinhoff, “Classification of orbit closures of $4$–dimensional complex Lie algebras”, Journal of Algebra, 214:2 (1999), 729–739 | DOI | MR | Zbl

[13] A. Calderón Martín, A. Fernández Ouaridi, I. Kaygorodov, “The classification of $n$-dimensional anticommutative algebras with $(n-3)$-dimensional annihilator,”, Communications in Algebra, 2018 | DOI | MR

[14] A. Calderón Martín, A. Fernández Ouaridi, I. Kaygorodov, The classification of bilinear maps with radical of codimension 2, arXiv: 1806.07009

[15] L. Camacho, E. Cañete, S. Gomez-Vidal, B. Omirov, “$p$-filiform Zinbiel algebras”, Linear Algebra Appl., 438:7 (2013), 2958–2972 | DOI | MR | Zbl

[16] S. Cicalo, W. De Graaf, C. Schneider, “Six-dimensional nilpotent Lie algebras”, Linear Algebra Appl., 436:1 (2012), 163–189 | DOI | MR | Zbl

[17] I. Darijani, H. Usefi, “The classification of 5-dimensional $p$-nilpotent restricted Lie algebras over perfect fields, I”, J. Algebra, 464 (2016), 97–140 | DOI | MR | Zbl

[18] W. De Graaf, “Classification of nilpotent associative algebras of small dimension”, Internat. J. Algebra Comput., 28:1 (2018), 133–161 | DOI | MR | Zbl

[19] W. De Graaf, “Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2”, J. Algebra, 309:2 (2007), 640–653 | DOI | MR | Zbl

[20] I. Dokas, “Zinbiel algebras and commutative algebras with divided powers”, Glasg. Math. J., 52:2 (2010), 303–313 | DOI | MR | Zbl

[21] A. Dzhumadildaev, “Zinbiel algebras under $q$-commutators”, Journal of Mathematical Sciences (New York), 144:2 (2007), 3909–3925 | DOI | MR

[22] A. Dzhumadildaev, K. Tulenbaev, “Nilpotency of Zinbiel algebras”, J. Dyn. Control Syst., 11:2 (2005), 195–213 | DOI | MR | Zbl

[23] A. Dzhumadildaev, N. Ismailov, F. Mashurov, Embeddable algebras into Zinbiel algebras via the commutator, arXiv: 1809.10550 | MR

[24] A. Hegazi, H. Abdelwahab, “Nilpotent evolution algebras over arbitrary fields”, Linear Algebra Appl., 486 (2015), 345–360 | DOI | MR | Zbl

[25] A. Hegazi, H. Abdelwahab, “Classification of five-dimensional nilpotent Jordan algebras”, Linear Algebra and its Applications, 494 (2016), 165–218 | DOI | MR | Zbl

[26] A. Hegazi, H. Abdelwahab, “The classification of $n$-dimensional non-associative Jordan algebras with $(n-3)$-dimensional annihilator”, Communications in Algebra, 46:2 (2018), 629–643 | DOI | MR | Zbl

[27] A. Hegazi, H. Abdelwahab, A. Calderón Martín, “The classification of $n$-dimensional non-Lie Malcev algebras with $(n-4)$-dimensional annihilator”, Linear Algebra Appl., 505 (2016), 32–56 | DOI | MR | Zbl

[28] A. Hegazi, H. Abdelwahab, A. Calderón Martín, “Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2”, Algebr. Represent. Theory, 21:1 (2018), 19–45 | DOI | MR | Zbl

[29] F. Grunewald, J.O'Halloran, “Varieties of nilpotent Lie algebras of dimension less than six”, Journal of Algebra, 112 (1988), 315–325 | DOI | MR | Zbl

[30] F. Grunewald, J. O'Halloran, “A Characterization of orbit closure and applications”, Journal of Algebra, 116 (1988), 163–175 | DOI | MR | Zbl

[31] F. Grunewald, J.O'Halloran, “Deformations of Lie algebras”, Journal of Algebra, 162:1 (1993), 210–224 | DOI | MR | Zbl

[32] N. Ismailov, I. Kaygorodov, Yu. Volkov, “The geometric classification of Leibniz algebras”, International Journal of Mathematics, 29:5 (2018) | DOI | MR | Zbl

[33] N. Ismailov, I. Kaygorodov, Yu. Volkov, Degenerations of Leibniz and anticommutative algebras, arXiv: 1808.00907

[34] I. Karimjanov, I. Kaygorodov, M. Ladra, Central extensions of filiform associative algebras, arXiv: 1809.00183

[35] I. Kaygorodov, Yu. Popov, A. Pozhidaev, Yu. Volkov, “Degenerations of Zinbiel and nilpotent Leibniz algebras”, Linear and Multilinear Algebra, 66:4 (2018), 704–716 | DOI | MR | Zbl

[36] I. Kaygorodov, Yu. Popov, Yu. Volkov, “Degenerations of binary-Lie and nilpotent Malcev algebras”, Communications in Algebra, 46:11 (2018), 4929–4941 | DOI | MR

[37] I. Kaygorodov, Yu. Volkov, “The variety of $2$-dimensional algebras over an algebraically closed field”, Canadian Journal of Mathematics | DOI

[38] I. Kaygorodov, Yu. Volkov, The degeneration level classification of algebras, arXiv: 1710.08943

[39] J.-L. Loday, “Cup-product for Leibniz cohomology and dual Leibniz algebras”, Math. Scand., 77:2 (1995), 189–196 | DOI | MR | Zbl

[40] A. Naurazbekova, U. Umirbaev, “Identities of dual Leibniz algebras”, TWMS J. Pure Appl. Math., 1:1 (2010), 86–91 | MR | Zbl

[41] C. Seeley, “Degenerations of 6-dimensional nilpotent Lie algebras over $\mathbb{C}$”, Communications in Algebra, 18 (1990), 3493–3505 | DOI | MR | Zbl

[42] T. Skjelbred, T. Sund, “Sur la classification des algebres de Lie nilpotentes”, C. R. Acad. Sci. Paris Ser. A-B, 286:5 (1978), 241–242 | MR | Zbl

[43] I. Rakhimov, M. Hassan, “On one-dimensional Leibniz central extensions of a filiform Lie algebra”, Bull. Aust. Math. Soc., 84:2 (2011), 205–224 | DOI | MR | Zbl

[44] D. Yau, “Deformation of dual Leibniz algebra morphisms”, Comm. Algebra, 35:4 (2007), 1369–1378 | DOI | MR | Zbl

[45] P. Zusmanovich, “Central extensions of current algebras”, Trans. Amer. Math. Soc., 334:1 (1992), 143–152 | DOI | MR | Zbl