| Title:
|
The variety of dual mock-Lie algebras (English) |
| Author:
|
Camacho, Luisa M. |
| Author:
|
Kaygorodov, Ivan |
| Author:
|
Lopatkin, Viktor |
| Author:
|
Salim, Mohamed A. |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
28 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
161-178 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We classify all complex $7$- and $8$-dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex $9$-dimensional dual mock-Lie algebras. (English) |
| Keyword:
|
Nilpotent algebra |
| Keyword:
|
mock-Lie algebra |
| Keyword:
|
dual mock-Lie algebra |
| Keyword:
|
anticommutative algebra |
| Keyword:
|
algebraic classification |
| Keyword:
|
geometric classification |
| Keyword:
|
central extension |
| Keyword:
|
degeneration |
| MSC:
|
14D06 |
| MSC:
|
14L30 |
| MSC:
|
17A30 |
| idZBL:
|
Zbl 07300188 |
| idMR:
|
MR4162928 |
| . |
| Date available:
|
2021-03-03T08:48:18Z |
| Last updated:
|
2021-03-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148701 |
| . |
| Reference:
|
[1] Abdelwahab, H., Calderón, A.J., Kaygorodov, I.: The algebraic and geometric classification of nilpotent binary Lie algebras.International Journal of Algebra and Computation, 29, 6, 2019, 1113-1129, MR 3996987, 10.1142/S0218196719500437 |
| Reference:
|
[2] Alvarez, M.A.: On rigid $2$-step nilpotent Lie algebras.Algebra Colloquium, 25, 02, 2018, 349-360, MR 3805328, 10.1142/S100538671800024X |
| Reference:
|
[3] Alvarez, M.A.: The variety of $7$-dimensional $2$-step nilpotent Lie algebras.Symmetry, 10, 1, 2018, 26, Multidisciplinary Digital Publishing Institute, |
| Reference:
|
[4] Burde, D., Fialowski, A.: Jacobi--Jordan algebras.Linear Algebra and its Applications, 459, 2014, 586-594, Elsevier, MR 3247244 |
| Reference:
|
[5] Burde, D., Steinhoff, C.: Classification of orbit closures of $4$-dimensional complex Lie algebras.Journal of Algebra, 214, 2, 1999, 729-739, Academic Press, MR 1680532, 10.1006/jabr.1998.7714 |
| Reference:
|
[6] Cicalò , S., Graaf, W. De, Schneider, C.: Six-dimensional nilpotent Lie algebras.Linear Algebra and its Applications, 436, 1, 2012, 163-189, Elsevier, MR 2859920, 10.1016/j.laa.2011.06.037 |
| Reference:
|
[7] Darijani, I., Usefi, H.: The classification of 5-dimensional p-nilpotent restricted Lie algebras over perfect fields, I.Journal of Algebra, 464, 2016, 97-140, Elsevier, MR 3533425, 10.1016/j.jalgebra.2016.06.011 |
| Reference:
|
[8] Graaf, W.A. De: Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not $2$.Journal of Algebra, 309, 2, 2007, 640-653, Elsevier, MR 2303198, 10.1016/j.jalgebra.2006.08.006 |
| Reference:
|
[9] Graaf, W.A. De: Classification of nilpotent associative algebras of small dimension.International Journal of Algebra and Computation, 28, 01, 2018, 133-161, World Scientific, MR 3768261 |
| Reference:
|
[10] Ouaridi, A. Fernandez, Kaygorodov, I., Khrypchenko, M., Yu. Volkov: Degenerations of nilpotent algebras.arXiv:1905.05361. |
| Reference:
|
[11] Gorshkov, I., Kaygorodov, I., Khrypchenko, M.: The geometric classification of nilpotent Tortkara algebras.Communications in Algebra, 48, 1, 2020, 204-209, Taylor & Francis, MR 4060024, 10.1080/00927872.2019.1635612 |
| Reference:
|
[12] Grunewald, F., O'Halloran, J.: Varieties of nilpotent Lie algebras of dimension less than six.Journal of Algebra, 112, 2, 1988, 315-325, Academic Press, MR 0926608, 10.1016/0021-8693(88)90093-2 |
| Reference:
|
[13] Grunewald, F., O'Halloran, J.: A characterization of orbit closure and applications.Journal of Algebra, 116, 1, 1988, 163-175, Elsevier, MR 0944153, 10.1016/0021-8693(88)90199-8 |
| Reference:
|
[14] Hegazi, A.S., Abdelwahab, H.: Classification of five-dimensional nilpotent Jordan algebras.Linear Algebra and its Applications, 494, 2016, 165-218, Elsevier, MR 3455692, 10.1016/j.laa.2016.01.015 |
| Reference:
|
[15] Hegazi, A.S., Abdelwahab, H., Martin, A.J. Calderon: The classification of $N$-dimensional non-Lie Malcev algebras with $(N-4)$-dimensional annihilator.Linear Algebra and its Applications, 505, 2016, 32-56, Elsevier, MR 3506483, 10.1016/j.laa.2016.04.029 |
| Reference:
|
[16] Ismailov, N., Kaygorodov, I., Mashurov, F.: The algebraic and geometric classification of nilpotent assosymmetric algebras.Algebras and Representation Theory, 2020, 14 pp, Springer, DOI: 10.1007/s10468-019-09935-y. MR 4207393, 10.1007/s10468-019-09935-y |
| Reference:
|
[17] Ismailov, N., Kaygorodov, I., Yu. Volkov: The geometric classification of Leibniz algebras.International Journal of Mathematics, 29, 05, 2018, Article 1850035, World Scientific, MR 3808051, 10.1142/S0129167X18500350 |
| Reference:
|
[18] Ismailov, N., Kaygorodov, I., Yu. Volkov: Degenerations of Leibniz and anticommutative algebras.Canadian Mathematical Bulletin, 62, 3, 2019, 539-549, Canadian Mathematical Society, MR 3998738, 10.4153/S0008439519000018 |
| Reference:
|
[19] Karimjanov, I., Kaygorodov, I., Khudoyberdiyev, A.: The algebraic and geometric classification of nilpotent Novikov algebras.Journal of Geometry and Physics, 143, 2019, 11-21, Elsevier, MR 3954151, 10.1016/j.geomphys.2019.04.016 |
| Reference:
|
[20] Kaygorodov, I., Khrypchenko, M., Lopes, S.: The algebraic and geometric classification of nilpotent anticommutative algebras.Journal of Pure and Applied Algebra, 224, 8, 2020, Article 106337, MR 4074577 |
| Reference:
|
[21] Kaygorodov, I., Yu. Popov, Yu. Volkov: Degenerations of binary Lie and nilpotent Malcev algebras.Communications in Algebra, 46, 11, 2018, 4928-4940, Taylor & Francis, MR 3864274, 10.1080/00927872.2018.1459647 |
| Reference:
|
[22] Kaygorodov, I., Yu. Volkov: The Variety of Two-dimensional Algebras Over an Algebraically Closed Field.Canadian Journal of Mathematics, 71, 4, 2019, 819-842, Canadian Mathematical Society, MR 3984022, 10.4153/S0008414X18000056 |
| Reference:
|
[23] Kaygorodov, I., Yu. Volkov: Complete classification of algebras of level two.Moscow Mathematical Journal, 19, 3, 2019, 485-521, MR 3993005, 10.17323/1609-4514-2019-19-3-485-521 |
| Reference:
|
[24] Okubo, S., Kamiya, N.: Jordan--Lie super algebra and Jordan--Lie triple system.Journal of Algebra, 198, 2, 1997, 388-411, Elsevier, MR 1489904, 10.1006/jabr.1997.7144 |
| Reference:
|
[25] Ren, B., Zhu, L.S.: Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.Czechoslovak Mathematical Journal, 67, 4, 2017, 953-965, Springer, MR 3736011, 10.21136/CMJ.2017.0253-16 |
| Reference:
|
[26] Seeley, C.: Degenerations of 6-dimensional nilpotent Lie algebras over $\mathbb {C}$.Communications in Algebra, 18, 10, 1990, 3493-3505, Taylor & Francis, MR 1063991, 10.1080/00927879008824088 |
| Reference:
|
[27] Skjelbred, T., Sund, T.: Sur la classification des algèbres de Lie nilpotentes.Comptes rendus de l'Académie des Sciences, 286, 5, 1978, A241-A242, MR 0498734 |
| Reference:
|
[28] Zhevlakov, K.A.: Solvability and nilpotency of Jordan rings.Algebra i Logika, 5, 3, 1966, 37-58, MR 0207786 |
| Reference:
|
[29] Zusmanovich, P.: Central extensions of current algebras.Transactions of the American Mathematical Society, 334, 1, 1992, 143-152, MR 1069751, 10.1090/S0002-9947-1992-1069751-2 |
| Reference:
|
[30] Zusmanovich, P.: Special and exceptional mock-Lie algebras.Linear Algebra and its Applications, 518, 2017, 79-96, Elsevier, MR 3598575, 10.1016/j.laa.2016.12.029 |
| . |
