×

zbMATH — the first resource for mathematics

Abdelwahab, Hani; Calderón Martín, Antonio J.; Fernández Ouaridi, Amir
Central extensions of 4-dimensional binary Lie algebras. (English) Zbl 07274818
Rocky Mt. J. Math. 50, No. 5, 1541-1559 (2020).
Summary: The class of binary Lie algebras contains the one of Malcev algebras (and so the one of Lie algebras). We provide a classification of central extensions of complex non-Malcev binary Lie algebras of dimensions either \(3\) or \(4\). From here, a classification of central extensions of complex non-Lie anticommutative \(\mathfrak{CD}\)-algebras of dimensions either \(3\) or \(4\) is also given.
MSC:
17A30 Nonassociative algebras satisfying other identities
17A60 Structure theory for nonassociative algebras
17D10 Mal’tsev rings and algebras
Keywords:
binary Lie algebra; \(\mathfrak{CD}\)-algebra; Malcev algebra; Lie algebra; central extension; classification theorem
PDF BibTeX XML Cite
\textit{H. Abdelwahab} et al., Rocky Mt. J. Math. 50, No. 5, 1541--1559 (2020; Zbl 07274818)
Full Text: DOI Euclid
References:
[1] J. K. Adashev, L. M. Camacho, and B. A. Omirov, “Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras”, J. Algebra 479 (2017), 461-486. Mathematical Reviews (MathSciNet): MR3627293
Zentralblatt MATH: 1421.17001
Digital Object Identifier: doi:10.1016/j.jalgebra.2017.02.003
· Zbl 1421.17001
[2] A. J. Calderón, A. F. Ouaridi, and I. Kaygorodov, “Classification of bilinear maps with radical of codimension 2”, 2018. arXiv: 1806.07009
[3] A. J. Calderón, A. Fernández Ouaridi, and I. Kaygorodov, “The classification of \(n\)-dimensional anticommutative algebras with \((n-3)\)-dimensional annihilator”, Comm. Algebra 47:1 (2019), 173-181. Mathematical Reviews (MathSciNet): MR3924774
Digital Object Identifier: doi:10.1080/00927872.2018.1468909
· Zbl 07051443
[4] S. Cicalò, W. A. de Graaf, and C. Schneider, “Six-dimensional nilpotent Lie algebras”, Linear Algebra Appl. 436:1 (2012), 163-189. Mathematical Reviews (MathSciNet): MR2859920
Zentralblatt MATH: 1250.17017
Digital Object Identifier: doi:10.1016/j.laa.2011.06.037
· Zbl 1250.17017
[5] I. Darijani and H. Usefi, “The classification of 5-dimensional \(p\)-nilpotent restricted Lie algebras over perfect fields I”, J. Algebra 464 (2016), 97-140. Mathematical Reviews (MathSciNet): MR3533425
Zentralblatt MATH: 1403.17022
Digital Object Identifier: doi:10.1016/j.jalgebra.2016.06.011
· Zbl 1403.17022
[6] W. A. De Graaf, “Classification of nilpotent associative algebras of small dimension”, Internat. J. Algebra Comput. 28:1 (2018), 133-161. Mathematical Reviews (MathSciNet): MR3768261
Zentralblatt MATH: 1416.16052
Digital Object Identifier: doi:10.1142/S0218196718500078
· Zbl 1416.16052
[7] V. T. Filippov, “Binary Lie algebras without strong zero divisors”, Algebra and Logic 30:2 (1991), 149-166. Mathematical Reviews (MathSciNet): MR1159151
Zentralblatt MATH: 0771.17029
Digital Object Identifier: doi:10.1007/BF01978835
· Zbl 0771.17029
[8] A. T. Gaĭnov, “Identical relations for binary Lie rings”, Uspehi Mat. Nauk \((\) N.S.\() 12\):3(75) (1957), 141-146. In Russian. Mathematical Reviews (MathSciNet): MR0094372
· Zbl 0079.04801
[9] A. T. Gaĭnov, “Binary Lie algebras of lower ranks”, Algebra i Logika Sem. 2:4 (1963), 21-40. Mathematical Reviews (MathSciNet): MR0159851
Zentralblatt MATH: 0161.03603
· Zbl 0161.03603
[10] W. A. de Graaf, “Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2”, J. Algebra 309:2 (2007), 640-653. Mathematical Reviews (MathSciNet): MR2303198
Zentralblatt MATH: 1137.17012
Digital Object Identifier: doi:10.1016/j.jalgebra.2006.08.006
· Zbl 1137.17012
[11] A. S. Hegazi and H. Abdelwahab, “Nilpotent evolution algebras over arbitrary fields”, Linear Algebra Appl. 486 (2015), 345-360. Mathematical Reviews (MathSciNet): MR3401767
Zentralblatt MATH: 1358.17041
Digital Object Identifier: doi:10.1016/j.laa.2015.07.041
· Zbl 1358.17041
[12] A. S. Hegazi and H. Abdelwahab, “Classification of five-dimensional nilpotent Jordan algebras”, Linear Algebra Appl. 494 (2016), 165-218. Mathematical Reviews (MathSciNet): MR3455692
Zentralblatt MATH: 1382.17021
Digital Object Identifier: doi:10.1016/j.laa.2016.01.015
· Zbl 1382.17021
[13] A. S. Hegazi and H. Abdelwahab, “The classification of \(N\)-dimensional non-associative Jordan algebras with \((N-3)\)-dimensional annihilator”, Comm. Algebra 46:2 (2018), 629-643. Mathematical Reviews (MathSciNet): MR3764884
Zentralblatt MATH: 06875437
Digital Object Identifier: doi:10.1080/00927872.2017.1327052
· Zbl 1447.17021
[14] A. S. Hegazi, H. Abdelwahab, and A. J. Calderon Martin, “The classification of \(n\)-dimensional non-Lie Malcev algebras with \((n-4)\)-dimensional annihilator”, Linear Algebra Appl. 505 (2016), 32-56. Mathematical Reviews (MathSciNet): MR3506483
Zentralblatt MATH: 1386.17029
Digital Object Identifier: doi:10.1016/j.laa.2016.04.029
· Zbl 1386.17029
[15] A. S. Hegazi, H. Abdelwahab, and A. J. Calderon Martin, “Classification of nilpotent Malcev algebras of small dimensions over arbitrary fields of characteristic not 2”, Algebr. Represent. Theory 21:1 (2018), 19-45. Mathematical Reviews (MathSciNet): MR3748352
Zentralblatt MATH: 1430.17102
Digital Object Identifier: doi:10.1007/s10468-017-9701-4
· Zbl 1430.17102
[16] I. Karimjanov, I. Kaygorodov, and M. Ladra, “Central extensions of filiform associative algebras”, 2018. arXiv: 1809.00183
Zentralblatt MATH: 07143596
Digital Object Identifier: doi:10.1080/03081087.2018.1501331
· Zbl 07143596
[17] I. Karimjanov, I. Kaygorodov, and A. Khudoyberdiyev, “The algebraic and geometric classification of nilpotent Novikov algebras”, J. Geom. Phys. 143 (2019), 11-21. Mathematical Reviews (MathSciNet): MR3954151
Zentralblatt MATH: 07068214
Digital Object Identifier: doi:10.1016/j.geomphys.2019.04.016
· Zbl 07068214
[18] I. Kaygorodov, Y. Popov, and Y. Volkov, “Degenerations of binary Lie and nilpotent Malcev algebras”, Comm. Algebra 46:11 (2018), 4928-4940. Mathematical Reviews (MathSciNet): MR3864274
Zentralblatt MATH: 1441.17027
Digital Object Identifier: doi:10.1080/00927872.2018.1459647
· Zbl 1441.17027
[19] I. Kaygorodov, Y. Popov, and Y. Volkov, “Degenerations of binary Lie and nilpotent Malcev algebras”, Comm. Algebra 46:11 (2018), 4928-4940. Mathematical Reviews (MathSciNet): MR3864274
Zentralblatt MATH: 1441.17027
Digital Object Identifier: doi:10.1080/00927872.2018.1459647
· Zbl 1441.17027
[20] I. Kaygorodov, A. Pozhidaev, and P. Saraiva, “On a ternary generalization of Jordan algebras”, Linear Multilinear Algebra 67:6 (2019), 1074-1102. Mathematical Reviews (MathSciNet): MR3937028
Zentralblatt MATH: 07046241
Digital Object Identifier: doi:10.1080/03081087.2018.1443426
· Zbl 07046241
[21] E. A. de Kerf, G. G. A. Bäuerle, and A. P. E. ten Kroode, Lie algebras, II: Finite and infinite dimensional Lie algebras and applications in physics, Studies in Mathematical Physics 7, North-Holland Publishing Co., Amsterdam, 1997. Mathematical Reviews (MathSciNet): MR1489232
· Zbl 1125.17301
[22] E. N. Kuzrime min, “Mal”cev algebras and their representations”, Algebra i Logika 7:4 (1968), 48-69. Mathematical Reviews (MathSciNet): MR0252468
[23] E. N. Kuzrime min, “Binary Lie algebras of small dimension”, Algebra i Logika 37:3 (1998), 320-328, 376. Mathematical Reviews (MathSciNet): MR1673053
Zentralblatt MATH: 0913.17018
· Zbl 0913.17018
[24] A. I. Malrime cev, “Analytic loops”, Mat. Sb. N.S. 36(78) (1955), 569-576. In Russian. Mathematical Reviews (MathSciNet): MR0069190
[25] I. S. Rakhimov and M. A. Hassan, “On one-dimensional Leibniz central extensions of a filiform Lie algebra”, Bull. Aust. Math. Soc. 84:2 (2011), 205-224. Mathematical Reviews (MathSciNet): MR2832616
Zentralblatt MATH: 1228.17003
Digital Object Identifier: doi:10.1017/S0004972711002371
· Zbl 1228.17003
[26] A. A. Sagle, “Malcev algebras”, Trans. Amer. Math. Soc. 101 (1961), 426-458. Mathematical Reviews (MathSciNet): MR143791
Zentralblatt MATH: 0101.02302
Digital Object Identifier: doi:10.1090/S0002-9947-1961-0143791-X
· Zbl 0101.02302
[27] A. V. Sidorov, “On Lie triple algebras”, Algebra i Logika 20:1 (1981), 101-108, 124. Mathematical Reviews (MathSciNet): MR635653
Zentralblatt MATH: 0492.17001
· Zbl 0492.17001
[28] T. Skjelbred and T. Sund, “Sur la classification des algèbres de Lie nilpotentes”, C. R. Acad. Sci. Paris Sér. A-B 286:5 (1978), A241-A242. Mathematical Reviews (MathSciNet): MR498734
Zentralblatt MATH: 0375.17006
· Zbl 0375.17006
[29] P. · Zbl 0777.17016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.