On some relationships between the generalized central series of Leibniz algebras
DOI:
https://doi.org/10.37863/umzh.v73i12.6739Keywords:
Leibniz algebra, Lie algebra, Schur's theorem, Baer's theorem, Hegarty's theorem, D-center, D-derived subalgebra, upper (lower) D-central seriesAbstract
UDC 512.554
The purpose of this article is to show a close relationship between the generalized central series of Leibniz algebras. Some analogues of the classical group-theoretical theorems by Schur and Baer for Leibniz algebras are proved.
References
R. Baer, Endlichkeitskriterien f¨ur Kommutatorgruppen, Math. Ann., 124, 161 – 177 (1952); DOI:10.1007/BF01343558. DOI: https://doi.org/10.1007/BF01343558
A. Blokh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR, 165, № 3, 471 – 473 (1965) (in Russian).
V. A. Chupordia, A. A. Pypka, N. N. Semko, V. S. Yashchuk, Leibniz algebras: a brief review of current results, Carpathian Math. Publ., 11, № 2, 250 – 257 (2019); DOI:10.15330/cmp.11.2.250-257. DOI: https://doi.org/10.15330/cmp.11.2.250-257
M. R. Dixon, L. A. Kurdachenko, A. A. Pypka, On some variants of theorems of Schur and Baer, Milan J. Math., 82, № 2, 233 – 241 (2014); DOI:10.1007/s00032-014-0215-9. DOI: https://doi.org/10.1007/s00032-014-0215-9
M. R. Dixon, L. A. Kurdachenko, A. A. Pypka, The theorems of Schur and Baer: a survey, Int. J. Group Theory, 4, № 1, 21 – 32 (2015); DOI:10.22108/ IJGT.2015.7376.
P. Hegarty, The absolute centre of a group, J. Algebra, 169, 929 – 935 (1994); DOI:10.1006/jabr.1994.1318. DOI: https://doi.org/10.1006/jabr.1994.1318
V. V. Kirichenko, L. A. Kurdachenko, A. A. Pypka, I. Ya. Subbotin, Some aspects of Leibniz algebra theory, Algebra and Discrete Math., 24, № 1, 1 – 33 (2017).
L. A. Kurdachenko, J. Otal, A. A. Pypka, Relationships between the factors of the canonical central series of Leibniz algebras, Eur. J. Math., 2, № 2, 565 – 577 (2016); DOI:10.1007/s40879-016-0093-5. DOI: https://doi.org/10.1007/s40879-016-0093-5
L. A. Kurdachenko, N. N. Semko, I. Ya. Subbotin, Applying group theory philosophy to Leibniz algebras: some new developments, Adv. Group Theory and Appl., 9, 71 – 121 (2020); DOI:10.32037/agta-2020-004.
L. A. Kurdachenko, I. Ya. Subbotin, A brief history of an important classical theorem, Adv. Group Theory and Appl., 2, 121 – 124 (2016); doi:10.4399/97888548970148.
J.-L. Loday, Une version non commutative des alg`ebres de Lie: les alg`ebras de Leibniz, Enseign. Math., 39, 269 – 293 (1993).
B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3), 1, № 1, 178 – 187 (1951); DOI:10.1112/plms/s3-1.1.178. DOI: https://doi.org/10.1112/plms/s3-1.1.178
I. N. Stewart, Verbal and marginal properties of non-associative algebras, Proc. Lond. Math. Soc. (3), 28, № 1, 129 – 140 (1974); DOI:10.1112/plms/s3-28.1.129. DOI: https://doi.org/10.1112/plms/s3-28.1.129
E. Stitzinger, R. Turner, Concerning derivations of Lie algebras, Linear and Multilinear Algebra, 45, № 4, 329 – 331 (1999); DOI:10.1080/03081089908818596. DOI: https://doi.org/10.1080/03081089908818596
M. R. Vaughan-Lee, Metabelian BFC p-groups, J. Lond. Math. Soc. (2), 5, № 4, 673 – 680 (1972); DOI:10.1112/jlms/s2 – 5.4.673. DOI: https://doi.org/10.1112/jlms/s2-5.4.673
B. A. F. Wehrfritz, Schur’s theorem and Wiegold’s bound, J. Algebra, 504, 440 – 444 (2018); DOI:10.1016/j.jalgebra.2018.02.023. DOI: https://doi.org/10.1016/j.jalgebra.2018.02.023
J. Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z., 89, № 4, 345 – 347 (1965); DOI:10.1007/BF01112166. DOI: https://doi.org/10.1007/BF01112166
