We prove the existence of a close relationship between the generalized central series of Leibniz algebras. We also prove some analogs of the classical Schur and Baer group-theoretic theorems for Leibniz algebras.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Baer, “Endlichkeitskriterien f¨ur Kommutatorgruppen,” Math. Ann., 124, 161–177 (1952); https://doi.org/10.1007/BF01343558.
A. Blokh, “On a generalization of the concept of Lie algebra,” Dokl. Akad. Nauk SSSR, 165, No. 3, 471–473 (1965).
V. A. Chupordia, A. A. Pypka, N. N. Semko, and V. S. Yashchuk, “Leibniz algebras: a brief review of current results,” Carpathian Math. Publ., 11, No. 2, 250–257 (2019); https://doi.org/10.15330/cmp.11.2.250-257.
M. R. Dixon, L. A. Kurdachenko, and A. A. Pypka, “On some variants of theorems of Schur and Baer,” Milan J. Math., 82, No. 2, 233–241 (2014); https://doi.org/10.1007/s00032-014-0215-9.
M. R. Dixon, L. A. Kurdachenko, and A. A. Pypka, “The theorems of Schur and Baer: a survey,” Int. J. Group Theory, 4, No. 1, 21–32 (2015); https://doi.org/10.22108/IJGT.2015.7376.
P. Hegarty, “The absolute centre of a group,” J. Algebra, 169, 929–935 (1994); https://doi.org/10.1006/jabr.1994.1318.
V. V. Kirichenko, L. A. Kurdachenko, A. A. Pypka, and I. Ya. Subbotin, “Some aspects of Leibniz algebra theory,” Algebra Discrete Math., 24, No. 1, 1–33 (2017).
L. A. Kurdachenko, J. Otal, and A. A. Pypka, “Relationships between the factors of the canonical central series of Leibniz algebras,” Europ. J. Math., 2, No. 2, 565–577 (2016); https://doi.org/10.1007/s40879-016-0093-5.
L. A. Kurdachenko, N. N. Semko, and I. Ya. Subbotin, “Applying group theory philosophy to Leibniz algebras: Some new developments,” Adv. Group Theory Appl., 9, 71–121 (2020); https://doi.org/10.32037/agta-2020-004.
L. A. Kurdachenko and I. Ya. Subbotin, “A brief history of an important classical theorem,” Adv. Group Theory Appl., 2, 121–124 (2016); https://doi.org/10.4399/97888548970148.
J.-L. Loday, “Une version non commutative des algèbres de Lie: les algèbras de Leibniz,” Enseign. Math., 39, 269–293 (1993).
B. H. Neumann, “Groups with finite classes of conjugate elements,” Proc. London Math. Soc. (3), 1, No. 1, 178–187 (1951); https://doi.org/10.1112/plms/s3-1.1.178.
I. N. Stewart, “Verbal and marginal properties of non-associative algebras,” Proc. London Math. Soc. (3), 28, No. 1, 129–140 (1974); https://doi.org/10.1112/plms/s3-28.1.129.
E. Stitzinger and R. Turner, “Concerning derivations of Lie algebras,” Lin, Multilin, Algebra, 45, No. 4, 329–331 (1999); https://doi.org/10.1080/03081089908818596.
M. R. Vaughan-Lee, “Metabelian BFC p-groups,” J. London Math. Soc. (2), 5, No. 4, 673–680 (1972); 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); https://doi.org/10.1016/j.jalgebra.2018.02.023.
J. Wiegold, “Multiplicators and groups with finite central factor-groups,” Math. Z., 89, No. 4, 345–347 (1965); https://doi.org/10.1007/BF01112166.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 12, pp. 1691–1697, December, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i12.6739.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Pypka, O.O. Some Relationships between the Generalized Central Series of Leibniz Algebras. Ukr Math J 73, 1958–1966 (2022). https://doi.org/10.1007/s11253-022-02040-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02040-2