Abstract
This paper concerns the study of Leibniz algebras, a natural generalization of Lie algebras, from the perspective of centralizers of elements. We study conditions on Leibniz algebras under which centralizers of all elements are ideals. We call a Leibniz algebra, a CL-algebra if centralizers of all elements are ideals. We discuss nilpotency of CL-algebras.
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References
S. Albeverio, Sh. A. Ayupov, and B. A. Omirov, On nilpotent and simple Leibniz algebras, Comm. in Algebra, 33(1) (2005), pp. 159–172.
S. Albeverio, B. A. Omirov, and I. S. Rakhimov, Classification of 4-dimensional nilpotent complex Leibniz algebras. Extracta mathematicae, 21(3) (2006), pp. 197–210.
A. R. Ashrafi, On finite groups with a given number of centralizers. Algebra Colloq., 7(2) (2000), 139–146.
Sh. A. Ayupov and B. A. Omirov, On some classes of nilpotent Leibniz algebras, Siberian Math. J., 42(1) (2001), 18–29.
Y. Barnea and I. M. Isaacs, Lie algebras with few centralizer dimensions, J. Algebra, 259 (2003), 284–299.
S. M. Belcastro and G. J. Sherman, Counting centralizers infinite groups. Math. Mag., 5 (1994), 111–114.
A. Bloh, On a generalization of Lie algebra notion, Math. in USSR Doklady, 165(3) (1965), 471–473.
J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, New York: Springer-Verlag, 1972.
L. Kurdachenko, N. Semko, and I. Subbotin, The Leibniz algebras whose subalgebras are ideals, Open Mathematics, 15(1) (2017), pp. 92–100.
J. L. Loday, Cup product for Leibniz cohomology and dual Leibniz algebras, Math. Scand., 77 (1995), 189–196.
J. L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 296 (1993), 139–158.
G. Mukherjee and R. Saha, Cup-Product for Equivariant Leibniz Cohomology and Zinbiel Algebras, Algebra Colloquium, 26(2) (2019), 271–284.
Saffarnia, Somayeh, Moghaddam, Mohammad Reza R., Rostamyari and Mohammad, Centralizers in Lie algebras, Indian J. Pure Appl. Math., 49(1) (2018), 39–49.
A. J. Zapirain, Centralizer sizes and nilpotency class in Lie algebras and finite p-groups, Proc. Am. Math. Soc., 133(10) (2005), 2817–2820.
Acknowledgement
The second author would like to thank Prof. Ayupov of Institute of Mathematics, Uzbekistan Academy of Sciences and Prof. Karimbergen of Karakalpak State University, Uzbekistan for valuable discussions on this problem in a CIMPA research school on “Non-associative algebra and applications” held at Tashkent, Uzbekistan. The second author is especially thankful to Dr. Abror Kh. Khudoyberdiyev for reading a draft version of the article and his useful suggestions. The second author expresses his gratitude to CIMPA, France for their financial help to attend the research school. The Authors would like to thank the esteemed referee for her/his useful comments on the earlier version of the manuscript that have improved the exposition.
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Das, P., Saha, R. On Leibniz Algebras Whose Centralizers Are Ideals. Indian J Pure Appl Math 51, 1555–1571 (2020). https://doi.org/10.1007/s13226-020-0481-x
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DOI: https://doi.org/10.1007/s13226-020-0481-x