On the derivations of cyclic Leibniz algebras

Authors

  • M.M. Semko University of the State Fiscal Service of Ukraine, 31 Universitetskaya str., 08205, Irpin, Ukraine
  • L.V. Skaskiv University of the State Fiscal Service of Ukraine, 31 Universytetska str., 08205, Irpin, Ukraine https://orcid.org/0000-0001-9090-6700
  • O.A. Yarovaya University of the State Fiscal Service of Ukraine, 31 Universytetska str., 08205, Irpin, Ukraine
https://doi.org/10.15330/cmp.14.2.345-353

Keywords:

(cyclic) Leibniz algebra, Lie algebra, ideal, derivation
Published online: 2022-08-07

Abstract

Let $L$ be an algebra over a field $F$. Then $L$ is called a left Leibniz algebra, if its multiplication operation $[-,-]$ additionally satisfies the so-called left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of a Leibniz algebra $L$ is called a derivation of an algebra $L$, if $f([a,b])=[f(a),b]+[a,f(b)]$ for all elements $a,b\in L$. It is well known that the set of all derivations $\mathrm{Der}(L)$ of a Leibniz algebra $L$ is a subalgebra of the Lie algebra $\mathrm{End}_{F}(L)$ of all linear transformations of an algebra $L$. The algebras of derivations of Leibniz algebras play an important role in the study of structure of Leibniz algebras. Their role is similar to that played by groups of automorphisms in the study of group structure.

In this paper, a complete description of the algebra of derivations of nilpotent cyclic Leibniz algebra is obtained. In particular, it was proved that this algebra is metabelian and supersoluble Lie algebra, and its dimension is equal to the dimension of an algebra $L$.

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How to Cite
(1)
Semko, M.; Skaskiv, L.; Yarovaya, . O. On the Derivations of Cyclic Leibniz Algebras. Carpathian Math. Publ. 2022, 14, 345-353.