On the structure of Leibniz algebras, whose subalgebras are ideals or core-free

Abstract

Let L be an algebra over a field F with the binary operations + and [,]. Then L is called a Leibniz algebra (more precisely, a left Leibniz algebra), if it satisfies the Leibniz identity [a, [b, c] [a, c]] for all a, b, c  L.Note that any Lie algebra is obviously a Leibniz algebra. Conversely, if L is a Leibniz algebra such that [a, a] = 0 for every element a  L, then L is a Lie algebra. Therefore, Lie algebras can be characterized as the Leibniz algebras, in which [a, a]  0 for every element a.Leibniz algebras appeared first in papers by A.M. Blokh [1-3], in which he called them the D-algebras. However, in that time, those works were not in demand, and they had not been properly developed. Only after two decades, a real interest in Leibniz algebras arose. It was happened due to the work by J.L. Loday [4] (see also [5, Section 10.6]), who "rediscovered" these algebras and used the term Leibniz algebras, since it was Leibniz who discovered and proved the Leibniz rule for the differentiation of functions.A Leibniz algebra, which is not a Lie algebra, has one specific ideal. By Leib(L), we denote the subspace generated by the elements [a, a], a  L. It is possible to prove that Leib(L) is an ideal of L. Moreover, L/Leib(L) is a Lie algebra. Conversely, if H is an ideal of L such that L/H is a Lie algebra, then Leib(L)  H.

On Leibniz algebras with maximal cyclic subalgebras

“…This approach turned out to be quite effective. We will not do any review of the results here, we just make links to surveys [6,11], and the papers [5,7,8,9,10,12,13,16,21].…”

On the endomorphisms and derivations of some Leibniz algebras

“…A lot has already been done in this direction. We will not review the related results here, we simply link to the surveys [6,7] and the papers [8][9][10][11][12][13][14][15][16][17][18]. When studying Leibniz algebras, the information about the endomorphisms and derivations of a Leibniz algebra is quite useful.…”