Leibniz Algebras Whose Semisimple Part is Related to $$sl_2$$ s l 2

Abstract

In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras sl 1 2 ⊕ sl 2 2 ⊕ · · · ⊕ sl s 2 ⊕ R, where R is a solvable radical. The classifications of such Leibniz algebras in the cases dimR = 2, 3 and dimI = 3 have been obtained. Moreover, we classify Leibniz algebras with L/I ∼ = sl 1 2 ⊕ sl 2 2 and some conditions on ideal I = id < [x, x] | x ∈ L > .

Leibniz algebras as non-associative algebras

“…Then we obtain from identity (2.2) that L x 2 (y) = x 2 y = x(xy) − x(xy) = 0 , which shows that L x 2 = 0. Every abelian (left or right) Leibniz algebra is a Lie algebra, but there are many Leibniz algebras that are not Lie algebras (see, for example, [20,4,5,27,1,2,3,39,18,31,30,34,16,15,17,22,21,23,24,25]). We will use the following three examples to illustrate the concepts introduced in this section.…”

“…In part, the missing hypothesis in Example 2.3 of [18] and in Example 1 at the end of Section 2 of [12] as well as the lack of any proof have been the motivation for writing this paper, and especially, for proving the simplicity criterion Theorem 2.3. Moreover, as far as I can see, most of the existing papers on (semi-)simple Leibniz algebras (for example, [18], [12], [4], [5], [2]) do not use a conceptual approach, and the latter is exactly what we will present here. In particular, in Section 2 we discuss how our results can be employed to prove some of the statements in previous papers on (semi-)simple Leibniz algebras or sometimes how to complete the proofs of these statements.…”

Semisimple Leibniz algebras, their derivations and automorphisms

“…In our notations we present an example that generalizes the one in [8,Theorem 4.2] as a semisimple Leibniz algebra which can not be decomposed into the direct sum of simple ideals.…”

“…From the classical theory of finite-dimensional Lie algebras it is known that an arbitrary semisimple Lie algebra is decomposed into a direct sum of simple ideals, which are completely classified [11]. In the paper [8] an example of semisimple Leibniz algebra, which can not be decomposed into a direct sum of simple ideals, is presented (see Example 3.6 also). This shows that the structure of semisimple Leibniz algebras is much more complicated than structure of semisimple Lie algebras.…”