On Description of Leibniz Algebras Corresponding to sl 2

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.…”

The geometric classification of Leibniz algebras

“…Then the description of all four dimensional solvable non-nilpotent Lebniz algebras was obtained in [5]. Finally, it was proved in [21] that there is only one non-solvable indecomposable Leibniz algebra whose dimension is less or equal to four, namely, the simple Lie algebra sl 2 .…”

Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type L4

“…Besides the algebra of right (left) multiplication operators is endowed with a structure of a Lie algebra by means of the commutator [9]. Also the quotient algebra by the two-sided ideal generated by the square elements of a Leibniz algebra is a Lie algebra [19], where such ideal is the minimal, abelian and in the case of non-Lie Leibniz algebras it is always non trivial.…”