Some remarks on semisimple Leibniz algebras

Abstract

Abstract. From the Levi's Theorem it is known that every finite dimensional Lie algebra over a field of characteristic zero is decomposed into semidirect sum of solvable radical and semisimple subalgebra. Moreover, semisimple part is the direct sum of simple ideals. In [5] the Levi's theorem is extended to the case of Leibniz algebras. In the present paper we investigate the semisimple Leibniz algebras and we show that the splitting theorem for semisimple Leibniz algebras is not true. Moreover, we consider some special classes of the semisimple Leibniz algebras and find a condition under which they decompose into direct sum of simple ideals.Mathematics Subject Classification 2010: 17A32, 17A60, 17B20.

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 second cohomology group of simple Leibniz algebras

“…Note that simple and a direct sum of simple Leibniz algebras are examples of semisimple Leibniz algebra. Unfortunately, there exist semisimple Leibniz algebras which do not decompose into a direct sum of simple algebras (for an example see [4]).…”

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

“…A lot of papers have been devoted to the description of finite-dimensional nilpotent Leibniz algebras [1], [2] so far. However, just a few works are related to the semisimple part of Leibniz algebras [5], [4], [10].…”

“…Moreover, the ideal I is abelian and hence, it is contained in the solvable radical. Thanks to result of Barnes in order to describe Leibniz algebras it is enough to investigate the relationship between products of a semisimple Lie algebra and the radical (see [5], [9] and [10]).…”