Description of some classes of Leibniz algebras

Abstract

In this paper we describe the isomorphism classes of finitedimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl 2 and a threedimensional solvable ideal. We choose a basis of the isomorphism classes' representatives and give explicit multiplication tables.

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

“…Theorem 2.7. [10] Let L be a Leibniz algebra whose quotient L/I ∼ = sl 2 ⊕ R, where R is a threedimensional solvable ideal and I is an irreducible right module over sl 2 (dimI = 3). Then there exists a basis {e, h, f, x 0 , x 1 , .…”

“…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]).…”

Local and 2-local automorphisms of some solvable Leibniz algebras

“…The problem of classification of finite-dimensional Leibniz algebras is fundamental and a very complicated problem. Last 30 years the Leibniz algebras has been actively investigated and a lot of papers have been devoted to the study of these algebras [4][5][6]20]. The analogue of the Levi-Malcev decomposition for Leibniz algebras was proved by D.W. Barnes [7], that asserts that any Leibniz algebra decomposes into a semidirect sum of its solvable radical and a semisimple Lie algebra.…”