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.

The paper deals with the classification of a subclass of finite-dimensional Zinbiel algebras: the naturally graded p-filiform Zinbiel algebras. A Zinbiel algebra is the dual to Leibniz algebra in Koszul sense. We prove that there exists, up to isomorphism, only one family of naturally graded p-filiform Zinbiel algebras under hypothesis n − p ≥ 4.

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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 > .

In this work, we investigate the structure of Leibniz algebras whose associated Lie algebra is a direct sum of sl 2 and the solvable radical. In particular, we obtain the description of such algebras when: the ideal generated by the squares of elements of a Leibniz algebra is irreducible over sl 2 and when the dimension of the radical is equal to two.

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We present the classification of a subclass of n-dimensional naturally graded Zinbiel algebras. This subclass has the nilindex n − 3 and the characteristic sequence (n − 3, 2, 1). In fact, this result completes the classification of naturally graded Zinbiel algebras of nilindex n − 3.