Leibniz algebras are certain generalization of Lie algebras. In this paper we survey the important results in Leibniz algebras which are analogs of corresponding results in Lie algebras. In particular we highlight the differences between Leibniz algebras and Lie algebras.

Abstract. Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of four dimensional non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms and some other techniques to obtain our result.

Leibniz algebras are certain generalization of Lie algebras. In this paper we give the classification of 5−dimensional complex non-Lie nilpotent Leibniz algebras. We use the canonical forms for the congruence classes of matrices of bilinear forms to classify the case dim(A 2 ) = 3 and dim(Leib(A)) = 1 which can be applied to higher dimensions. The remaining cases are classified via direct method.

Abstract. Leibniz algebras are certain generalization of Lie algebras. In this paper we give classification of non-Lie solvable (left) Leibniz algebras of dimension ≤ 8 with one dimensional derived subalgebra. We use the canonical forms for the congruence classes of matrices of bilinear forms to obtain our result. Our approach can easily be extended to classify these algebras of higher dimensions. We also revisit the classification of three dimensional non-Lie solvable (left) Leibniz algebras.

This article is a contribution to the improvement of classification theory in Leibniz algebras. We extend the method of congruence classes of matrices of bilinear forms that was used to classify complex nilpotent Leibniz algebras with one dimensional derived algebra. In this work we focus on applying this method to the classification of 6− dimensional complex nilpotent Leibniz algebras with two dimensional derived algebra.