The present paper is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz $n$-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz $n$-algebra and Cartan subalgebras and regular elements of the corresponding factor $n$-Lie algebra is established.
In this paper we initiate the study of the maximal subalgebras of exceptional simple classical Lie algebras g over algebraically closed fields k of positive characteristic p, such that the prime characteristic is good for g. In this paper we deal wit
h what is surely the most unnatural case; that is, where the maximal subalgebra in question is a simple subalgebra of non-classical type. We show that only the first Witt algebra can occur as a subalgebra of g and give explicit details on when it may be maximal in g.
In this paper we prove some general results on Leibniz 2-cocycles for simple Leibniz algebras. Applying these results we establish the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose assoc
iated Lie algebra is isomorphic to $mathfrak{sl}_2$.
In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras $sl_2^1oplus sl_2^2oplus dots oplus sl_2^soplus R,$ where $R$ is a solvable radical. The classifications of such Leibniz algebras in t
he cases $dim R=2, 3$ and $dim I eq 3$ have been obtained. Moreover, we classify Leibniz algebras with $L/Icong sl_2^1oplus sl_2^2$ and some conditions on ideal $I=id<[x,x] | xin L>.$
In this paper we present the classification of a subclass of naturally graded Leibniz algebras. These $n$-dimensional Leibniz algebras have the characteristic sequence equal to (n-3,3). For this purpose we use the software Mathematica.
We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.