In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra $mathfrak{D}$ and the ideal generated by the squares of elements (further denoted by $I$) is a right $mathfrak{D}$-module. Using d
escription cite{Cas} of representations of algebra $mathfrak{D}$ in $mathfrak{sl}(3,{mathbb{C}})$ and $mathfrak{sp}(4,{mathbb{F}})$ where ${mathbb{F}}={mathbb{R}}$ or ${mathbb{C}}$ we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra $mathfrak{D}.$ Classification of Leibniz algebras with corresponding Lie algebra $mathfrak{D}$ and with the ideal $I$ as a Fock right $mathfrak{D}$-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces $BL^2$ and $ZL^2$ are constructed, as well.
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 $mathfrak{sl}_2$.
The descriptions (up to isomorphism) of naturally graded $p$-filiform Leibniz algebras and $p$-filiform ($pleq 3$) Leibniz algebras of maximum length are known. In this paper we study the gradation of maximum length for $p$-filiform Leibniz algebras.
The present work aims at the classification of complex $p$-filiform ($p geq 4$) Leibniz algebras of maximum length.
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 $I=id<[x,x] | xin L>.$
We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We
establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(alpha)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibniz algebras. Among these deformations we found one new rigid algebra.
In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a
Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m eq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra $NF^{n}$. Similar results for the case of Leibniz superalgebras are obtained.
The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras wi
th three-dimensional rigid nilradical. We show that the Grunewald-OHallorans conjecture any $n$-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991 Gorbatsevichs paper, is indicated.
In this paper we show that the method for describing solvable Lie algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case of Leibniz algebras. Using this method we extend the classif
ication of solvable Lie algebras with naturally graded filiform Lie algebra to the case of Leibniz algebras. Namely, the classification of solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra is obtained.
In this paper we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct su
m of null-filiform ideals, and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra, as well.
We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $Gtimes G$ to a ground field and give a multiplication of the algebra which depends on $f$. We study the basic properties of such algebras. In particular
, we find a condition on $f$ under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup $hat G$ of $G$ we define a $hat G$-periodic algebra, which corresponds to a $hat G$-periodic function $f,$ we establish a criterion for the right nilpotency of a $hat G$-periodic algebra. In addition, for $G=mathbb Z$ we describe all $2mathbb Z$- and $3mathbb Z$-periodic algebras. Some properties of $nmathbb Z$-periodic algebras are obtained.
