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Along this paper we show that under certain conditions the method for describing of solvable Lie and Leibniz algebras with maximal codimension of nilradical is also extensible to Lie and Leibniz superalgebras, respectively. In particular, we totally determine the solvable Lie and Leibniz superalgebras with maximal codimension of model filiform and model nilpotent nilradicals. Finally, it is established that the superderivations of the obtained superalgebras are inner.

The present article is a part of the study of solvable Leibniz algebras with a given nilradical. In this paper solvable Leibniz algebras, whose nilradicals is naturally graded $p$-filiform non-Lie Leibniz algebra $(n-p\geq4)$ and the complemented space to nilradical has maximal dimension, are described up to isomorphism. Moreover, among obtained algebras we indicate the rigid and complete algebras, Comment: 14. arXiv admin note: text overlap with arXiv:1605.00848

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.

We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated algebras, mainly the nilpotency and solvability in the case of Leibniz algebras.

In this paper we construct a minimal faithful representation of the $(2m+2)$-dimensional complex general Diamond Lie algebra, $\mathfrak{D}_m(\mathbb{C})$, which is isomorphic to a subalgebra of the special linear Lie algebra $\mathfrak{sl}(m+2,\mathbb{C})$. We also construct a faithful representation of the general Diamond Lie algebra $\mathfrak{D}_m$ which is isomorphic to a subalgebra of the special symplectic Lie algebra $\mathfrak{sp}(2m+2,\mathbb{R})$. Furthermore, we describe Leibniz algebras with corresponding $(2m+2)$-dimensional general Diamond Lie algebra $\mathfrak{D}_m$ and ideal generated by the squares of elements giving rise to a faithful representation of $\mathfrak{D}_m$., Comment: 9

In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-Lie Leibniz algebras are described up to isomorphism. It is shown that $k$-dimensional central extensions ($k\geq 5$) of these algebras are split., Comment: 18 pages

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}.$ We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$., Comment: 15 pages

We introduce and provide a classification theorem for the class of Heisenberg-Fock Leibniz algebras. This category of algebras is formed by those Leibniz algebras $L$ whose corresponding Lie algebras are Heisenberg algebras $H_n$ and whose ${H_n}$-modules $I$, where $I$ denotes the ideal generated by the squares of elements of $L$, are isomorphic to Fock modules. We also consider the three-dimensional Heisenberg algebra $H_3$ and study three classes of Leibniz algebras with $H_3$ as corresponding Lie algebra, by taking certain generalizations of the Fock module. Moreover, we describe the class of Leibniz algebras with $H_n$ as corresponding Lie algebra and such that the action $I \times H_n \to I$ gives rise to a minimal faithful representation of $H_n$. The classification of this family of Leibniz algebras for the case of $n=3$ is given.

In this paper we study subalgebras of complex finite dimensional evolution algebras. We obtain the classification of nilpotent evolution algebras whose any subalgebra is an evolution subalgebra with a basis which can be extended to a natural basis of algebra. Moreover, we formulate three conjectures related to description of such non-nilpotent algebras., Comment: 14 pages

The descriptions (up to isomorphism) of naturally graded $p$-filiform Leibniz algebras and $p$-filiform ($p\leq 3$) 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., Comment: 12 pages

In this paper we identify the structure of complex finite-dimensional Leibniz algebras with associated Lie algebras $sl_2^1\oplus sl_2^2\oplus \dots \oplus sl_2^s\oplus R,$ where $R$ is a solvable radical. The classifications of such Leibniz algebras in the cases $dim R=2, 3$ and $dim I\neq 3$ have been obtained. Moreover, we classify Leibniz algebras with $L/I\cong sl_2^1\oplus sl_2^2$ and some conditions on ideal $I=id<[x,x] \ | \ x\in L>.$, Comment: 11 pages

This work completes the study of the solvable Leibniz algebras, more precisely, it completes the classification of the $3$-filiform Leibniz algebras of maximum length \cite{3-filiform}. Moreover, due to the good structure of the algebras of maximum length, we also tackle some of their cohomological properties. Our main tools are the previous result of Cabezas and Pastor \cite{Pastor}, the construction of appropriate homogeneous basis in the considered connected gradation and the computational support provided by the two programs implemented in the software \textit{Mathematica}., Comment: 13 pages

In this paper we continue the description of solvable Leibniz algebras whose nilradical is a filiform algebra. In fact, solvable Leibniz algebras whose nilradical is a naturally graded filiform Leibniz algebra are described in \cite{Campo} and \cite{SolGFil}. Here we extend the description to solvable Leibniz algebras whose nilradical is a filiform algebra. We establish that solvable Leibniz algebras with filiform Lie nilradical are Lie algebras., Comment: 15 pages

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 $n-3.$

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 $n-p \geq 4.$, Comment: 12 pages

From the theory of finite dimensional Lie algebras it is known that every finite dimensional Lie algebra is decomposed into a semidirect sum of semisimple subalgebra and solvable radical. Moreover, due to work of Mal'cev the study of solvable Lie algebras is reduced to the study of nilpotent ones. For the finite dimensional Leibniz algebras the analogues of the mentioned results are not proved yet. In order to get some idea how to establish the results we examine the Leibniz algebra for which the quotient algebra with respect to the ideal generated by squares elements of the algebra (denoted by $I$) is a semidirect sum of semisimple Lie algebra and the maximal solvable ideal. In this paper the class of complex Leibniz algebras, for which quotient algebras by the ideal $I$ are isomorphic to the semidirect sum of the algebra $sl_2$ and two-dimensional solvable ideal $R$, are described., Comment: 11 pages

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., Comment: 14 pages

The $n$-dimensional $p$-filiform Leibniz algebras of maximum length have already been studied with $0\leq p\leq 2$. For Lie algebras whose nilindex is equal to $n-2$ there is only one characteristic sequence, $(n-2,1,1)$, while in Leibniz theory we obtain two possibilities: $(n-2,1,1)$ and $(n-2,2)$. The first case (the 2-filiform case) is already known. The present paper deals with the second case, i.e., quasi-filiform non Lie Leibniz algebras of maximum length. Therefore this work completes the study of maximum length of Leibniz algebras with nilindex $n-p$ with $0 \leq p \leq 2$., Comment: 16 pages

In this work we investigate the derivations of $n-$dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank $n-1$ are described., Comment: 13 pages

The paper is devoted to the study of finite dimensional complex evolution algebras. The class of evolution algebras isomorphic to evolution algebras with Jordan form matrices is described. For finite dimensional complex evolution algebras the criteria of nilpotency is established in terms of the properties of corresponding matrices. Moreover, it is proved that for nilpotent $n-$dimensional complex evolution algebras the possible maximal nilpotency index is $1+2^{n-1}.$ The criteria of planarity for finite graphs is formulated by means of evolution algebras defined by graphs., Comment: 14 pages

In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n_1,...,n_k|m)$ and nilindex n+m, where $n=n_1+...+n_k,$ n and m (m is not equal to zero) are dimensions of even and odd parts, respectively. Such superalgebras with condition n_1 > n-2 were classified in \cite{FilSup}--\cite{C-G-O-Kh}. Here we prove that in the case $n_1 < n-1$ the Leibniz superalgebras have nilindex less than $n+m.$ Thus, we get the classification of Leibniz superalgebras with characteristic sequence $(n_1, ...,n_k|m)$ and nilindex n+m., Comment: 13 pages

We present the description up to isomorphism of Leibniz superalgebras with characteristic sequence $(n|m_1,...,m_k)$ and nilindex $n+m,$ where $m=m_1+ >...+m_k,$ $n$ and $m$ ($m\neq 0$) are dimensions of even and odd parts, respectively., Comment: 13 pages

The aim of this work is to present the description up to isomorphism of Leibniz superalgebras with characteristic sequence (n | m-1, 1) and nilindex n+m., Comment: 21 pages. submitted

The Leibniz algebras appear as a generalization of the Lie algebras \cite{loday}. The classification of naturally graded $p$-filiform Lie algebras is known \cite{C-G-JM}, \cite{J.Lie.Theory}, \cite{AJM}, \cite{Ve}. In this work we deal with the classification of 2-filiform Leibniz algebras. The study of $p$-filiform Leibniz non Lie algebras is solved for $p=0$ (trivial) and $p=1$ \cite{Omirov1}. In this work we get the classification of naturally graded non Lie 2-filiform Leibniz algebras., Comment: 15 pages