arXiv subjects
Publications and source records attributed to B. A. Omirov.
In this paper we study the $p$-adic equation $x^q=a$ over the field of $p$-adic numbers. We construct an algorithm of calculation of criteria of solvability in the case of $q=p^m$ and present a computer program to compute the criteria for fixed value of $m \leq p-1$. Moreover, using this solvability criteria for $q=2,3,4,5,6$, we classify $p$-adic 6-dimensional filiform Leibniz algebras.
In the present work we introduce notions such as $k$-solvability, $s$- and $K_1$-nilpotency and the corresponding radicals. We prove that these radicals are invariant under derivations of Leibniz $n$-algebras. The Frattini and Cartan subalgebras of Leibniz $n$-algebras are studied. In particular, we construct examples that show that a classical result on conjugacy of Cartan subalgebras of Lie algebras, which also holds in Leibniz algebras and Lie $n$-algebras, is not true for Leibniz $n$-algebras.
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.
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 $0 \leq p \leq 2$.
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.
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.
The structural constants of an evolution algebra is given by a quadratic matrix $A$. In this work we establish equivalence between nil, right nilpotent evolution algebras and evolution algebras, which are defined by upper triangular matrix $A$. The classification of 2-dimensional complex evolution algebras is obtained. For an evolution algebra with a special form of the matrix $A$ we describe all its isomorphisms and their compositions. We construct an algorithm running under Mathematica which decides if two finite dimensional evolution algebras are isomorphic.
In this paper we investigate the description of the complex Leibniz superalgebras with nilindex n+m, where n and m ($m\neq 0$) are dimensions of even and odd parts, respectively. In fact, such superalgebras with characteristic sequence equal to $(n_1, ..., n_k | m_1, ..., m_s)$ (where $n_1+... +n_k=n, m_1+ ... + m_s=m$) for $n_1\geq n-1$ and $(n_1, ..., n_k | m)$ were classified in works \cite{FilSup}--\cite{C-G-O-Kh1}. Here we prove that in the case of $(n_1, ..., n_k| m_1, ..., m_s)$, where $n_1\leq n-2$ and $m_1 \leq m-1$ the Leibniz superalgebras have nilindex less than n+m. Thus, we complete the classification of Leibniz superalgebras with nilindex n+m.
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.
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.
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.
In this work the classification of filiform Leibniz superalgebras of nilindex $n+m,$ where $n$ and $m$ ($m \ne 0$) are dimensions of even and odd parts, respectively, is obtained.
In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m.$ We prove that such superalgebras with the condition $m_2\neq0$ have nilindex less than $n+m$. Therefore the complete classification of Leibniz algebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m$ is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to $n+m,$ which was provided in \cite{AOKh} and \cite{GKh}.
In this work the description up to isomorphism of complex naturally graded quasi-filiform Zinbiel algebras is obtained.
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 work $n$-dimensional filiform Leibniz algebras admitting a gradation of length $(n-1)$ are classified. Derivations of such algebras are also described.
In this work nul-filiform and filiform Zinbiel algebras are described up to isomorphism. Moreover, the classification of complex Zinbiel algebras is extended from dimensions $\leq 3$ up to the dimension $4.$
The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras dimensions less than 3 is extended to the dimension four. {\it AMS Subject Classifications}: 16D70, 17A30, 17A60, 17B30 {\it Key words:} Leibniz algebra, associative algebra, nilpotence, nulfiliform Leibniz algebra, filiform Leibniz algebra.