Related papers: Finite Gr\"obner basis algebra with unsolvable nil…
This paper investigates the algebraic structure of indecomposable $\mathbb{N}$-graded vertex algebras $V = \bigoplus_{n=0}^{\infty} V_n$, emphasizing the intricate interactions between the commutative associative algebra $V_0$, the Leibniz…
Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads…
We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra…
We present a functorial construction which, starting from a congruence $\alpha$ of finite index in an algebra A, yields a new algebra C with the following properties: the congruence lattice of C is isomorphic to the interval of congruences…
In this paper we study the possibility to define irreducible representations of the symmetric groups with the help of finitely many relations. The existence of finite bases is established for the classes of representations corresponding to…
Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases,…
Throughout the current paper, we extend the study of Zinbiel algebras to Zinbiel superalgebras. In particular, we show that all the Zinbiel superalgebras over an arbitrary field are nilpotent in the same way as occurs for Zinbiel algebras.…
The present article is part of a research program the aim of which is to find all indecomposable solvable extensions of a given class of nilpotent Lie algebras. Specifically in this article we consider a nilpotent Lie algebra n that is…
Let $f_1,\ldots,f_m$ be elements in a quotient $R^n / N$ which has finite dimension as a $K$-vector space, where $R = K[X_1,\ldots,X_r]$ and $N$ is an $R$-submodule of $R^n$. We address the problem of computing a Gr\"obner basis of the…
We study the link between stably finiteness and stably projectionless-ness for $C^*$-algebras of solvable Lie groups. We show that these two properties are equivalent if the dimension of the group is not divisible by $4$; otherwise, they…
Let $K$ be an infinite field and $K< X> =K< X_1,...,X_n>$ the free associative algebra generated by $X=\{X_1,...,X_n\}$ over $K$. It is proved that if $I$ is a two-sided ideal of $K< X>$ such that the $K$-algebra $A=K< X> /I$ is almost…
We prove that all definable pre-orders are atomic, in a finitely generated free algebra of a discriminator variety of finite similarity type which is generated by its finite members.
Given a finite connected bipartite graph, finite-dimensional indecomposable semisimple Leibniz algebras are constructed. Furthermore, any finite-dimensional indecomposable semisimple Leibniz algebra admits a similar construction.
We consider the Schur multipliers of finite dimensional nilpotent Lie algebras. If the algebra has dimension greater than one, then the Schur multiplier is non-zero. We give a direct proof of an upper bound for the dimension of the Schur…
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 quasi-filiform algebra and the complemented space to the…
It is well known that algebraic power series are differentially finite (D-finite): they satisfy linear differential equations with polynomial coefficients. The converse problem, whether a given D-finite power series is algebraic or…
We construct an infinite finitely generated recursively presented residually finite algorithmically finite group $G$ answering thereby a question of Myasnikov and Osin. Moreover, $G$ is "very infinite" and "very algorithmically finite" in…
We show that if a countably generated Lie algebra $H$ does not contain isomorphic copies of certain finite-dimensional nilpotent Lie algebras $A$ and $B$ (satisfying some mild conditions), then $H$ embeds into a quotient of $A \ast B$ that…
We develop a method for approximating the Gr\"obner basis of the ideal of polynomials which vanish at a finite set of points, when the coordinates of the points are known with only limited precision. The method consists of a preprocessing…
In the paper the class of all solvable extensions of a filiform Leibniz algebra in the infinite-dimensional case is classified. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of residually solvable Leibniz algebra.…