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We introduce and study a non-abelian tensor product of two algebras with bracket with compatible actions on each other. We investigate its applications to the universal central extensions and the low-dimensional homology of perfect algebras with bracket., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2307.15636

Working over an arbitrary field of characteristic different from $2$, we extend the Skjelbred-Sund method to compatible Lie algebras and give a full classification of nilpotent compatible Lie algebras up to dimension $4$. In case the base field is cubically closed, we find that there are three isomorphism classes and a one-parameter family in dimension $3$, and $12$ isomorphism classes, $6$ one-parameter families and two $2$-parameter families in dimension $4$, Comment: To appear in Rendiconti del Circolo Matematico di Palermo Series 2

We study the extensibility problem of a pair of derivations associated with an abelian extension of algebras with bracket, and derive an exact sequence of the Wells type. We introduce crossed modules for algebras with bracket and prove their equivalence with internal categories in the category of algebras with bracket. We interpret the set of equivalence classes of crossed extensions as the second cohomology. Finally, we construct an eight term exact sequence in the cohomology of algebras with bracket., Comment: 25 pages

Given a non-negative integer $q$, we study two different notions of the $q$-capability of Lie algebras via the non-abelian $q$-exterior product of Lie algebras. The first is related to the $q$-crossed modules and inner $q$-derivations, and the second is the Lie algebra version of the $q$-capability of groups proposed by Ellis in 1995., Comment: 16 pages

In this paper, we give a natural braiding on the universal central extension of a crossed module of Lie algebras with a given braiding and construct the universal central extension of a braided crossed module of Lie algebras, showing that, when one of the constructions exists, both exist and coincide., Comment: 19 Pages

The paper is devoted to study new classes of chains of evolution algebras and their time-depending dynamics. Moreover, we construct some Rote-Baxter operators of such algebras., Comment: 29 pages

We study the capability property of Leibniz algebras via the non-abelian exterior product., Comment: arXiv admin note: text overlap with arXiv:1612.07910

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform associative algebras., Comment: arXiv admin note: text overlap with arXiv:1803.07698

In this paper we classify filiform associative algebras of degree $k$ over a field of characteristic zero. Moreover, we also classify naturally graded complex filiform and quasi-filiform nilpotent associative algebras which are described by the characteristic sequence $C(\mathcal{A})=(n-2,1,1)$ or $C(\mathcal{A})=(n-2,2)$., Comment: 17 pages

We give the description of homogeneous Rota-Baxter operators, Reynolds operators, Nijenhuis operators, Average operators and differential operator of weight 1 of null-filiform associative algebras of arbitrary dimension.

In this paper we study the category of braided categorical Leibniz algebras and braided crossed modules of Leibniz algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras using the Loday-Pirashvili category., Comment: 33 pages

This paper is a survey on the study of the behaviour of the composition of polynomials on the computation of Gr\"obner bases. This survey brings together some works published between 1995 and 2007. The authors of these papers gave answers to some questions in this subject for several types of Gr\"obner bases, over different monomials orderings and over different polynomials rings. Some of these answers are complete and some are not. Some papers of them were given to answer some open questions given in the first published paper of these papers and other some were given to generalize previous work. Note that, there are many other works in this subject over other special cases and types of Gr\"obner bases under the usual definition over arbitrary fields, over other types of bases, under the non-commutative case and with other methods of proofs and calculations., Comment: 38 pages

In this paper we study the categories of braided categorical associative algebras and braided crossed modules of associative algebras and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras., Comment: 24 pages

The present paper is devoted to provide conditions for the Levi--Malcev theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not conjugate by an inner automorphism) in the context of finite-dimensional Leibniz algebras over a field of characteristic zero. Particularly, in the case of the field $\mathbb{C}$ of complex numbers, we consider all possible cases in which Levi subalgebras are conjugate and not conjugate., Comment: 11 pages

In this paper, we present a methodology for off-line handwritten character recognition. The proposed methodology relies on a new feature extraction technique based on structural characteristics, histograms and profiles. As novelty, we propose the extraction of new eight histograms and four profiles from the $32\times 32$ matrices that represent the characters, creating 256-dimension feature vectors. These feature vectors are then employed in a classification step that uses a $k$-means algorithm. We performed experiments using the NIST database to evaluate our proposal. Namely, the recognition system was trained using 1000 samples and 64 classes for each symbol and was tested on 500 samples for each symbol. We obtain promising accuracy results that vary from 81.74\% to 93.75\%, depending on the difficulty of the character category, showing better accuracy results than other methods from the state of the art also based on structural characteristics., Comment: Revised version with a number of improvements and update references, 9 pages

The universal enveloping algebra functor between Leibniz and associative algebras defined by Loday and Pirashvili is extended to crossed modules. We prove that the universal enveloping crossed module of algebras of a crossed module of Leibniz algebras is its natural generalization. Then we construct an isomorphism between the category of representations of a Leibniz crossed module and the category of left modules over its universal enveloping crossed module of algebras. Our approach is particularly interesting since the actor in the category of Leibniz crossed modules does not exist in general, so the technique used in the proof for the Lie case cannot be applied. Finally we move on to the framework of the Loday-Pirashvili category $\mathcal{LM}$ in order to comprehend this universal enveloping crossed module in terms of the Lie crossed modules case.

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of $\mathfrak{sl}(m, n, D)$ when $m+n \geq 3$ and $D$ is a superdialgebra, solving in particular the problem when $D$ is an associative algebra, superalgebra or dialgebra. To accomplish this task we use a different method than the standard studied in the literature. We introduce and use the non-abelian tensor square of Leibniz superalgebras and its relations with the universal central extension., Comment: 13 pages

We prove that the class of nilpotent by finite, solvable by finite, polycyclic by finite, nilpotent of nilpotency class $n$ and supersolvable groups are closed under the formation of the non-abelian tensor product. We provide necessary and sufficient conditions for the non-abelian tensor product of finitely generated groups to be finitely generated. We prove that central extensions of most finite simple groups have trivial Bogomolov multiplier., Comment: 16 pages

Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are constructed. The well-known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules., Comment: 17 pages

We introduce the non-abelian tensor product of Lie superalgebras, study some of its properties including nilpotency, solvability and Engel, and we use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.

We complete the solution of the problem of finding the universal central extension of the matrix superalgebras $\mathfrak{sl}(m, n, A)$ where $A$ is an associative superalgebra and computing $H_2\big(\mathfrak{sl}(m, n, A)\big)$. The Steinberg Lie superalgebra $\mathfrak{st}(m, n, A)$ has a very important role and we will also find out $H_2\big(\mathfrak{st}(m, n, A)\big)$. In Chen and Sun (arXiv:1311.7079, 2013) it is solved the problem where $m+n \geq 5$ and in Chen and Guay (Algebr. Represent. Theory, 2013) it is solved when $n=0$, so here we work out the three remaining cases $\mathfrak{sl}(2, 1, A), \mathfrak{sl}(3, 1, A)$ and $\mathfrak{sl}(2, 2, A)$., Comment: 16 pages

In this paper we study the universal central extension of a Lie--Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie--Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension., Comment: 23 pages

For any finitely generated group $G$, two complexity functions $\alpha_G$ and $\beta_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its inverse. Restricting attention to free groups $F_r$, the exact asymptotic behaviour of $\alpha_2$ and $\beta_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $\alpha_r$ and $\beta_r$, and the existence of a polynomial upper bound is proved for $\beta_r$., Comment: 24 pages; To appear in Collectanea Mathematica

The notions of left-right noncommutative Poisson algebra ($\NP^{lr}$-algebra) and left-right algebra with bracket $\AWB^{lr}$ are introduced. These algebras are special cases of $\NLP$-algebras and algebras with bracket $\AWB$, respectively, studied earlier. An $\NP^{lr}$-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of the new algebras are studied. The constructions of free objects in the corresponding categories are given. The relations between the properties of $\NP^{lr}$-algebras, the underlying $\AWB^{lr}$, associative and Leibniz algebras are investigated. In the categories $\AWB^{lr}$ and $\NP^{lr}$-algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding well-known notions in categories of groups with operations. The cohomologies of $\NP^{lr}$-algebras and $\AWB^{lr}$ (resp. of $\NP^r$-algebras and $\AWB^r$) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases $P$ is a free $\NP^{r}$ or $\NP^{l}$-algebra, the Hochschild or/and Leibniz cohomological dimension of $P$ is $\leq n$ are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions., Comment: 25 pages

We give interpretations of some known key agreement protocols in the framework of category theory and in this way we give a method of constructing of many new key agreement protocols.

The purpose of this paper is two fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor product such as the finiteness of the product when each factor is finite. This would give yet another proof of Ellis's theorem on the finiteness of the nonabelian tensor product of groups when each factor is finite. Secondly, using the methods developed in proving the finiteness of the box-tensor product, we prove the finiteness of Inassaridze's tensor product under some additional hypothesis which generalizes his results on the finiteness of his product. In addition, we prove an Ellis like finiteness theorem under weaker assumptions, which is a generalization of his theorem on the finiteness of nonabelian tensor product. As a consequence, we prove the finiteness of low-dimensional nonabelian homology groups.

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras., Comment: 14 pages

We define a category $\galt$ of g-alternative algebras over a field $F$ and present the category of alternative algebras $\alt$ as a full subcategory of $\galt$; in the case $\ch F\neq 2$, we have $\alt=\galt$. For any g-alternative algebra $A$ we give a construction of a universal strict general actor $\cB(A)$ of $A$. We define the subset $\asoci(A)$ of $A$, and show that it is a $\cB(A)$-substructure of $A$. We prove that if $\asoci(A)=0$, then there exists an actor of $A$ in $\galt$ and $\act(A)=\cB(A)$. In particular, we obtain that if $A$ is anticommutative and $\ann(A)=0$, then there exists an actor of $A$ in $\galt$; from this, under the same conditions, we deduce the existence of an actor in $\alt$.

The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.