Manuel Ladra

• Mathematics
• University of Santiago de Compostela

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.

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...

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.

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 dimen...

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 equiv...

Given a non-negative integer [Formula: see text], we study two different notions of the [Formula: see text]-capability of Lie algebras via the non-abelian [Formula: see text]-exterior product of Lie algebras. The first is related to the [Formula: see text]-crossed modules and inner [Formula: see text]-derivations, and the second is the Lie algebra...

We show the interplay between compatible Leibniz algebras and compatible associative dialgebras by means of a commutative diagram. We construct a homology with trivial coefficients for compatible Leibniz algebras and use it to construct the universal central extension of a perfect compatible Leibniz algebra. Furthermore, we conjecture that the cate...

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 equiv...

In this paper, we consider two-dimensional cellular automata (CA) with the von Neumann neighborhood. We study the characterization of 2D linear cellular automata defined by the von Neumann neighborhood with new type of boundary conditions over the field ℤp. Furthermore, we investigate the rule matrices of 2D von Neumann CA by applying the group of...

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.

In this paper, we give a natural braiding on the universal central extension of a Lie crossed module with a given braiding in the category of Lie crossed modules. We also construct the universal central extension of a braided Lie crossed module in the category of braided Lie crossed modules, showing that, when one of these constructions exists, bot...

In this paper we give the description of some non-strongly nilpotent Leibniz algebras. We pay our attention to the subclass of nilpotent Leibniz algebras, which is called filiform. Note that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three disjoint families. We describe the pre-derivations of filiform Leibniz alg...

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras.

We consider an evolution algebra identifying the coefficients of SIS–SIR worm propagation models as the structure constants of the algebra. The basic properties of this algebra are studied. We prove that it is a commutative (and hence flexible), not associative and baric algebra. We describe the full set of idempotent elements and the full set of a...

The paper is devoted to studying new classes of chains of evolution algebras and their time-depending dynamics and property transition.

In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \set...

In the present paper, we consider non-Volterra cubic stochastic operators defined on a finite-dimensional simplex depending on a permutation π and a parameter α. We showed that for any permutation π, except the identity permutation, the set of limit points of the trajectories corresponding to the operators converges to a periodic trajectory. The tr...

In the algebraic-geometry-based theory of automated proving and discovery, it is often required that the user includes, as complementary hypotheses, some intuitively obvious non-degeneracy conditions. Traditionally there are two main procedures to introduce such conditions into the hypotheses set. The aim of this paper is to present these two appro...

We study the capability property of Leibniz algebras via the non-abelian exterior product.

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on a parameter α and study their trajectory behaviors. We find all fixed and periodic points for a non-Volterra quadratic stochastic operator on a finite-dimensional simplex. A complete description of the set of limit points is given, and we show that...

In this paper, we classify naturally graded complex quasi-filiform nilpotent associative algebras described using the characteristic sequence \(C(\mathcal {A})= (n-2,1,1)\) or \(C(\mathcal {A})=(n-2,2)\).

In this article, solvable Leibniz algebras, whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to the nilradical is proved. Communicated by K. C. Misra

In this paper, we classify filiform associative algebras of degree p over a field of characteristic zero. Moreover, over an algebraically closed field of characteristic zero, we also classify filiform nilpotent associative algebras and naturally graded quasi-filiform nilpotent associative algebras, described through the characteristic sequence C(A)...

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.

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.

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.

We construct HNN-extensions of Lie superalgebras and prove that every Lie superalgebra embeds into any of its HNN-extensions. Then as an application we show that any Lie superalgebra with at most countable dimension embeds into a two-generator Lie superalgebra.

In this paper, we describe central extensions (up to isomorphism) of all complex null-filiform and filiform associative algebras.

In this paper we construct HNN-extensions of diassociative algebras and Leibniz algebras and prove that every Leibniz algebra embeds into any of its HNN-extensions. We also prove that every Leibniz algebra with at most countable dimension embeds into a two-generator Leibniz algebra.

We present the category of alternative algebras as a category of interest. This kind of approach enables us to describe derived actions in this category, study their properties and construct a universal strict general actor of any alternative algebra. We apply the results obtained in this direction to investigate the problem of the existence of an...

We show that the Schur multiplier of a Noetherian group need not be finitely generated. We prove that the non-abelian tensor product of a polycyclic (resp. polycyclic-by-finite) group and a Noetherian group is a polycyclic (resp. polycyclic-by-finite) group. We also prove new versions of Schur's theorem.

We study the capability property of Leibniz algebras via the non-abelian exterior product.

In this paper we investigate pre-derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three non-intersected families. We describe the pre-derivation of filiform Leibniz algebras for the first and second families. We found sufficient conditions under which filiform Leibn...

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform associative algebras.

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 $C(\mathcal{A})=(n-2,2)$.

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.

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.

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...

In this paper solvable Leibniz algebras whose nilradical is quasi-filiform Lie algebra of maximum length, are classified. The rigidity of such Leibniz algebras with two-dimensional complemented space to nilradical is proved.

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 representa...

We consider a class of Volterra cubic stochastic operators. We describe the set of fixed points, the invariant sets and construct several Lyapunov functions to use them in the study of the asymptotical behavior of the given Volterra cubic stochastic operators. A complete description of the set of limit points is given, and we show that such operato...

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.

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.

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 c...

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 c...

In this paper we consider an equiprobable strictly non-Volterra quadratic stochastic operator defined on a finite-dimensional simplex. We show that such an operator has a unique fixed point, which is an attracting fixed point. Furthermore, we construct a Lyapunov function and use it in order to prove that for any initial point the set of limit poin...

We show that the non-abelian tensor product of nilpotent, solvable and Engel multiplicative Lie rings is nilpotent, solvable and Engel, respectively. The six term exact sequence in homology of multiplicative Lie rings is obtained. We also prove a new version of Stallings' theorem.

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...

We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multipli...

We consider Markov processes of cubic stochastic (in a fixed sense) matrices which are also called quadratic stochastic process (QSPs). A QSP is a particular case of a continuous-time dynamical system whose states are stochastic cubic matrices satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are several kinds of multipli...

Recently, we introduced the notion of flow (depending on time) of finite-dimensional algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are finite-dimensional algebras with (cubic) matrices of structural constants satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are...

Recently, we introduced the notion of flow (depending on time) of finite-dimensional algebras. A flow of algebras (FA) is a particular case of a continuous-time dynamical system whose states are finite-dimensional algebras with (cubic) matrices of structural constants satisfying an analogue of the Kolmogorov-Chapman equation (KCE). Since there are...

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of \(\mathfrak {sl}(m, n, D)\) when \(m+n \ge 3\) 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 standar...

We introduce the notion of identically distributed strictly non-Volterra cubic stochastic operator. We show that any identically distributed strictly non-Volterra cubic stochastic operator has a unique fixed point and that such operator has the property of being regular.

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.

There is a mistake in the description of complex solvable Leibniz algebras whose nilradical is a naturally graded filiform algebra [2, Theorem 4.3]. Namely, in the case where the dimension of the solvable Leibniz algebra with nilradical F-n(1) is equal to n+2, it was asserted that there is no such algebra. However, it was possible for us to find a...

We generate an algebra on blood phenotypes with multiplication based on the human ABO-blood group inheritance pattern. We assume that gametes are not chosen randomly during meiosis. We investigate some of the properties of this algebra, namely, the set of idempotents, lattice of ideals and the associative enveloping algebra.

We extend some general properties of automorphisms and derivations known for the Lie algebras to finite-dimensional complex Leibniz algebras. The analogs of the Jordan – Chevalley decomposition for derivations and the multiplicative decomposition for automorphisms of finite-dimensional complex Leibniz algebras are obtained.

We consider algebras of -cubic matrices (with ). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on...

We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. S...

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...

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...

In this paper we find minimal faithful representations of several classes filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras are filiform Lie algebras such that the action $I \times L \to I$ gives rise to a minimal faithful representation of a filiform Lie algebra....

We introduce a notion of flow (depending on time) of finite-dimensional algebras over the field of the real numbers. The sequence of cubic matrices of structural constants for this flow of algebras satisfies an analogue of Kolmogorov-Chapman equation. These flows of algebras (FA) can be considered as deformations of algebras with the rule (the evol...

Each finite-dimensional algebra can be identified to the cubic matrix given by structural constants defining the multiplication between the basis elements of the algebra. In this paper we introduce the notion of flow (depending on time) of finite-dimensional algebras. This flow can be considered as a particular case of (continuous-time) dynamical s...

In this paper we find minimal faithful representations of several classes filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras are filiform Lie algebras such that the action $I \times L \to I$ gives rise to a minimal faithful representation of a filiform Lie algebra....

In this paper solvable Leibniz algebras with naturally graded non-Lie $p$-filiform $(n-p\geq4)$ nilradical and with one-dimensional complemented space of nilradical are described. Moreover, solvable Leibniz algebras with abelian nilradical and extremal (minimal, maximal) dimensions of complemented space nilradical are studied. The rigidity of solva...

In this paper solvable Leibniz algebras with naturally graded non-Lie $p$-filiform $(n-p\geq4)$ nilradical and with one-dimensional complemented space of nilradical are described. Moreover, solvable Leibniz algebras with abelian nilradical and extremal (minimal, maximal) dimensions of complemented space nilradical are studied. The rigidity of solva...

For a given basis of a vector space L over a field K and a multiplication table which is defined by a multilinear map [-,...,-]:L×n→L, we present an algorithm and develop a computer program on Mathematica in order to test if the given multiplication table corresponds to a Lie n-algebra or a non-Lie Leibniz n-algebra or neither. The algorithm is bas...

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 representa...

The universal enveloping algebra functor UL: Lb → Alg, defined by Loday and Pirashvili [1], is extended to crossed modules. 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. Note that the procedure followed i...

In this paper we present a decomposition of HLn(L, L) into a direct sum of some subspaces for a finite dimensional complex semisimple Leibniz algebra L. Furthermore, we provide a more specific decomposition in case n = 2 into two subspaces. We verify that one of those subspaces annihilates for specific Leibniz algebras with liezation 2 and some oth...

In this paper we classify Leibniz algebras whose associated Lie algebra is four-dimensional Diamond Lie algebra 𝕯 and the ideal generated by squares of elements is represented by one of the finite-dimensional indecomposable D-modules Un 1, Un 2 or Wn 1 or Wn 2.

The description of complex solvable Leibniz algebras whose nilradical is a naturally graded filiform algebra is already known. Unfortunately, a mistake was made in that description. Namely, in the case where the dimension of the solvable Leibniz algebra with nilradical $F_n^1$ is equal to $n+2$, it was asserted that there is no such algebra. Howeve...

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 st...

We consider the evolution algebra of a free population generated by an F-quadratic stochastic operator. We prove that this algebra is commutative, not associative and necessarily power-associative. We show that this algebra is not conservative, not stationary, not genetic and not train algebra, but it is a Banach algebra. The set of all derivations...

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...

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.

We introduce the concept of conditional cubic stochastic operator in this study. We show that any conditional cubic stochastic operator has a unique fixed point and such an operator has the property of being regular.

We define a non-abelian tensor product of multiplicative Lie rings. This is a new concept providing a common approach to the non-abelian tensor product of groups defined by Brown and Loday and to the non-abelian tensor product of Lie rings defined by Ellis. We also prove an analogue of Miller’s theorem for multiplicative Lie rings.

In this paper we consider a uniform extremal Volterra quadratic stochastic operator defined on an even-dimensional unit simplex, which corresponds to a balanced digraph, and show that such operator has the non-ergodicity property. We also reinterpret this result in the framework of zero-sum games obtaining that these operators correspond to rock-pa...

In the present paper we indicate some Leibniz algebras whose closures of orbits under the natural action of $\GL_n$ form an irreducible component of the variety of complex $n$-dimensional Leibniz algebras. Moreover, for these algebras we calculate the bases of their second groups of cohomologies.

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 associated Lie algebra is isomorphic to $\mathfrak{sl}_2$.

This work is a continuation of the description of some classes of nilpotent Zinbiel algebras. We focus on the study of Zinbiel algebras with restrictions to gradation and characteristic sequence. Namely, the classification of naturally graded Zinbiel algebras with characteristic sequence equal to $ (n-p, p)$ is obtained.

We generate an algebra on blood phenotypes with multiplication based on human ABO-blood type inheritance pattern. We assume that during meiosis gametes are not chosen randomly. For this algebra we investigate its algebraic properties. Namely, the lattice of ideals and the associative enveloping algebra are described.

In this paper, we continue the investigation of complex finite-dimensional solvable Leibniz algebras with nilradical NF1 circle plus NF2 circle plus ... circle plus NFs, where NFi are ideals of maximal nilindex of the nilradical. The multiplication tables of such solvable algebras with restrictions to structural constants are obtained. In the case...

We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power-associative, in general. We prove t...

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...

We find conditions on ideals of an algebra under which the algebra is dibaric. Dibaric algebras have not non-zero homomorphisms to the set of the real numbers. We introduce a concept of bq-homomorphism (which is given by two linear maps $f, g$ of the algebra to the set of the real numbers) and show that an algebra is dibaric if and only if it admit...

In this paper we investigate the derivations of filiform Leibniz algebras. Recall that the set of filiform Leibniz algebras of fixed dimension is decomposed into three non-intersected families. We found sufficient conditions under which filiform Leibniz algebras of the first family are characteristically nilpotent. Moreover, for the first family we...

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...

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(\ma...