# Manuel LadraUniversity of Santiago de Compostela | USC · Departamento de Álgebra

Manuel Ladra

Mathematics

## About

163

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## Publications

Publications (163)

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...

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.

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.

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.

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...

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...

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...

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 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...

We construct a pair of adjoint functors between the categories of crossed modules of groups and associative algebras and establish an equivalence of categories between module structures over a crossed module of groups and its respective crossed module of associative algebras.

We study the excision property for Hochschild and cyclic homologies in the category of simplicial algebras. We extend Wodzicki's notion of H-unital algebras to simplicial algebras and then show that a simplicial algebra I
* satisfies excision in Hochschild and cyclic homologies if and only if it is H-unital. We use this result in the category of cr...

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 L...

In this paper we study Baer invariants of precrossed modules relative to the subcategory of crossed modules, following Fröhlich and Furtado-Coelho’s general theory on Baer invariants in varieties of Ω-groups and Modi’s theory on higher dimensional Baer invariants. Several homological invariants of precrossed and crossed modules were defined in the...

We construct a pair of adjoint functors between the categories of crossed modules of Lie and associative algebras, which extends the classical one between the categories of Lie and associative algebras. This result is used to establish an equivalence of categories of modules over a Lie crossed module and its universal enveloping crossed module.

The cotriple homology of crossed 2-cubes of Lie algebras is constructed and investigated. Namely, we calculate the cotriple homology of an inclusion crossed 2-cube of Lie algebras in terms of the bi-relative Chevalley-Eilenberg homologies. We also define in a natural way the Chevalley-Eilenberg homology of crossed 2-cubes of Lie algebras and study...

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. For such algebras in terms of its structure
constants we calculate right and plenary periods of generator elements. Some
results on subalgebras of EACP and ideals on low...

The present paper is devoted to the description of rigid solvable Leibniz
algebras. In particular, we prove that solvable Leibniz algebras under some
conditions on the nilradical are rigid and we describe four-dimensional
solvable Leibniz algebras with three-dimensional rigid nilradical. We show that
the Grunewald-O'Halloran's conjecture "any $n$-d...

We construct the zero and first non-abelian cohomologies of Leibniz algebras with coefficients in crossed modules, which differ from those of Gnedbaye and generalize the zero and first Leibniz cohomologies of Loday and Pirashvili. We also introduce the second non-abelian Leibniz cohomology and describe its relationship with extensions of Leibniz al...

We define the category of Lie–Leibniz algebras and its full subcategory LL1, which are categories of interest. We describe derived action conditions in LL1; according to the general construction for categories of interest, we construct the universal strict general actor (A) and define the algebra of triderivations Trider(A) of any object A in LL1....

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...

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 h...

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. Proper...

An evolution algebra corresponds to a quadratic matrix $A$ of structural
constants. It is known the equivalence between nil, right nilpotent evolution
algebras and evolution algebras which are defined by upper triangular matrices
$A$. We establish a criterion for an $n$-dimensional nilpotent evolution
algebra to be with maximal nilpotent index $2^{...

For any object L in the category of precrossed modules in Lie algebras PXLie, we construct the object Act(L), which we call the actor of this object. From this construction, we derive the notions of action, center, semidirect product, derivation, commutator, and abelian precrossed module in PXLie. We show that the notion of action is equivalent to...

We study the nilpotency of Leibniz n-algebras related with the adapted version of Engel's theorem to Leibniz n-algebras. We also deal with the characterization of finite-dimensional nilpotent complex Leibniz n-algebras.

We propose an algorithm using Gröbner bases that decides in terms of the existence of a non singular matrix P if two Leibniz algebra structures over a finite dimensional CC-vector space are representative of the same isomorphism class.We apply this algorithm in order to obtain a reviewed classification of the 3-dimensional Leibniz algebras given by...

In this paper we show that the method for describing solvable Lie algebras
with given nilradical by means of non-nilpotent outer derivations of the
nilradical is also applicable to the case of Leibniz algebras. Using this
method we extend the classification of solvable Lie algebras with naturally
graded filiform Lie algebra to the case of Leibniz a...

In this paper we classify solvable Leibniz algebras whose nilradical is a
null-filiform algebra. We extend the obtained classification to the case when
the solvable Leibniz algebra is decomposed as a direct sum of its nilradical,
which is a direct sum of null-filiform ideals, and a one-dimensional
complementary subspace. Moreover, in this case we e...

We show that one cannot construct a cyclic homology theory with coefficients that would be related to the Hochschild homology by the Connes periodicity exact sequence. We show that this is impossible even if the ideals of a given algebra have been taken as coefficients. Despite this, the cyclic homology with coefficients can be defined by restricti...

We show that the degree of derived functors of a group-valued functor defined on some distinguished category is less than or equal to the degree of the initial functor. Some illustrative examples of distinguished categories and applications to our main result are given.

We investigate the Hochschild and cyclic homologies of crossed modules of algebras in some special cases. We prove that the cotriple cyclic homology of a crossed module of algebras (I, A, ρ) is isomorphic to HC*(ρ): HC*(I) → HC*(A), provided I is H-unital and the ground ring is a field with characteristic zero. We also calculate the Hochschild and...

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...

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