On the nilpotent Lie superalgebras of small superbreadth
In this paper, we classify finite-dimensional nilpotent Lie superalgebras of superbreadth at most two.
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arXiv subjects
M. Ladra
Publications and source records attributed to M. Ladra.
Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras
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 Leibniz algebras are strongly nilpotent. Moreover, for the first and second families, we give the description of characteristically nilpotent algebras which are non-strongly nilpotent.
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Solvable Leibniz algebras with quasi-filiform Lie algebras of maximum length nilradicals
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.
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Markov processes of cubic stochastic matrices: {\it Quadratic stochastic processes}
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 multiplications between cubic matrices we have to fix first a multiplication and then consider the KCE with respect to the fixed multiplication. Moreover, the notion of stochastic cubic matrix also varies depending on the real models of application. The existence of a stochastic (at each time) solution to the KCE provides the existence of a QSP. In this paper, our aim is to construct QSPs for two specially chosen notions of stochastic cubic matrices and two multiplications of such matrices (known as Maksimov's multiplications). We construct a wide class of QSPs and give some time-dependent behavior of such processes. We give an example with applications to the Biology, constructing a QSP which describes the time behavior (dynamics) of a population with the possibility of twin births.
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Construction of flows of finite-dimensional algebras
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 several kinds of multiplications between cubic matrices one has fix a multiplication first and then consider the KCE with respect to the fixed multiplication. The existence of a solution for the KCE provides the existence of an FA. In this paper our aim is to find sufficient conditions on the multiplications under which the corresponding KCE has a solution. Mainly our conditions are given on the algebra of cubic matrices (ACM) considered with respect to a fixed multiplication of cubic matrices. Under some assumptions on the ACM (e.g. power associative, unital, associative, commutative) we describe a wide class of FAs, which contain algebras of arbitrary finite dimension. In particular, adapting the theory of continuous-time Markov processes, we construct a class of FAs given by the matrix exponent of cubic matrices. Moreover, we remarkably extend the set of FAs given with respect to the Maksimov's multiplications of our previous paper (J. Algebra 470 (2017) 263--288). For several FAs we study the time-dependent behavior (dynamics) of the algebras. We derive a system of differential equations for FAs.
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Algebras of cubic matrices
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. Such a multiplication depends on an associative binary operation on the set of size $m$. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff $m=1$. We introduce a notion of accompanying algebra (which is $m^2$-dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed.
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Flow of finite-dimensional algebras
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 system whose states are finite-dimensional algebras with matrices of structural constants satisfying an analogue of Kolmogorov-Chapman equation. These flows of algebras (FA) can also be considered as deformations of algebras with the rule (the evolution equation) given by Kolmogorov-Chapman equation. We mainly use the multiplications of cubic matrices which were introduced by Maksimov and consider Kolmogorov-Chapman equation with respect to these multiplications. If all cubic matrices of structural constants are stochastic (there are several kinds of stochasticity) then the corresponding FA is called stochastic FA (SFA). We define SFA generated by known quadratic stochastic processes. For some multiplications of cubic matrices we reduce Kolmogorov-Chapman equation given for cubic matrices to the equation given for square matrices. Using this result many FAs are given (time homogenous, time non-homogenous, periodic, etc.). For a periodic FA we construct a continuum set of finite-dimensional algebras and show that the corresponding discrete time FA is dense in the set. Moreover, we give a construction of an FA which contains algebras of arbitrary (finite) dimension. For several FAs we describe the time depending behavior (dynamics) of the properties to be baric, limit algebras, commutative, evolution algebras or associative algebras.
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Leibniz algebras constructed by representations of General Diamond Lie algebras
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 $\mathfrak{D}_m$ which is isomorphic to a subalgebra of the special symplectic Lie algebra $\mathfrak{sp}(2m+2,\mathbb{R})$. Furthermore, we describe Leibniz algebras with corresponding $(2m+2)$-dimensional general Diamond Lie algebra $\mathfrak{D}_m$ and ideal generated by the squares of elements giving rise to a faithful representation of $\mathfrak{D}_m$.
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Minimal linear representations of filiform Lie algebras and their application for construction of Leibniz algebras
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. The classification up to isomorphism of such Leibniz algebras is given for low-dimensional cases.
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Solvable Leibniz algebras with naturally graded non-Lie $p$-filiform nilradicals
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 solvable Leibniz algebras with abelian nilradical and maximal dimension of its complemented space is proved.
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Solvable Leibniz algebra with non-Lie and non-split naturally graded filiform nilradical and its rigidity
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. However, it was possible for us to find a unique $(n+2)$-dimensional solvable Leibniz algebra with nilradical $F_n^1$. In addition, we establish the triviality of the second group of cohomology for this algebra with coefficients in itself, which implies its rigidity.
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Some irreducible components of the variety of complex $n+1$-dimensional Leibniz algebras
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.
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On the second cohomology group of simple Leibniz algebras
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 $\mathfrak{sl}_2$.
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On nilpotent index and dibaricity of evolution algebras
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 $2^{n-1}+1$. We give the classification of finite-dimensional complex evolution algebras with maximal nilpotent index. Moreover, for any $s=1,...,n-1$ we construct a wide class of $n$-dimensional evolution algebras with nilpotent index $2^{n-s}+1$. We show that nilpotent evolution algebras are not dibaric and establish a criterion for two-dimensional real evolution algebras to be dibaric.
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The classification of naturally graded Zinbiel algebras with characteristic sequence equal to $(n-p,p).$
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.
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On Algebraic properties of ABO-Blood Type Inheritance Pattern
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.
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Evolution algebra of a "chicken" population
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. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions on the structural constants it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.
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On dibaric and evolution algebras
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 admits a non-zero bq-homomorphism. Using the pair $(f,g)$ we define conservative algebras and establish criteria for a dibaric algebra to be conservative. Moreover, the notions of a Bernstein algebra and an algebra induced by a linear operator are introduced and relations between these algebras are studied. For dibaric algebras we describe a dibaric algebra homomorphism and study their properties by bq-homomorphisms of the dibaric algebras. We apply the results to the (dibaric) evolution algebra of a bisexual population. For this dibaric algebra we describe all possible bq-homomorphisms and find conditions under which the algebra of a bisexual population is induced by a linear operator. Moreover, some properties of dibaric algebra homomorphisms of such algebras are studied.
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