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En el presente artı́culo trataremos diversos tópicos del álgebra lineal (y más concretamente del álgebra matricial) tanto desde una perspectiva histórica en la que se mostrará la evolución de diversos conceptos como desde su aplicación a la resolución de problemas económicos. En relación al recorrido histórico del álgebra lineal, expondremos los inicios de la misma y los principales hitos alcanzados en relación al estudio de las matrices, aunque no seremos exhaustivos por motivo de extensión. Con respecto al uso del álgebra lineal para resolver cuestiones económicas, mostraremos algunas de las aplicaciones más habituales y tradicionales a este respecto, haciendo especial énfasis en el análisis input-output y la teorı́a de juegos para la toma de decisiones.

Even if it has been less than a decade and a half since Tian introduced his concept of evolution algebras to represent algebraically non-Mendelian rules in Genetics, their study is becoming increasingly widespread mainly due to their applications to many scientific disciplines. In order to facilitate further research on the topic, this paper deals with the past and present research on these kind of algebras, together with the most relevant topics regarding them Junta de Andalucía FQM-016

This paper studies the link between isomorphic digraphs and isomorphic Leibniz algebras, determining in detail this fact when using (psuedo)digraphs of 2 and 3 vertices associated with Leibniz algebras according to their isomorphism classes. Moreover, we give the complete list with all the combinatorial structures of 3 vertices associated with Leibniz algebras, studying their isomorphism classes. We also compare our results with the current classifications of 2- and 3-dimensional Leibniz algebras. Finally, we introduce and implement the algorithmic procedure used for our goals and devoted to decide

In this paper, the maximal abelian dimension is algorithmically and computationally studied for the Lie algebra hn, of n×n upper-triangular matrices. More concretely, we define an algorithm to compute abelian subalgebras of hn besides programming its implementation with the symbolic computation package MAPLE. The algorithm returns a maximal abelian subalgebra of hn and, hence, its maximal abelian dimension. The order n of the matrices hn is the unique input needed to obtain these subalgebras. Finally, a computational study of the algorithm is presented and we explain and comment some suggestions and comments related to how it works.

A well-known result on Lie Theory states that every finite-dimensional complex solvable Lie algebra can be represented as a matrix Lie algebra, with upper-triangular square matrices as elements. However, this result does not specify which is the minimal order of the matrices involved in such representations. Hence, the main goal of this paper is to revisit and implement a method to compute both that minimal order and a matrix representative for a given solvable Lie algebra. As application of this procedure, we compute representatives for each solvable Lie algebra with dimension less than $6$., 19 pages, 6 tables

In this paper, we study how two important ideals of a given Lie algebra $$\mathfrak {g}$$ (namely, the center $$Z(\mathfrak {g})$$ and the derived Lie algebra $$\mathcal {D}(\mathfrak {g})$$ ) can be translated into the language of Graph Theory. In this way, we obtain some criteria and characterizations of these ideals using Graph Theory.

This paper deals with several operations on graphs and combinatorial structures linking them with their associated Lie algebras. More concretely, our main goal is to obtain some criteria to determine when there exists a Lie algebra associated with a combinatorial structure arising from those operations. Additionally, we show an algorithmic method for one of those operations.

In this paper, we revise some ideas on Graph Theory and develop them to explain and determine three usual concepts of Input–Output Analysis: fundamental products, autonomous sets, and indecomposable matrices. Firstly, we define a digraph associated with the given economic system (i.e., with its technical coefficients matrix). Secondly, we consider topological properties of this graph to study the economical concepts previously mentioned. Thirdly, we describe different, original algorithms to make the calculations easier. Finally, we give an example from a real economy, Andalusia (a region in Spain). Along this paper, we also discuss the advantages of this new approach and present some ideas to ease its application in the future research.

This paper shows a characterization of digraphs of three vertices associated with Lie algebras, as well as determining the list of isomorphism classes for Lie algebras associated with these digraphs. Additionally, we introduce and implement two algorithmic procedures related to this study: the first is devoted to draw, if exists, the digraph associated with a given Lie algebra; whereas the other corresponds to the converse problem and allows us to test if a given digraph is associated or not with a Lie algebra. Finally, we give the complete list of all non-isomorphic combinatorial structures of three vertices associated with Lie algebras and we study the type of Lie algebra associated with each configuration.

This paper is devoted to study and compare two algebraic algorithms related to the computation of Lie algebras by using statistical techniques. These techniques allow us to decide which of them is more suitable and less costly depending on several variables, like the dimension of the considered algebra.

In this paper, we study the structure and properties of those n -dimensional Lie algebras associated with either summed structures of complete graphs or some families of digraphs, having into consideration that all these combinatorial structures are made up of n vertices. Our main goal is to obtain criteria determining when a Lie algebra is associated with some of combinatorial structures considered in this paper, as well as to study the properties of those structures in order to use them as a tool for classifying the types of Lie algebras associated with them.

This paper describes an algorithm to compute the law of the Lie algebra \(\mathfrak{g}_{n}\) associated with the Lie group Gn, formed of all the n×n upper-unitriangular matrices. The goal of this paper is to show the algorithm that computes the law of \(\mathfrak{g}_{n}\) and its implementation using the symbolic computation package MAPLE. In addition, the complexity of the algorithm is described.

This paper is devoted to show and explain some applications of Lie Theory to solve some problems in Economics and Mathematical Finance. So we put forward and discuss mathematical aspects and approaches for several economic problems which have been previously considered in the literature. Besides we also show our advances on this topic, mentioning some open problems for future research.

This paper deals with the maximal abelian dimension of the Lie algebra hn, of n×n upper‐triangular matrices. Regarding this, we obtain an algorithm which computes abelian subalgebras of hn as well as its implementation (and a computational study) by using the symbolic computation package MAPLE, where the order n of the matrices in hn is the unique input needed. Let us note that the algorithm also allows us to obtain a maximal abelian subalgebra of hn.

The main goal of this paper is to compute a minimal matrix representation for each non-isomorphic nilpotent Lie algebra of dimension less than $6$. Indeed, for each of these algebras, we search the natural number $n\in\mathsf{N}\setminus\{1\}$ such that the linear algebra $\mathfrak{g}_n$, formed by all the $n \times n$ complex strictly upper-triangular matrices, contains a representation of this algebra. Besides, we show an algorithmic procedure which computes such a minimal representation by using the Lie algebras $\mathfrak{g}_n$. In this way, a classification of such algebras according to the dimension of their minimal matrix representations is obtained. In this way, we improve some results by Burde related to the value of the minimal dimension of the matrix representations for nilpotent Lie algebras.

According to Ado and Cartan Theorems, every Lie algebra of finite dimension can be represented as a Lie subalgebra of the Lie algebra associated with the general linear group of matrices. We show in this paper a method to obtain the simply connected Lie group associated with a nilpotent Lie algebra, by using unipotent matrices. Two cases are distinguished, according to the nilpotent Lie algebra is or not filiform.

This work shows how to associate the Lie algebra h n , of upper triangular matrices, with a specific combinatorial structure of dimension 2 , for n ∈ N . The properties of this structure are analyzed and characterized. Additionally, the results obtained here are applied to obtain faithful representations of solvable Lie algebras.