Ángel F. Tenorio

Complete triangular structures and Lie algebras

In this paper, we study the families of n-dimensional Lie algebras associated with a combinatorial structure made up of n vertices and with its edges forming a complete simple, undirected graph. Moreover, some properties are characterized for these structures using Lie theory, giving some examples and representations. Furthermore, we also study the type of Lie algebras associated with them in order to get their classification. Finally, we also show an implementation of the algorithmic method used to associate Lie algebras with complete triangular structures.

Lie Theory: Applications to problems in Mathematical Finance and Economics

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.

Combinatorial structures of three vertices and Lie algebras

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.

Algorithmic method to obtain abelian subalgebras and ideals in Lie algebras

In this paper, we show an algorithmic procedure to compute abelian subalgebras and ideals of finite-dimensional Lie algebras, starting from the non-zero brackets in its law. In order to implement this method, we use the symbolic computation package MAPLE 12. Moreover, we also give a brief computational study considering both the computing time and the memory used in the two main routines of the implementation. Finally, we determine the maximal dimension of abelian subalgebras and ideals for non-decomposable solvable non-nilpotent Lie algebras of dimension 6 over both the fields ℝ and ℂ, showing the differences between these fields.

Combinatorial structures and Lie algebras of upper triangular matrices

Abstract 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.

COMPUTING THE LAW OF A FAMILY OF SOLVABLE LIE ALGEBRAS

This paper shows an algorithm which computes the law of the Lie algebra associated with the complex Lie group of n × n upper-triangular matrices with exponential elements in their main diagonal. For its implementation two procedures are used, respectively, to define a basis of the Lie algebra and the nonzero brackets in its law with respect to that basis. These brackets constitute the final output of the algorithm, whose unique input is the matrix order n. Besides, its complexity is proved to be polynomial and some complementary computational data relative to its implementation are also shown.

Algorithm to compute the maximal abelian dimension of Lie algebras

In this paper, the maximal abelian dimension is computationally obtained for an arbitrary finite-dimensional Lie algebra, defined by its nonzero brackets. More concretely, we describe and implement an algorithm which computes such a dimension by running it in the symbolic computation package MAPLE. Finally, we also show a computational study related to this implementation, regarding both the computing time and the memory used.

A method to obtain the lie group associated with a nilpotent lie algebra

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.

Algorithmic method to obtain combinatorial structures associated with Leibniz algebras

In this paper, we introduce an algorithmic process to associate Leibniz algebras with combinatorial structures. More concretely, we have designed an algorithm to automatize this method and to obtain the restrictions over the structure coefficients for the law of the Leibniz algebra and so determine its associated combinatorial structure. This algorithm has been implemented with the symbolic computation package Maple. Moreover, we also present another algorithm (and its implementation) to draw the combinatorial structure associated with a given Leibniz algebra, when such a structure is a (pseudo)digraph. As application of these algorithms, we have studied what (pseudo)digraphs are associated with low-dimensional Leibniz algebras by determination of the restrictions over edge weights (i.e. structure coefficients) in the corresponding combinatorial structures.

REPRESENTING FILIFORM LIE ALGEBRAS MINIMALLY AND FAITHFULLY BY STRICTLY UPPER-TRIANGULAR MATRICES

In this paper, we compute minimal faithful representations of filiform Lie algebras by means of strictly upper-triangular matrices. To obtain such representations, we use nilpotent Lie algebras n, of n × n strictly upper-triangular matrices, because any given (filiform) nilpotent Lie algebra admits a Lie-algebra isomorphism with a subalgebra of n for some n ∈ ℕ\{1}. In this sense, we search for the lowest natural integer n such that the Lie algebra n contains the filiform Lie algebra as a subalgebra. Additionally, we give a representative of each representation.