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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 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.

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 deals with the maximal abelian dimension of a Lie algebra, that is, the maximal value for the dimensions of its abelian Lie subalgebras. Indeed, we compute the maximal abelian dimension for every nilpotent Lie algebra of dimension less than 7 and for the Heisenberg algebra $\mathfrak{H}_k$ , with $k\in\mathbb{N}$ . In this way, an algorithmic procedure is introduced and applied to compute the maximal abelian dimension for any arbitrary nilpotent Lie algebra with an arbitrary dimension. The maximal abelian dimension is also given for some general families of nilpotent Lie algebras.

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 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.

We compute the largest dimension of the Abelian Lie subalgebras contained in the Lie algebra % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-bc8NnaaBaaaleaa% tCvAUfKttLearCqr1ngBPrgaiyGacqGFUbGBaeqaaaaa!4900! $$\mathfrak{g}_n $$ of n×n strictly upper triangular matrices, where n ∈ ℕ \ {1}. We do this by proving a conjecture, which we previously advanced, about this dimension. We introduce an algorithm and use it first to study the two simplest particular cases and then to study the general case.

In this paper, we compute minimal faithful unitriangular matrix representations of filiform Lie algebras. To do it, we use the nilpotent Lie algebra, gn, formed of n×n strictly upper-triangular matrices. More concretely, we search the lowest natural number n such that the Lie algebra gn contains a given filiform Lie algebra, also computing a representative of this algebra. All the computations in this paper have been done using MAPLE 9.5.

Abelian subalgebras play an important role in the study of Lie algebras and their properties and structures. In this paper, the historical evolution of this concept is shown, analyzing the current status for the research on this topic. So, the main results obtained from previous years are indicated and commented here. Additionally, a list of some related open problems is also given.

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 [Formula: see text]n, of n × n strictly upper-triangular matrices, because any given (filiform) nilpotent Lie algebra [Formula: see text] admits a Lie-algebra isomorphism with a subalgebra of [Formula: see text]n for some n ∈ ℕ\{1}. In this sense, we search for the lowest natural integer n such that the Lie algebra [Formula: see text]n contains the filiform Lie algebra [Formula: see text] as a subalgebra. Additionally, we give a representative of each representation.

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.