Combinatorial structures and Lie algebras of upper triangular matrices (original) (raw)

Triangular Configurations and Strictly Upper-Triangular Matrices Lie Algebras

This paper shows how to associate the Lie algebra gn (n ∈ N), of n×n strictly upper-triangular matrices, with a specific 2-dimensional triangular configuration. The properties of those configu-rations are analyzed here by using Graph Theory and, finally, applied to obtain some results about Representation Theory for nilpotent Lie algebras.

Study of Lie algebras by using combinatorial structures

Linear Algebra and its Applications, 2012

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.

Lie algebras associated with triangular configurations

Linear Algebra and its Applications, 2005

A new class of Lie algebras of finite dimension, those which are associated with a certain combinatorial configuration made up by triangles of weighted and non-directed edges, is introduced and a characterization theorem for them is proved. Moreover, some subclasses of such Lie algebras are classified.

Combinatorial structures associated with Lie algebras of finite dimension

2004

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.

Minimal Faithful Upper-Triangular Matrix Representations for Solvable Lie Algebras

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 666.

The computation of abelian subalgebras in the Lie algebra of upper-triangular matrices

2008

This paper deals with the computation of abelian subalgebras of the solvable Lie algebra hn, of n × n upper-triangular matrices. Firstly, we construct an algorithm to find abelian Lie subalgebras in a given Lie algebra hn. This algorithm allows us to compute an abelian subalgebra up to a certain dimension. Such a dimension is proved to be equal to the maximum for abelian subalgebras of hn.

Representing Filiform Lie Algebras Minimally and Faithfully by Strictly Upper-Triangular Matrices

Journal of Algebra and Its Applications, 2013

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.

Combinatorial structures of three vertices and Lie algebras

International Journal of Computer Mathematics, 2012

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

Invariants of triangular Lie algebras

Journal of Physics A: Mathematical and Theoretical, 2007

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.

Combinatorial representation theory of Lie algebras. Richard Stanley’s work and the way it was continued

The Mathematical Legacy of Richard P. Stanley, 2016

Dedicated to Richard Stanley on the occasion of his seventieth birthday. Representation theory is a fundamental tool for studying group symmetry − geometric, analytic, or algebraic − by means of linear algebra, which has important applications to other areas of mathematics and mathematical physics. One very successful trend in this field in recent decades involves using combinatorial objects to model the representations, which allows one to apply combinatorial methods for studying them, e.g., for concrete computations. This trend led to the emergence of combinatorial representation theory, which has now become a thriving area. Richard Stanley played a crucial role, through his work and his students, in the development of this new area. In the early stages, he has the merit to have pointed out to combinatorialists, in [34, 36], the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I will describe some of Richard's contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work. Acknowledgement. Meeting Richard was a defining moment of my life and career, as well as of my evolution since then. Therefore, I always said that I consider him my mentor. I admire him both as a person and as a scientist, while his vast work in combinatorics and on its many relationships with other areas of mathematics is a continuous inspiration for me.