Representing Filiform Lie Algebras Minimally and Faithfully by Strictly Upper-Triangular Matrices (original) (raw)

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.

Minimal linear representations of the low-dimensional nilpotent Lie algebras

MATHEMATICA SCANDINAVICA, 2008

The main goal of this paper is to compute a minimal matrix representation for each non-isomorphic nilpotent Lie algebra of dimension less than 666. Indeed, for each of these algebras, we search the natural number ninmathsfNsetminus1n\in\mathsf{N}\setminus\{1\}ninmathsfNsetminus1 such that the linear algebra mathfrakgn\mathfrak{g}_nmathfrakgn, formed by all the ntimesnn \times nntimesn 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 mathfrakgn\mathfrak{g}_nmathfrakgn. 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.

Minimal Matrix Representations for Six-Dimensional Nilpotent Lie Algebras

2018

This paper is concerned with finding minimal dimension linear representations for six-dimensional real, indecomposable nilpotent Lie algebras. It is known that all such Lie algebras can be represented in gl(6, R). After discussing the classification of the 24 such Lie algebras, it is shown that only one algebra can be represented in gl(4, R). A Theorem is then presented that shows that 13 of the algebras can be represented in gl(5, R). The special case of filiform Lie algebras is considered, of which there are five, and it is shown that each of them can be represented in gl(6, R) and not gl(5, R). Of the remaining five algebras, four of them can be represented minimally in gl(5, R). That leaves one difficult case that is treated in detail in an Appendix.

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.

Filiform Lie algebras of order 3

Journal of Mathematical Physics, 2014

The aim of this work is to generalize a very important type of Lie algebras and superalgebras, i.e., filiform Lie (super)algebras, into the theory of Lie algebras of order F. Thus, the concept of filiform Lie algebras of order F is obtained. In particular, for F = 3 it has been proved that by using infinitesimal deformations of the associated model elementary Lie algebra it can be obtained families of filiform elementary lie algebras of order 3, analogously as that occurs into the theory of Lie algebras [M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes,” Bull. Soc. Math. France 98, 81–116 (1970)]. Also we give the dimension, using an adaptation of the \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(2,\mathbb {C})$\end{document}sl(2,C)-module Method, and a basis of such infinitesimal deformations in some generic cases.

A computational study of a family of nilpotent Lie algebras

The Journal of Supercomputing

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

Matrix Representation for Seven-Dimensional Nilpotent Lie Algebras

Journal of Physical Mathematics, 2016

This paper is concerned with finding linear representations for seven-dimensional real, indecomposable nilpotent Lie algebras. We consider the first 39 algebras presented in Gong's classification which was based on the upper central series dimensions. For each algebra, we give a corresponding matrix Lie group, a representation of the Lie algebra in terms of left-invariant vector field and left-invariant one forms.

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

Computers & Mathematics With Applications, 2006

According to Ado and Cm'tan 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. (~)

Completable filiform Lie algebras

Linear Algebra and its Applications, 2003

We determine the solvable complete Lie algebras whose nilradical is isomorphic to a filiform Lie algebra. Moreover we show that for any positive integer n there exists a solvable complete Lie algebras whose second cohomology group with values in the adjoint module has dimension at least n.