scholarly journals Minimal linear representations of the low-dimensional nilpotent Lie algebras

2008 ◽  
Vol 102 (1) ◽  
pp. 17 ◽  
Author(s):  
J. C. Benjumea ◽  
J. Núnez ◽  
A. F. Tenorio
Keyword(s):  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Matrix Representations ◽  
Triangular Matrices ◽  
Nilpotent Lie Algebras ◽  
Linear Representations ◽  
Minimal Dimension ◽  
The Matrix ◽  
Low Dimensional

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.

KAC-Moody Lie Algebras and the Classification of Nilpotent Lie Algebras of Maximal Rank

1982 ◽  
Vol 34 (6) ◽  
pp. 1215-1239 ◽  
Author(s):  
L. J. Santharoubane
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Maximal Rank ◽  
Killing Form ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Infinite Dimensional ◽  
Dimensional Version

Introduction. The natural problem of determining all the Lie algebras of finite dimension was broken in two parts by Levi's theorem:1) the classification of semi-simple Lie algebras (achieved by Killing and Cartan around 1890)2) the classification of solvable Lie algebras (reduced to the classification of nilpotent Lie algebras by Malcev in 1945 (see [10])).The Killing form is identically equal to zero for a nilpotent Lie algebra but it is non-degenerate for a semi-simple Lie algebra. Therefore there was a huge gap between those two extreme cases. But this gap is only illusory because, as we will prove in this work, a large class of nilpotent Lie algebras is closely related to the Kac-Moody Lie algebras. These last algebras could be viewed as infinite dimensional version of the semisimple Lie algebras.

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

2013 ◽  
Vol 12 (04) ◽  
pp. 1250196 ◽  
Author(s):  
MANUEL CEBALLOS ◽  
JUAN NÚÑEZ ◽  
ÁNGEL F. TENORIO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Algebra Isomorphism ◽  
Triangular Matrices ◽  
Nilpotent Lie Algebras ◽  
Upper Triangular Matrices ◽  
Filiform Lie Algebras ◽  
Filiform Lie Algebra

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.

scholarly journals Μελέτη ιδεωδών σε ειδικές lie άλγεβρες

2002 ◽  
Author(s):  
Θεόδουλος Ταπανίδης
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Open Problems ◽  
Nilpotent Lie Algebras ◽  
Central Sequence ◽  
Special Values ◽  
The Third

In this paper we study special properties of Nilpotent Lie Algebras of dimension eight over the field K of characteristic zero. The complete classification of these Lie Algebras has been done recently and there exist a great number of open problems. The problems, which have been solved in the thesis, are the following: i. There is not an Algebra of this category, which has two maximum abellian ideals of different dimension. ii. Extension of a Nilpotent Lie Algebra to others of bigger dimension. iii. Determination of Nilpotent Lie Algebras from another category iv. Determination of characteristic Nilpotent Lie Algebras from this category of Nilpotent Lie Algebras of dimensions eight. This thesis has three chapters. Each of them is analyzed as follows. The first chapter contains basic elements of the theory of Nilpotent Lie Algebras. This has eleven paragraphs; each of them consists of the following. The first paragraph has a general theory of algebra. Basic elements about Lie Algebras are given in the second paragraph. The structure constants of a Lie algebra are also given in this paragraph and also some relations between them. Finally it contains the determination of a Lie Algebra by constant structure and conversely. The third paragraph includes mappings between Lie Algebras. The notions of homomorphic and isomorphic Lie Algebras are defined by these mappings. The definitions of subalgebras and ideals of Lie Algebras are given in the fourth paragraph. It also contains some of their properties. Finally it has the notion of quotient Lie Algebra. The derivations of a Lie Algebra are contain in the fifth paragraph. It also contains some of their properties. The sixth paragraph includes some basic subsets of Lie Algebra. These basic sets play an important role in the theory of Lie Algebras. From a Lie Algebra g we can form sequences of ideals of g. Two basic ideals are the central sequence and the derived sequence. These are in the seventh paragraph. The eighth paragraph contains some elements of solvable Lie Algebras. Some elements of Nilpotent Lie Algebras are included in the ninth paragraph. The tenth paragraph contains basic elements of simple and semi-simple Lie Algebras. Finally the problem of classification of Lie Algebras is included in the last paragraph. The purpose of the second chapter is to study some properties of Nilpotent Lie Algebras of dimension eight. The whole chapter contains three paragraphs; each of them is analyzed as follows. The first paragraph describes the maximum abelian ideals of a Nilpotent Lie Algebra. The Nilpotent Lie Algebras of dimension eight are studied in the second paragraph. It is given their separation in categories according to the number of parameters, which have the none zero Lie brackets. Special categories of Nilpotent Lie Algebras of dimension eight are determined in the third paragraph. Furthermore some basic problems are studied for which we have some solutions. One of them is to determine a Nilpotent Lie Algebra of dimension eight which has two maximum abelian ideals of different dimension. The answer to this problem is negative, that mean there exists no such Lie Algebra of dimension eight, which has two maximum abelian ideals of different dimension. In this paragraph is also given the theory of extension of a Nilpotent Lie Algebra of bigger dimensions. The third chapter contains the study of Nilpotent Lie Algebras of dimension eight which are characteristically Nilpotent for all the parameters. Another category of Nilpotent Lie Algebras is determined which is characteristically Nilpotent for special values of parameters. The chapter has two paragraphs. The first paragraph gives special elements for characteristically Nilpotent Lie Algebras, which are necessary for the next paragraph. In the second paragraph we determine the category of Nilpotent Lie Algebras of dimension eight which are characteristically Nilpotent. We also determine other such Nilpotent Lie Algebras of dimension eight for special values of the parameters.

scholarly journals Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1455
Author(s):  
Alina Dobrogowska ◽  
Karolina Wojciechowicz
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Poisson Structures ◽  
Triangular Matrices ◽  
Dual Spaces ◽  
Upper Triangular Matrices ◽  
Low Dimensional ◽  
Algebra Level

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

Central automorphisms of free nilpotent Lie algebras

2017 ◽  
Vol 16 (11) ◽  
pp. 1750205
Author(s):  
Özge Öztekin ◽  
Naime Ekici
Keyword(s):  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Sufficient Conditions ◽  
Nilpotency Class ◽  
Nilpotent Lie Algebra ◽  
Central Automorphism ◽  
Nilpotent Lie Algebras ◽  
Central Automorphisms

Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

scholarly journals k-step nilpotent Lie algebras

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Michel Goze ◽  
Elisabeth Remm
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Deformation Process ◽  
Nilpotent Lie Algebras ◽  
Early Stages ◽  
Finite Dimensional ◽  
Contraction Process ◽  
Finite Dimensional Lie Algebras

AbstractThe classification of complex or real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example, the nilpotent Lie algebras are classified only up to dimension 7. Moreover, to recognize a given Lie algebra in the classification list is not so easy. In this work, we propose a different approach to this problem. We determine families for some fixed invariants and the classification follows by a deformation process or a contraction process. We focus on the case of 2- and 3-step nilpotent Lie algebras. We describe in both cases a deformation cohomology for this type of algebras and the algebras which are rigid with respect to this cohomology. Other

scholarly journals Pseudo-metric 2-step nilpotent Lie algebras

2018 ◽  
Vol 18 (2) ◽  
pp. 237-263 ◽  
Author(s):  
Christian Autenried ◽  
Kenro Furutani ◽  
Irina Markina ◽  
Alexander Vasiľev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Nilpotent Lie Algebra ◽  
Rational Structure ◽  
Lie Triple System ◽  
Nilpotent Lie Algebras ◽  
Lie Bracket ◽  
Orthogonal Lie Algebra ◽  
Structure Constants ◽  
Scalar Products

Abstract The metric approach to studying 2-step nilpotent Lie algebras by making use of non-degenerate scalar products is realised. We show that a 2-step nilpotent Lie algebra is isomorphic to its standard pseudo-metric form, that is a 2-step nilpotent Lie algebra endowed with some standard non-degenerate scalar product compatible with the Lie bracket. This choice of the standard pseudo-metric form allows us to study the isomorphism properties. If the elements of the centre of the standard pseudo-metric form constitute a Lie triple system of the pseudo-orthogonal Lie algebra, then the original 2-step nilpotent Lie algebra admits integer structure constants. Among particular applications we prove that pseudo H-type algebras have bases with rational structure constants, which implies that the corresponding pseudo H-type groups admit lattices.

scholarly journals Almost inner derivations of Lie algebras

2018 ◽  
Vol 17 (11) ◽  
pp. 1850214
Author(s):  
Dietrich Burde ◽  
Karel Dekimpe ◽  
Bert Verbeke
Keyword(s):  
Lie Algebras ◽  
The Other ◽  
Large Space ◽  
Nilpotent Lie Algebras ◽  
Other Hand ◽  
Low Dimensional ◽  
Fixed Basis ◽  
Basis Vectors

We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for [Formula: see text]-step nilpotent Lie algebras determined by graphs, free [Formula: see text] and [Formula: see text]-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand, we also exhibit families of nilpotent Lie algebras having an arbitrary large space of non-inner almost inner derivations.

COMPUTING THE LAW OF A FAMILY OF SOLVABLE LIE ALGEBRAS

2009 ◽  
Vol 19 (03) ◽  
pp. 337-345 ◽  
Author(s):  
JUAN C. BENJUMEA ◽  
JUAN NÚÑEZ ◽  
ÁNGEL F. TENORIO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Computational Data ◽  
Solvable Lie Algebras ◽  
Final Output ◽  
The Matrix ◽  
The Law

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.

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