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

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

2007 ◽  
Vol 17 (03) ◽  
pp. 527-555 ◽  
Author(s):  
YOU'AN CAO ◽  
DEZHI JIANG ◽  
JUNYING WANG
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Nilpotent Lie Algebras ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Positive Roots ◽  
Nilpotent Subalgebra

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

ON THETA PAIRS FOR A MAXIMAL SUBALGEBRA

2012 ◽  
Vol 11 (01) ◽  
pp. 1250001 ◽  
Author(s):  
ALI REZA SALEMKAR ◽  
SARA CHEHRAZI ◽  
SOMAIEH ALIZADEH NIRI
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Nonzero Ideal ◽  
Maximal Subgroups ◽  
Maximal Subalgebra ◽  
Nilpotent Lie Algebras ◽  
Finite Dimensional ◽  
A Finite Group ◽  
Number Of Authors

Given a maximal subalgebra M of a finite-dimensional Lie algebra L, a θ-pair for M is a pair (A, B) of subalgebras such that A ≰ M, B is an ideal of L contained in A ∩ M, and A/B includes properly no nonzero ideal of L/B. This is analogous to the concept of θ-pairs associated to maximal subgroups of a finite group, which has been studied by a number of authors. A θ-pair (A, B) for M is said to be maximal if M has no θ-pair (C, D) such that A < C. In this paper, we obtain some properties of maximal θ-pairs and use them to give some characterizations of solvable, supersolvable and nilpotent Lie algebras.

scholarly journals Almost nilpotent Lie algebras

1987 ◽  
Vol 29 (1) ◽  
pp. 7-11 ◽  
Author(s):  
David A. Towers
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebras ◽  
Semisimple Lie Algebras ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

Throughout we shall consider only finite-dimensional Lie algebras over a field of characteristic zero. In [3] it was shown that the classes of solvable and of supersolvable Lie algebras of dimension greater than two are characterised by the structure of their subalgebra lattices. The same is true of the classes of simple and of semisimple Lie algebras of dimension greater than three. However, it is not true of the class of nilpotent Lie algebras. We seek here the smallest class containing all nilpotent Lie algebras which is so characterised.

scholarly journals Coclosed G2-structures inducing nilsolitons

2018 ◽  
Vol 30 (1) ◽  
pp. 109-128 ◽  
Author(s):  
Leonardo Bagaglini ◽  
Marisa Fernández ◽  
Anna Fino
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Nilpotent Lie Algebras ◽  
Almost Complex ◽  
Metric Structures ◽  
Trivial Center ◽  
Nilsoliton Metric

Abstract We show obstructions to the existence of a coclosed {\mathrm{G}_{2}} -structure on a Lie algebra {\mathfrak{g}} of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from {\mathfrak{g}} to a six-dimensional Lie algebra {\mathfrak{h}} , with the kernel contained in the center of {\mathfrak{g}} , then any coclosed {\mathrm{G}_{2}} -structure on {\mathfrak{g}} induces a closed and stable three form on {\mathfrak{h}} that defines an almost complex structure on {\mathfrak{h}} . As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed {\mathrm{G}_{2}} -structures. We also prove that each one of these Lie algebras has a coclosed {\mathrm{G}_{2}} -structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed {\mathrm{G}_{2}} -structures. The existence of contact metric structures is also studied.

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.

scholarly journals Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

2015 ◽  
Vol 15 (02) ◽  
pp. 1650029 ◽  
Author(s):  
Leandro Cagliero ◽  
Fernando Szechtman
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Weight Space ◽  
Arbitrary Field ◽  
Nilpotent Radical ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
A Chain ◽  
Algebra Over A Field

Let 𝔤 be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical 𝔯 and nilpotent radical 𝔫 = [𝔤, 𝔯]. Given a finite-dimensional 𝔤-module U, its nilpotency series 0 ⊂ U(1) ⊂ ⋯ ⊂ U(m) = U is defined so that U(1) is the 0-weight space of 𝔫 in U, U(2)/U(1) is the 0-weight space of 𝔫 in U/U(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial 𝔤/𝔫-module, i.e. its 𝔤/𝔫-submodules form a chain. Every uniserial 𝔤-module is linked, every linked 𝔤-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked 𝔤-modules when 𝔤 = 〈x〉 ⋉ 𝔞 and ad x acts diagonalizably on the abelian Lie algebra 𝔞. Moreover, we identify and classify all uniserial 𝔤-modules amongst them.

Lie algebras whose maximal subalgebras are modular

1983 ◽  
Vol 94 (1-2) ◽  
pp. 9-13 ◽  
Author(s):  
V. R. Varea
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Maximal Subalgebras ◽  
Modular Element ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

SynopsisA subalgebra M of a Lie algebra L is called modular in L if M is a modular element in the lattice of the subalgebras of L. Our aim is to study the finite-dimensional Lie algebras all of whose maximal subalgebras are modular. We characterize these algebras over any field of characteristic zero.

scholarly journals MULTISTRING VERTICES AND HYPERBOLIC KAC-MOODY ALGEBRAS

1996 ◽  
Vol 11 (03) ◽  
pp. 429-514 ◽  
Author(s):  
R.W. GEBERT ◽  
H. NICOLAI ◽  
P.C. WEST
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complete Information ◽  
Liouville Theory ◽  
Root Lattice ◽  
Root Space ◽  
Respective Group ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Finite Dimensional Lie Algebras

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac-Moody algebras, in particular E10. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds more generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain “decoupling polynomials.” This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a nontrivial root space of E10· Because the N-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as E10 by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras.

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.

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