scholarly journals Some remarks on graded nilpotent Lie algebras and the Toral Rank Conjecture

2014 ◽  
Vol 14 (02) ◽  
pp. 1550024
Author(s):  
Guillermo Ames ◽  
Leandro Cagliero ◽  
Mónica Cruz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Algebraic Version ◽  
Finite Dimensional ◽  
Rank Conjecture ◽  
Algebra Over A Field ◽  
Toral Rank

If 𝔫 is a [Formula: see text]-graded nilpotent finite-dimensional Lie algebra over a field of characteristic zero, a well-known result of Deninger and Singhof states that dim H*(𝔫) ≥ L(p) where p is the polynomial associated to the grading and L(p) is the sum of the absolute values of the coefficients of p. From this result they derived the Toral Rank Conjecture (TRC) for 2-step nilpotent Lie algebras. An algebraic version of the TRC states that dim H*(𝔫) ≥ 2 dim (ℨ) for any finite-dimensional nilpotent Lie algebra 𝔫 with center ℨ. The TRC is more than 25 years old and remains open even for [Formula: see text]-graded 3-step nilpotent Lie algebras. Investigating to what extent the bound given by Deninger and Singhof could help to prove the TRC in this case, we considered the following two questions regarding a nilpotent Lie algebra 𝔫 with center ℨ: (A) If 𝔫 admits a [Formula: see text]-grading [Formula: see text], such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a ℤ+-grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? (B) If 𝔫 is r-step nilpotent admitting a grading 𝔫 = 𝔫1 ⊕ 𝔫2 ⊕ ⋯ ⊕ 𝔫k such that its associated polynomial p satisfies L(p) > 2 dim ℨ, does 𝔫 admit a grading [Formula: see text] such that its associated polynomial p′ satisfies L(p′) > 2 dim ℨ? In this paper we show that the answer to (A) is yes, but the answer to (B) is no.

scholarly journals On the cohomology of a class of nilpotent Lie algebras

1996 ◽  
Vol 54 (3) ◽  
pp. 517-527 ◽  
Author(s):  
Grant F. Armstrong ◽  
Stefan Sigg
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Betti Numbers ◽  
Jordan Decomposition ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Combinatorial Description ◽  
Finite Dimensional ◽  
Abelian Ideal

Let g denote a finite dimensional nilpotent Lie algebra over ℂ containing an Abelian ideal a of codimension 1, with z ∈ g/a. We give a combinatorial description of the Betti numbers of g in terms of the Jordan decomposition induced by ad(z)|a. As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.

scholarly journals On Marginal Automorphisms of Free Nilpotent Lie Algebras

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Özge Öztekin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Rank ◽  
Characteristic Zero ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras

Let L be the free nilpotent Lie algebra of finite rank over a field of characteristic zero. We define the concepts of marginal ideals and marginal automorphisms of L, and we give some results on marginal automorphisms.

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.

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].

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.

Euclidean Lie Algebras

1969 ◽  
Vol 21 ◽  
pp. 1432-1454 ◽  
Author(s):  
Robert V. Moody
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Cartan Matrix ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
Generalized Cartan Matrix

Our aim in this paper is to study a certain class of Lie algebras which arose naturally in (4). In (4), we showed that beginning with an indecomposable symmetrizable generalized Cartan matrix (A ij) and a field Φ of characteristic zero, we could construct a Lie algebra E((A ij)) over Φ patterned on the finite-dimensional split simple Lie algebras. We were able to show that E((A ij)) is simple providing that (A ij) does not fall in the list given in (4, Table). We did not prove the converse, however.The diagrams of the table of (4) appear in Table 2. Call the matrices that they represent Euclidean matrices and their corresponding algebras Euclidean Lie algebras. Our first objective is to show that Euclidean Lie algebras are not simple.

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 FAITHFUL REPRESENTATIONS OF MINIMAL DIMENSION OF CURRENT HEISENBERG LIE ALGEBRAS

2009 ◽  
Vol 20 (11) ◽  
pp. 1347-1362 ◽  
Author(s):  
LEANDRO CAGLIERO ◽  
NADINA ROJAS
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Polynomial Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Abelian Subalgebras ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Minimal Dimension

Given a Lie algebra 𝔤 over a field of characteristic zero k, let μ(𝔤) = min{dim π : π is a faithful representation of 𝔤}. Let 𝔥m be the Heisenberg Lie algebra of dimension 2m + 1 over k and let k [t] be the polynomial algebra in one variable. Given m ∈ ℕ and p ∈ k [t], let 𝔥m, p = 𝔥m ⊗ k [t]/(p) be the current Lie algebra associated to 𝔥m and k [t]/(p), where (p) is the principal ideal in k [t] generated by p. In this paper we prove that [Formula: see text]. We also prove a result that gives information about the structure of a commuting family of operators on a finite dimensional vector space. From it is derived the well-known theorem of Schur on maximal abelian subalgebras of 𝔤𝔩(n, k ).

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