scholarly journals On Some Structural Components of Nilsolitons

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Hulya Kadioglu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Sufficient Conditions ◽  
Structural Components ◽  
Nilpotent Lie Algebras ◽  
Structure Constants ◽  
Nilsoliton Metric

In this paper, we study nilpotent Lie algebras that admit nilsoliton metric with simple pre-Einstein derivation. Given a Lie algebra η , we would like to compute as much of its structure as possible. The structural components we consider in this study are the structure constants, the index, and the rank of the nilsoliton derivations. For this purpose, we prove necessary or sufficient conditions for an algebra to admit such metrics. Particularly, we prove theorems for the computation of the Jacobi identity for a given algebra so that we can solve the system of the equation(s) and find the structure constants of the nilsoliton.

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 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.

scholarly journals Post-Lie algebra structures for nilpotent Lie algebras

2018 ◽  
Vol 28 (05) ◽  
pp. 915-933
Author(s):  
Dietrich Burde ◽  
Christof Ender ◽  
Wolfgang Alexander Moens
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Necessary And Sufficient

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

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 Hom-structures on semi-simple Lie algebras

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Wenjuan Xie ◽  
Quanqin Jin ◽  
Wende Liu
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Jacobi Identity ◽  
Algebra Homomorphism ◽  
Closed Field ◽  
Simple Lie Algebras ◽  
3 Dimensional ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Reduction Methods

AbstractA Hom-structure on a Lie algebra (g,[,]) is a linear map σ W g σ g which satisfies the Hom-Jacobi identity: [σ(x), [y,z]] + [σ(y), [z,x]] + [σ(z),[x,y]] = 0 for all x; y; z ∈ g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. This paper aims to determine explicitly all the Homstructures on the finite-dimensional semi-simple Lie algebras over an algebraically closed field of characteristic zero. As a Hom-structure on a Lie algebra is not necessarily a Lie algebra homomorphism, the method developed for multiplicative Hom-structures by Jin and Li in [J. Algebra 319 (2008): 1398–1408] does not work again in our case. The critical technique used in this paper, which is completely different from that in [J. Algebra 319 (2008): 1398– 1408], is that we characterize the Hom-structures on a semi-simple Lie algebra g by introducing certain reduction methods and using the software GAP. The results not only improve the earlier ones in [J. Algebra 319 (2008): 1398– 1408], but also correct an error in the conclusion for the 3-dimensional simple Lie algebra sl2. In particular, we find an interesting fact that all the Hom-structures on sl2 constitute a 6-dimensional Jordan algebra in the usual way.

scholarly journals Graded modules over simple Lie algebras

2019 ◽  
Vol 53 (supl) ◽  
pp. 45-86
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov ◽  
Abdallah Shihadeh
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Simple Lie Algebras ◽  
Simple Modules ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Necessary And Sufficient

The paper is devoted to the study of graded-simple modules and gradings on simple modules over finite-dimensional simple Lie algebras. In general, a connection between these two objects is given by the so-called loop construction. We review the main features of this construction as well as necessary and sufficient conditions under which finite-dimensional simple modules can be graded. Over the Lie algebra sl2(C), we consider specific gradings on simple modules of arbitrary dimension.

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.

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