scholarly journals Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 465
Author(s):  
Javier de Lucas ◽  
Daniel Wysocki
Keyword(s):  
Vector Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Matrix Representations ◽  
Lie Bialgebras ◽  
Higher Dimensional ◽  
Trivial Center ◽  
Darboux Polynomial

This work introduces a new concept, the so-called Darboux family, which is employed to determine coboundary Lie bialgebras on real four-dimensional, indecomposable Lie algebras, as well as geometrically analysying, and classifying them up to Lie algebra automorphisms, in a relatively easy manner. The Darboux family notion can be considered as a generalisation of the Darboux polynomial for a vector field. The classification of r-matrices and solutions to classical Yang–Baxter equations for real four-dimensional indecomposable Lie algebras is also given in detail. Our methods can further be applied to general, even higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix representations of certain Lie algebras with a non-trivial center is developed.

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.

On Automorphisms and Derivations of a Lie Algebra

2006 ◽  
Vol 13 (01) ◽  
pp. 119-132 ◽  
Author(s):  
V. R. Varea ◽  
J. J. Varea
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Nilpotent Lie Algebra ◽  
Identity Component ◽  
Trivial Center

We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

scholarly journals On classification of (n+6)-dimensional nilpotent n-Lie algebras of class 2 with n≥4

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mehdi Jamshidi ◽  
Farshid Saeedi ◽  
Hamid Darabi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Design Methodology ◽  
Central Element ◽  
Complete Understanding ◽  
Content Type ◽  
Low Dimension

PurposeThe purpose of this paper is to determine the structure of nilpotent (n+6)-dimensional n-Lie algebras of class 2 when n≥4.Design/methodology/approachBy dividing a nilpotent (n+6)-dimensional n-Lie algebra of class 2 by a central element, the authors arrive to a nilpotent (n+5) dimensional n-Lie algebra of class 2. Given that the authors have the structure of nilpotent (n+5)-dimensional n-Lie algebras of class 2, the authors have access to the structure of the desired algebras.FindingsIn this paper, for each n≥4, the authors have found 24 nilpotent (n+6) dimensional n-Lie algebras of class 2. Of these, 15 are non-split algebras and the nine remaining algebras are written as direct additions of n-Lie algebras of low-dimension and abelian n-Lie algebras.Originality/valueThis classification of n-Lie algebras provides a complete understanding of these algebras that are used in algebraic studies.

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 Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

Symmetry ◽  
2019 ◽  
Vol 11 (11) ◽  
pp. 1354 ◽  
Author(s):  
Hassan Almusawa ◽  
Ryad Ghanam ◽  
Gerard Thompson
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Solvable Lie Groups ◽  
Related System ◽  
Geodesic Equations ◽  
Solvable Lie Algebras ◽  
Symmetry Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized.

scholarly journals LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION

2008 ◽  
Vol 10 (02) ◽  
pp. 221-260 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symmetric Solution ◽  
Baxter Equation ◽  
Symmetric Part ◽  
Lie Bialgebra ◽  
Lie Bialgebras ◽  
Poisson Lie Groups ◽  
Lagrangian Subalgebras

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

scholarly journals Rings with involution whose symmetric elements are central

1980 ◽  
Vol 3 (2) ◽  
pp. 247-253
Author(s):  
Taw Pin Lim
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
Semisimple Lie Algebra ◽  
Direct Sum ◽  
Rings With Involution ◽  
Torsion Free

In a ringRwith involution whose symmetric elementsSare central, the skew-symmetric elementsKform a Lie algebra over the commutative ringS. The classification of such rings which are2-torsion free is equivalent to the classification of Lie algebrasKoverSequipped with a bilinear formfthat is symmetric, invariant and satisfies[[x,y],z]=f(y,z)x−f(z,x)y. IfSis a field of char≠2,f≠0anddimK>1thenKis a semisimple Lie algebra if and only iffis nondegenerate. Moreover, the derived algebraK′is either the pure quaternions overSor a direct sum of mutually orthogonal abelian Lie ideals ofdim≤2.

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.

λ-Mappings Between Representation Rings of Lie Algebras

1983 ◽  
Vol 35 (5) ◽  
pp. 898-960 ◽  
Author(s):  
R. V. Moody ◽  
A. Pianzola
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Cartan Subalgebra ◽  
Complete Classification ◽  
Representation Rings ◽  
Mathematical Formalization ◽  
Intuitive Idea ◽  
Image Position

In [10] Patera and Sharp conceived a new relation, subjoining, between semisimple Lie algebras. Our objective in this paper is twofold. Firstly, to lay down a mathematical formalization of this concept for arbitrary Lie algebras. Secondly, to give a complete classification of all maximal subjoinings between Lie algebras of the same rank, of which many examples were already known to the above authors.The notion of subjoining is a generalization of the subalgebra relation between Lie algebras. To give an intuitive idea of what is involved we take a simple example. Suppose is a complex simple Lie algebra of type B2. Let be a Cartan subalgebra of and Δ the corresponding root system. We have the standard root diagramInside B2 there lies the subalgebra A1 × A1 which can be identified with the sum of and the root spaces corresponding to the long roots of B2.

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.

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