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.

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 RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

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 SOME PROPERTIES ON THE TENSOR SQUARE OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250085 ◽  
Author(s):  
PEYMAN NIROOMAND
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Tensor Square

In the present paper we extend the results of [2, 4] for the tensor square of Lie algebras. More precisely, for any Lie algebra L with L/L2 of finite dimension, we prove L ⊗ L ≅ L □ L ⊕ L ∧ L and Z∧(L) ∩ L2 = Z⊗(L). Moreover, we show that L ∧ L is isomorphic to derived subalgebra of a cover of L, and finally we give a free presentation for it.

Some studies on central derivation of nilpotent Lie superalgebra

2018 ◽  
Vol 13 (04) ◽  
pp. 2050068
Author(s):  
Rudra Narayan Padhan ◽  
K. C. Pati
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Nilpotent Lie Algebra ◽  
New Concepts

Many theorems and formulas of Lie superalgebras run quite parallel to Lie algebras, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case as the later type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, Saeedi and Sheikh-Mohseni [A characterization of stem algebras in terms of central derivations, Algebr. Represent. Theory 20 (2017) 1143–1150; On [Formula: see text]-derivations of Filippov algebra, to appear in Asian-Eur. J. Math.; S. Sheikh-Mohseni, F. Saeedi and M. Badrkhani Asl, On special subalgebras of derivations of Lie algebras, Asian-Eur. J. Math. 8(2) (2015) 1550032] recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclinism in Lie superalgebra is being recently introduced.

Nilpotent fuzzy lie ideals

2020 ◽  
Vol 39 (3) ◽  
pp. 4071-4079
Author(s):  
E. Mohammadzadeh ◽  
G. Muhiuddin ◽  
J. Zhan ◽  
R.A. Borzooei
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Ideal ◽  
Nilpotent Lie Algebra

In this paper, we introduce a new definition for nilpotent fuzzy Lie ideal, which is a well-defined extension of nilpotent Lie ideal in Lie algebras, and we name it a good nilpotent fuzzy Lie ideal. Then we prove that a Lie algebra is nilpotent if and only if any fuzzy Lie ideal of it, is a good nilpotent fuzzy Lie ideal. In particular, we construct a nilpotent Lie algebra via a good nilpotent fuzzy Lie ideal. Also, we prove that with some conditions, every good nilpotent fuzzy Lie ideal is finite. Finally, we define an Engel fuzzy Lie ideal, and we show that every Engel fuzzy Lie ideal of a finite Lie algebra is a good nilpotent fuzzy Lie ideal. We think that these notions could be useful to solve some problems of Lie algebras with nilpotent fuzzy Lie ideals.

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.

Camina Lie algebras

2018 ◽  
Vol 11 (05) ◽  
pp. 1850063
Author(s):  
S. Sheikh-Mohseni ◽  
F. Saeedi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Finite Dimension ◽  
Proper Ideal ◽  
Closed Field ◽  
Algebraically Closed Field

Let [Formula: see text] be a Lie algebra and [Formula: see text] be a proper ideal of [Formula: see text]. Then [Formula: see text] is called a Camina pair if [Formula: see text] for all [Formula: see text]. Also, [Formula: see text] is called a Camina Lie algebra if [Formula: see text] is a Camina pair. In this paper, we give some properties of Camina Lie algebras. Moreover, we show that a nilpotent Camina Lie algebra of finite dimension over an algebraically closed field is nilpotent with nilindex at most [Formula: see text].

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 Certain functors of nilpotent Lie algebras with the derived subalgebra of dimension two

2019 ◽  
Vol 19 (01) ◽  
pp. 2050012
Author(s):  
Farangis Johari ◽  
Peyman Niroomand
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Nilpotent Lie Algebra ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Tensor Square ◽  
Exterior Square

By considering the nilpotent Lie algebra with the derived subalgebra of dimension [Formula: see text], we compute some functors including the Schur multiplier, the exterior square and the tensor square of these Lie algebras. We also give the corank of such Lie algebras.

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