On the Derivation Algebras of Lie Algebras

LetLbe a Lie algebra over a field of characteristic 0 and letD(L)be the derivation algebra ofL, that is, the Lie algebra of all derivations ofL. Then it is natural to ask the following questions: What is the structure ofD(L)?What are the relations of the structures ofD(L)andL? It is the main purpose of this paper to present some results onD(L)as the answers to these questions in simple cases.Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebraLis not completely determined by the structure ofD(L). However, there is still some intimate connection between the structure ofD(L)and that ofL.

ON SOME EMBEDDING OF LIE ALGEBRAS

Journal of Algebra and Its Applications ◽

10.1142/s0219498811005439 ◽

2012 ◽

Vol 11 (01) ◽

pp. 1250017 ◽

Author(s):

E. CHIBRIKOV

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Free Generators ◽

In this paper we prove that every recursively presented Lie algebra over a field which is finite extention of its simple subfield can be embedded in a recursively presented Lie algebra defined by relations which are equalities of (nonassociative) words of generators and α + β = γ(α, β, γ are free generators).

Palindromes in the free metabelian Lie algebras

International Journal of Algebra and Computation ◽

10.1142/s0218196719500334 ◽

2019 ◽

Vol 29 (05) ◽

pp. 885-891

Author(s):

Şehmus Fındık ◽

Nazar Şahi̇n Öğüşlü

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Characteristic Zero ◽

Reverse Order ◽

Generating Set ◽

Linear Basis ◽

Free Metabelian Lie Algebra ◽

Definition Of ◽

A palindrome, in general, is a word in a fixed alphabet which is preserved when taken in reverse order. Let [Formula: see text] be the free metabelian Lie algebra over a field of characteristic zero generated by [Formula: see text]. We propose the following definition of palindromes in the setting of Lie algebras: An element [Formula: see text] is called a palindrome if it is preserved under the change of generators; i.e. [Formula: see text]. We give a linear basis and an explicit infinite generating set for the Lie subalgebra of palindromes.

Outer Derivations and Classical-Albert-Zassenhaus lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1984-054-7 ◽

1984 ◽

Vol 36 (6) ◽

pp. 961-972 ◽

Author(s):

David J. Winter

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Maximal Torus ◽

Cartan Subalgebra ◽

One Dimensional ◽

Derivation Algebra ◽

Image Position

This paper is concerned with the structure of the derivation algebra Der L of the Lie algebra L with split Cartan subalgebra H. The Fitting decompositionof Der L with respect to ad ad H leads to a decompositionwhereThis decomposition is studied in detail in Section 2, where the centralizer of ad L∞ in D0(H) is shown to bewhich is Hom(L/L2, Center L) when H is Abelian. When the root-spaces La (a nonzero) are one-dimensional, this leads to the decomposition of Der L aswhere T is any maximal torus of D0(H).

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

Journal of Algebra and Its Applications ◽

10.1142/s0219498816500298 ◽

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 ◽

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.

Subalgebras of free restricted Lie algebras

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700034936 ◽

2005 ◽

Vol 72 (1) ◽

pp. 147-156 ◽

Cited By ~ 4

Author(s):

R.M. Bryant ◽

L.G. Kovács ◽

Ralph Stöhr

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Main Step ◽

Free Lie Algebra ◽

Generating Set ◽

Restricted Lie Algebra ◽

Corrected Proof ◽

Restricted Lie Algebras ◽

Remarkable Result ◽

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshov's proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshov's argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

On the Dimension of Non-Abelian Tensor Squares of $n$-Lie Algebras

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.52.2021.3373 ◽

2021 ◽

Vol 52 ◽

Author(s):

Farshid Saeedi ◽

Nafiseh Akbarossadat

Keyword(s):

Group Theory ◽

Exact Sequence ◽

Lie Algebra ◽

Lie Algebras ◽

Short Exact Sequence ◽

Upper Bound ◽

Tensor Square ◽

Let $L$ be an $n$-Lie algebra over a field $\F$. In this paper, we introduce the notion of non-abelian tensor square $L\otimes L$ of $L$ and define the central ideal $L\square L$ of it. Using techniques from group theory and Lie algebras, we show that that $L\square L\cong L^{ab}\square L^{ab}$. Also, we establish the short exact sequence\[0\lra\M(L)\lra\frac{L\otimes L}{L\square L}\lra L^2\lra0\]and apply it to compute an upper bound for the dimension of non-abelian tensor square of $L$.

Uniqueness theorems for left-symmetric enveloping algebras

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410007479x ◽

1996 ◽

Vol 120 (2) ◽

pp. 193-206

Author(s):

J. R. Bolgar

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Characteristic Zero ◽

Automorphism Groups ◽

Uniqueness Theorems ◽

Enveloping Algebras ◽

Enveloping Algebra ◽

Uniqueness Results ◽

Symmetric Algebras ◽

AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

The Second Cohomology of Current Algebras of General Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-2008-038-6 ◽

2008 ◽

Vol 60 (4) ◽

pp. 892-922 ◽

Cited By ~ 11

Author(s):

Karl-Hermann Neeb ◽

Friedrich Wagemann

Keyword(s):

Vector Space ◽

Lie Algebra ◽

Lie Algebras ◽

Associative Algebra ◽

Characteristic Zero ◽

Cohomology Space ◽

Trivial Module ◽

Commutative Associative Algebra ◽

Locally Convex ◽

AbstractLet A be a unital commutative associative algebra over a field of characteristic zero, a Lie algebra, and a vector space, considered as a trivial module of the Lie algebra . In this paper, we give a description of the cohomology space in terms of easily accessible data associated with A and . We also discuss the topological situation, where A and are locally convex algebras.

Invariant Symmetric Bilinear Forms and Derivation Algebras of Unitary Lie Algebras

Algebra Colloquium ◽

10.1142/s1005386716000377 ◽

2016 ◽

Vol 23 (03) ◽

pp. 361-384 ◽

Author(s):

Yelong Zheng ◽

Jiwen Gao ◽

Zhihua Chang ◽

Yun Gao

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Associative Algebra ◽

Bilinear Forms ◽

Special Linear ◽

Invariant symmetric bilinear forms and derivation algebras of a unitary Lie algebra L over R are characterized: ℬ(L) ≅ (R+/([R,R] ∩ R+))* and Der (L) = Inn (L)+ Der (L)0 = Inn (L)+ SDer (R), which recover what of the special linear Lie algebra and Steinberg Lie algebra over R, where R is a unital involutory associative algebra over a field 𝔽.

ON A FORMULA FOR THE PI-EXPONENT OF LIE ALGEBRAS

Journal of Algebra and Its Applications ◽

10.1142/s0219498813500692 ◽

2013 ◽

Vol 13 (01) ◽

pp. 1350069 ◽

Author(s):

A. S. GORDIENKO

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Simple Formula ◽

Natural Generalization ◽

Rational Action ◽

Semisimple Lie Algebra ◽

Arbitrary Group ◽

Solvable Radical ◽

Finite Dimensional ◽

We prove that one of the conditions in Zaicev's formula for the PI-exponent and in its natural generalization for the Hopf PI-exponent, can be weakened. Using the modification of the formula, we prove that if a finite-dimensional semisimple Lie algebra acts by derivations on a finite-dimensional Lie algebra over a field of characteristic 0, then the differential PI-exponent coincides with the ordinary one. Analogously, the exponent of polynomial G-identities of a finite-dimensional Lie algebra with a rational action of a connected reductive affine algebraic group G by automorphisms, coincides with the ordinary PI-exponent. In addition, we provide a simple formula for the Hopf PI-exponent and prove the existence of the Hopf PI-exponent itself for H-module Lie algebras whose solvable radical is nilpotent, assuming only the H-invariance of the radical, i.e. under weaker assumptions on the H-action, than in the general case. As a consequence, we show that the analog of Amitsur's conjecture holds for G-codimensions of all finite-dimensional Lie G-algebras whose solvable radical is nilpotent, for an arbitrary group G.