Some Properties of ID∗-n-Lie-derivations of Leibniz algebras

2021 ◽  
pp. 2250054
Author(s):  
G. R. Biyogmam ◽  
C. Tcheka ◽  
D. A. Kamgam
Keyword(s):  
Lie Algebra ◽  
Leibniz Algebra ◽  
Central Series ◽  
Algebra Structure ◽  
Leibniz Algebras ◽  
Lie Derivations

The concepts of [Formula: see text]-derivations and [Formula: see text]-central derivations have been recently presented in [G. R. Biyogmam and J. M. Casas, [Formula: see text]-central derivations, [Formula: see text]-centroids and [Formula: see text]-stem Leibniz algebras, Publ. Math. Debrecen 97(1–2) (2020) 217–239]. This paper studies the notions of [Formula: see text]-[Formula: see text]-derivation and [Formula: see text]-[Formula: see text]-central derivation on Leibniz algebras as generalizations of these concepts. It is shown that under some conditions, [Formula: see text]-[Formula: see text]-central derivations of a non-Lie-Leibniz algebra [Formula: see text] coincide with [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations, that is, [Formula: see text]-[Formula: see text]-derivations in which the image is contained in the [Formula: see text]th term of the lower [Formula: see text]-central series of [Formula: see text] and vanishes on the upper [Formula: see text]-central series of [Formula: see text] We prove some properties of these [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations. In particular, it is shown that the Lie algebra structure of the set of [Formula: see text]-[Formula: see text]-[Formula: see text]-derivations is preserved under [Formula: see text]-[Formula: see text]-isoclinism.

Lie triple systems and Leibniz algebras

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Revaz Kurdiani
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Triple System ◽  
Leibniz Algebra ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Lie Triple System ◽  
Triple Systems ◽  
Leibniz Algebras ◽  
Universal Central Extensions

AbstractThe present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra.

scholarly journals The second cohomology group of simple Leibniz algebras

2018 ◽  
Vol 17 (12) ◽  
pp. 1850222 ◽  
Author(s):  
J. Q. Adashev ◽  
M. Ladra ◽  
B. A. Omirov
Keyword(s):  
Lie Algebra ◽  
Cohomology Group ◽  
Leibniz Algebra ◽  
Leibniz Algebras ◽  
Leibniz Cohomology

In this paper, we prove some general results on Leibniz 2-cocyles for simple Leibniz algebras. Applying these results, we prove the triviality of the second Leibniz cohomology for a simple Leibniz algebra with coefficients in itself, whose associated Lie algebra is isomorphic to [Formula: see text].

scholarly journals On universal central extensions of Hom-Leibniz algebras

2014 ◽  
Vol 13 (08) ◽  
pp. 1450053 ◽  
Author(s):  
J. M. Casas ◽  
M. A. Insua ◽  
N. Pacheco Rego
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Chain Complex ◽  
Leibniz Algebra ◽  
Central Extensions ◽  
Leibniz Algebras ◽  
Leibniz Homology ◽  
Universal Central Extensions

In the category of Hom-Leibniz algebras we introduce the notion of Hom-co-representation as adequate coefficients to construct the chain complex from which we compute the Leibniz homology of Hom-Leibniz algebras. We study universal central extensions of Hom-Leibniz algebras and generalize some classical results, nevertheless it is necessary to introduce new notions of α-central extension, universal α-central extension and α-perfect Hom-Leibniz algebra due to the fact that the composition of two central extensions of Hom-Leibniz algebras is not central. We also provide the recognition criteria for these kind of universal central extensions. We prove that an α-perfect Hom-Lie algebra admits a universal α-central extension in the categories of Hom-Lie and Hom-Leibniz algebras and we obtain the relationships between both of them. In case α = Id we recover the corresponding results on universal central extensions of Leibniz algebras.

scholarly journals Some Theorems on Leibniz n-Algebras from the Category Un (Lb)

2021 ◽  
Vol 28 (01) ◽  
pp. 155-168
Author(s):  
Min Soo Kim ◽  
Rustam Turdibaev
Keyword(s):  
Lie Algebra ◽  
Leibniz Algebra ◽  
Simple Lie Algebra ◽  
Leibniz Algebras ◽  
Leibniz Formula

We study the Leibniz [Formula: see text]-algebra [Formula: see text], whose multiplication is defined via the bracket of a Leibniz algebra [Formula: see text] as [Formula: see text]. We show that [Formula: see text] is simple if and only if [Formula: see text] is a simple Lie algebra. An analog of Levi's theorem for Leibniz algebras in [Formula: see text] is established and it is proven that the Leibniz [Formula: see text]-kernel of [Formula: see text] for any semisimple Leibniz algebra [Formula: see text] is the [Formula: see text]-algebra [Formula: see text].

scholarly journals Isomorphisms of Twisted Hilbert Loop Algebras

2017 ◽  
Vol 69 (02) ◽  
pp. 453-480
Author(s):  
Timothée Marquis ◽  
Karl-Hermann Neeb
Keyword(s):  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Energy ◽  
Algebra Structure ◽  
Affine Lie Algebras ◽  
Subsequent Work ◽  
Highest Weight ◽  
Loop Algebras ◽  
Highest Weight Representations

Abstract The closest infinite-dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e., real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras , also called affinisations of . They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the 7 families for some infinite set J. To each of these types corresponds a “minimal ” affinisation of some simple Hilbert-Lie algebra , which we call standard. In this paper, we give for each affinisation g of a simple Hilbert-Lie algebra an explicit isomorphism from g to one of the standard affinisations of . The existence of such an isomorphism could also be derived from the classiffication of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists that is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of g. In subsequent work, this paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of .

scholarly journals Algorithm for Testing the Leibniz Algebra Structure

2009 ◽  
pp. 177-186 ◽  
Author(s):  
José Manuel Casas ◽  
Manuel A. Insua ◽  
Manuel Ladra ◽  
Susana Ladra
Keyword(s):  
Leibniz Algebra ◽  
Algebra Structure

scholarly journals Decompositions of algebras and post-associative algebra structures

2019 ◽  
Vol 30 (03) ◽  
pp. 451-466
Author(s):  
Dietrich Burde ◽  
Vsevolod Gubarev
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Algebra Structure ◽  
Associative Algebras ◽  
Simple Lie Algebra ◽  
Reductive Lie Algebra ◽  
Baxter Operator

We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

scholarly journals Solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum length

2019 ◽  
Vol 47 (4) ◽  
pp. 1578-1594
Author(s):  
Kobiljon K. Abdurasulov ◽  
Jobir Q. Adashev ◽  
José M. Casas ◽  
Bakhrom A. Omirov
Keyword(s):  
Maximum Length ◽  
Leibniz Algebra ◽  
Leibniz Algebras ◽  
Filiform Leibniz Algebra
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