scholarly journals On the influence of ideals and self-idealizing subalgebras on the structure of Leibniz algebras

2021 ◽  
pp. 12-17
Author(s):  
L.A. Kurdachenko ◽  
A.A. Pypka ◽  
I.Ya. Subbotin
Keyword(s):  
Leibniz Algebra ◽  
Nilpotent Radical ◽  
Leibniz Algebras ◽  
Locally Nilpotent ◽  
Dimension 2

The subalgebra A of a Leibniz algebra L is self-idealizing in L, if A = IL (A) . In this paper we study the structure of Leibniz algebras, whose subalgebras are either ideals or self-idealizing. More precisely, we obtain a description of such Leibniz algebras for the cases where the locally nilpotent radical is Abelian non-cyclic, non-Abelian noncyclic, and cyclic of dimension 2.

Finite-dimensional Leibniz algebras and combinatorial structures

2017 ◽  
Vol 20 (01) ◽  
pp. 1750004 ◽  
Author(s):  
M. Ceballos ◽  
J. Núñez ◽  
Á. F. Tenorio
Keyword(s):  
Leibniz Algebra ◽  
Combinatorial Structure ◽  
Combinatorial Structures ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Dimension 2

Given a finite-dimensional Leibniz algebra with certain basis, we show how to associate such algebra with a combinatorial structure of dimension 2. In some particular cases, this structure can be reduced to a digraph or a pseudodigraph. In this paper, we study some theoretical properties about this association and we determine the type of Leibniz algebra associated to each of them.

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.

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

A locally nilpotent radical of nonassociative rings

1971 ◽  
Vol 10 (4) ◽  
pp. 219-224 ◽  
Author(s):  
G. V. Dorofeev
Keyword(s):  
Nilpotent Radical ◽  
Locally Nilpotent

scholarly journals Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

2021 ◽  
Author(s):  
Lucio Centrone ◽  
Chia Zargeh
Keyword(s):  
Hopf Algebra ◽  
Free Algebra ◽  
Hopf Algebras ◽  
Leibniz Algebra ◽  
Module Structure ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Filiform Leibniz Algebra ◽  
Algebra Over A Field ◽  
Cocommutative Hopf Algebra

AbstractLet L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

scholarly journals On a locally nilpotent radical Jacobson for special Lie algebras

2021 ◽  
Vol 22 (1) ◽  
pp. 234-272
Author(s):  
Olga Alexandrovna Pikhtilkova ◽  
Elena Vladimirovna Mescherina ◽  
Anna Nikolaevna Blagovisnava ◽  
Elena Vladislavovna Pronina ◽  
Olga Alekseevna Evseeva
Keyword(s):  
Lie Algebras ◽  
Nilpotent Radical ◽  
Locally Nilpotent

Classical Radicals of Invariants of Algebraic Skew Derivations

2022 ◽  
Vol 29 (01) ◽  
pp. 53-66
Author(s):  
Jeffrey Bergen ◽  
Piotr Grzeszczuk
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Nilpotent Radical ◽  
Enveloping Algebra ◽  
Locally Nilpotent ◽  
Nil Radical

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

scholarly journals Racks, Leibniz algebras and Yetter–Drinfel'd modules

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Ulrich Krähmer ◽  
Friedrich Wagemann
Keyword(s):  
Hopf Algebra ◽  
Leibniz Algebra ◽  
Unified Framework ◽  
Leibniz Algebras ◽  
Linear Maps

AbstractA Hopf algebra object in Loday and Pirashvili's category of linear maps entails an ordinary Hopf algebra and a Yetter–Drinfel'd module. We equip the latter with a structure of a braided Leibniz algebra. This provides a unified framework for examples of racks in the category of coalgebras discussed recently by Carter, Crans, Elhamdadi and Saito.

scholarly journals A study of n-Lie-isoclinic Leibniz algebras

2019 ◽  
Vol 19 (01) ◽  
pp. 2050013
Author(s):  
G. R. Biyogmam ◽  
J. M. Casas
Keyword(s):  
Leibniz Algebra ◽  
Leibniz Algebras

In this paper, we introduce the concept of [Formula: see text]-[Formula: see text]-isoclinism on non-Lie Leibniz algebras. Among the results obtained, we provide several characterizations of [Formula: see text]-[Formula: see text]-isoclinic classes of Leibniz algebras. Also, we provide a characterization of [Formula: see text]-[Formula: see text]-stem Leibniz algebras, and prove that every [Formula: see text]-[Formula: see text]-isoclinic class of Leibniz algebras contains a [Formula: see text]-[Formula: see text]-stem Leibniz algebra.

Sign in / Sign up

Export Citation Format

Share Document