On Some Classes of Leibniz Algebras

Ayupov, Shavkat; Omirov, B. A.; Rakhimov, I. S.

doi:10.1201/9780429344336-4

Cited by 6 publications

(9 citation statements)

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“…Similarly to the case of nilpotent Leibniz algebras [2] it is easy to check that a Leibniz superalgebra is null-filiform if and only if it is single-generated. Moreover, a null-filiform superalgebra has the maximal nilindex.…”

Section: Definition 23 An N-dimensional Leibniz Superalgebra Is Said To Be Null-filiform If Dimmentioning

confidence: 99%

“…The most interesting examples of non-Lie Leibniz algebras are cyclic Leibniz algebras, those Leibniz algebras generated by one element and do not appear in Lie algebras, except one-dimensional abelian algebra. In nilpotent algebras case such algebras are called null-filiform Leibniz algebras [2], while solvable one-generated Leibniz algebras are called cyclic algebras [16].…”

Section: Introductionmentioning

confidence: 99%

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Complex cyclic Leibniz superalgebras

2020

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Since Loday introduction of Leibniz algebras as a generalisation of Lie algebras, many results of the theory of Lie algebras have been extended to Leibniz algebras. Cyclic Leibniz algebras, which are generated by one element, have no equivalent into Lie algebras, though. This fact provides cyclic Leibniz algebras with important properties. Throughout the present paper we extend the concept of cyclic to Leibniz superalgebras, obtaining then the definition, as long as the description and classification of finite-dimensional complex cyclic Leibniz superalgebras. Furthermore, we prove that any cyclic Leibniz superalgebra can be obtained by means of infinitesimal deformations of the null-filiform Leibniz superalgebra. We also obtain a description of irreducible components in the variety of Leibniz algebras and superalgebras.

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Section: Definition 23 An N-dimensional Leibniz Superalgebra Is Said To Be Null-filiform If Dimmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Complex cyclic Leibniz superalgebras

2020

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“…Since Loday's introduction of Leibniz algebras in 1993, many results of the theory of Lie algebras have been extended to Leibniz algebras. Nevertheless, a great deal of the results have been devoted to (co)homological problems [11,[16][17][18] or to the classification problems of nilpotent part and its subclasses [2,3,6,[20][21][22][23][24]. This is in contrast with the semisimple part of Leibniz algebras, which has been less studied.…”

Section: Introductionmentioning

confidence: 99%

On Leibniz Superalgebras with Even Part Corresponding to $\mathfrak {sl}_{2}$

Camacho

Navarro

2020

Algebr Represent Theor

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In this paper we describe finite-dimensional complex Leibniz superalgebras whose even part is the simple Leibniz algebra corresponding to sl 2 , i.e. its quotient algebra with respect to the Leibniz kernel I is isomorphic to sl 2 . We classify these Leibniz superalgebras in several cases with arbitrary dimensions in which the odd part is essentially a Leibniz irreducible (sl 2 I )-module or a finite direct sum of them.

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“…Since Leibniz algebras are a generalization of Lie algebras [30], consequently many of the features of Leibniz superalgebras are generalization of Lie superalgebras [3,11,12,19]. Likewise, the study of nilpotent Leibniz algebras [2,6,7] can be very useful to study nilpotent Leibniz superalgebras.…”

Section: Introductionmentioning

confidence: 99%

On Naturally Graded Lie and Leibniz Superalgebras

Camacho

Navarro

Sánchez

2019

Bull. Malays. Math. Sci. Soc.

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In general, the study of gradations has always represented a cornerstone in the study of non-associative algebras. In particular, natural gradation can be considered to be the first and most relevant gradation of nilpotent Leibniz (resp. Lie) algebras. In fact, many families of relevant solvable Leibniz (resp. Lie) algebras have been obtained by extensions of naturally graded algebras, i.e., solvable algebras with a wellstructured nilradical. Thus, the aim of this work is introducing the concept of natural gradation for Lie and Leibniz superalgebras. Moreover, after having defined naturally graded Lie and Leibniz superalgebras, we characterize natural gradations on a very important class of each of them, that is, those with maximal supernilindex.

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On Some Classes of Leibniz Algebras

Cited by 6 publications

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Complex cyclic Leibniz superalgebras

Complex cyclic Leibniz superalgebras

On Leibniz Superalgebras with Even Part Corresponding to $\mathfrak {sl}_{2}$

On Naturally Graded Lie and Leibniz Superalgebras

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