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Cohomology of Leibniz Algebras
Jahresbericht der Deutschen Mathematiker-Vereinigung, 2023The paper under review is a survey of recent results on the cohomology of Leibniz algebras which are due to the author and the reviewer [J. Algebra 569, 276--317 (2021; Zbl 1465.17006); Indag. Math., New Ser. 35, No. 1, 87--113 (2024; Zbl 1543.17003)].
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Leibniz algebras in characteristic
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2001The paper under review presents a definition of a restricted Leibniz algebra \(Q\) in characteristic \(p\), and then presents a condition for the non-vanishing of the Leibniz cohomology of \(Q\). In particular, let \(k\) be an algebraically closed field of characteristic \(p > 0\), and let \(Q\) be a (left) Leibniz algebra over \(k\).
Dzhumadil'daev, Askar S. +1 more
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From Leibniz Algebras to Lie 2-algebras
Algebras and Representation Theory, 2015The authors construct a Lie 2-algebra associated to every Leibniz algebra via the skew-symmetrization.
Sheng, Yunhe, Liu, Zhangju
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VARIETIES OF METABELIAN LEIBNIZ ALGEBRAS
Journal of Algebra and Its Applications, 2002In this paper we commence the systematic study of T-ideals of the free Leibniz algebra or, equivalently, varieties of Leibniz algebras, over a field of characteristic 0. We give a description of the free metabelian (i.e. solvable of class 2) Leibniz algebras, a complete list of all left-nilpotent of class 2 varieties and the asymptotic description of ...
Drensky, Vesselin +1 more
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R-Matrices for Leibniz Algebras
Letters in Mathematical Physics, 2003The authors introduce \(R\)-matrices for Leibniz algebras as a direct generalization of the classical \(R\)-matrices. The linear mapping \(R_{\pm}: L\to L\) of a Leibniz algebra \(L\) is called an \(R_{\pm}\)-matrix if the new bilinear operator defined by \([X,Y]_{R_{\pm}}=[RX,Y]\pm [X,RY]\), \(X,Y\in L\), satisfies the Jacobi-Leibniz identity.
Felipe, Raúl +2 more
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Mathematical Notes, 2006
A Leibniz algebra \(L\) is said to be \textit{thin} if \(\dim(L^1/L^2)=2\) and \(\dim(L_i/L_{i+1})=1\) for all \(i\geq 2\). Here \(L^1=L\) and \(L^{n+1}=[L^n,L]\). The author proves that there are three classes of non-Lie thin Leibniz algebras.
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A Leibniz algebra \(L\) is said to be \textit{thin} if \(\dim(L^1/L^2)=2\) and \(\dim(L_i/L_{i+1})=1\) for all \(i\geq 2\). Here \(L^1=L\) and \(L^{n+1}=[L^n,L]\). The author proves that there are three classes of non-Lie thin Leibniz algebras.
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ON NILPOTENT LEIBNIZ n-ALGEBRAS
Journal of Algebra and Its Applications, 2012We study the nilpotency of Leibniz n-algebras related with the adapted version of Engel's theorem to Leibniz n-algebras. We also deal with the characterization of finite-dimensional nilpotent complex Leibniz n-algebras.
Camacho, L. M. +4 more
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2004
Leibniz algebras are possible non-(anti)commutative analogs of Lie algebras. These algebras have appeared in [55] under the name “D-algebras”. In [221, 222, 223] J.-L. Loday and T. Pirashvili studied these analogs from the point of view of homological algebra.
Alexander A. Mikhalev +2 more
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Leibniz algebras are possible non-(anti)commutative analogs of Lie algebras. These algebras have appeared in [55] under the name “D-algebras”. In [221, 222, 223] J.-L. Loday and T. Pirashvili studied these analogs from the point of view of homological algebra.
Alexander A. Mikhalev +2 more
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On some “minimal” Leibniz algebras
Journal of Algebra and Its Applications, 2017The aim of this paper is to describe some “minimal” Leibniz algebras, that are the Leibniz algebras whose proper subalgebras are Lie algebras, and the Leibniz algebras whose proper subalgebras are abelian.
Chupordia, V. A. +2 more
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Mathematical Notes
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