scholarly journals Differential Graded Lie Algebras and Leibniz Algebra Cohomology

2020 ◽  
Author(s):  
Jacob Mostovoy
Keyword(s):  
Lie Algebras ◽  
Leibniz Algebra ◽  
Graded Lie Algebras ◽  
Leibniz Algebras ◽  
Leibniz Cohomology ◽  
Differential Graded Lie Algebras

Abstract In this note, we interpret Leibniz algebras as differential graded (DG) Lie algebras. Namely, we consider two fully faithful functors from the category of Leibniz algebras to that of DG Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley–Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in [ 9].

scholarly journals Duality Hierarchies and Differential Graded Lie Algebras

2021 ◽  
Vol 382 (1) ◽  
pp. 277-315
Author(s):  
Roberto Bonezzi ◽  
Olaf Hohm
Keyword(s):  
Lie Algebras ◽  
Scalar Fields ◽  
Vector Spaces ◽  
Graded Lie Algebras ◽  
Yang Mills ◽  
Leibniz Algebras ◽  
Duality Relations ◽  
Graded Lie Algebra ◽  
Differential Graded Lie Algebra ◽  
Differential Graded Lie Algebras

AbstractThe gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.

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].

Complete Differential Graded Lie Algebras

2020 ◽  
pp. 71-91
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo ◽  
Daniel Tanré
Keyword(s):  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Differential Graded Lie Algebras

scholarly journals On split regular Hom-Leibniz algebras

2018 ◽  
Vol 17 (10) ◽  
pp. 1850185 ◽  
Author(s):  
Yan Cao ◽  
Liangyun Chen ◽  
Bing Sun
Keyword(s):  
Lie Algebras ◽  
Natural Generalization ◽  
Leibniz Algebra ◽  
Maximal Length ◽  
Abelian Subalgebra ◽  
Leibniz Algebras

We introduce the class of split regular Hom-Leibniz algebras as the natural generalization of split Leibniz algebras and split regular Hom-Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz algebra [Formula: see text] is of the form [Formula: see text] with [Formula: see text] a subspace of the abelian subalgebra [Formula: see text] and any [Formula: see text], a well described ideal of [Formula: see text], satisfying [Formula: see text] if [Formula: see text]. Under certain conditions, in the case of [Formula: see text] being of maximal length, the simplicity of the algebra is characterized.

Non-degenerate Invariant Bilinear Forms on Leibniz Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 711-720
Author(s):  
Song Wang ◽  
Linsheng Zhu
Keyword(s):  
Lie Algebras ◽  
Bilinear Form ◽  
Leibniz Algebra ◽  
Bilinear Forms ◽  
Sufficient Condition ◽  
Invariant Bilinear Form ◽  
Leibniz Algebras ◽  
Double Extension

In this paper, we study Leibniz algebras [Formula: see text] with a non-degenerate Leibniz-symmetric [Formula: see text]-invariant bilinear form B, such a pair [Formula: see text] is called a quadratic Leibniz algebra. Our first result generalizes the notion of double extensions to quadratic Leibniz algebras. This notion was introduced by Medina and Revoy to study quadratic Lie algebras. In the second theorem, we give a sufficient condition for a quadratic Leibniz algebra to be a quadratic Leibniz algebra by double extension.

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