scholarly journals From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets

2012 ◽  
Vol 62 (12) ◽  
pp. 2389-2400 ◽  
Author(s):  
Branislav Jurčo
Keyword(s):  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Differential Graded Lie Algebras

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 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 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 Cohomology jump loci of differential graded Lie algebras

2015 ◽  
Vol 151 (8) ◽  
pp. 1499-1528 ◽  
Author(s):  
Nero Budur ◽  
Botong Wang
Keyword(s):  
Lie Algebras ◽  
Vector Bundles ◽  
Fundamental Groups ◽  
Higgs Bundles ◽  
Infinitesimal Deformation ◽  
Graded Lie Algebras ◽  
Local Systems ◽  
Graded Lie Algebra ◽  
Differential Graded Lie Algebra ◽  
Differential Graded Lie Algebras

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

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