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.

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.

Parabolic Higgs Bundles for Real Reductive Lie Groups

2018 ◽  
pp. 653-680 ◽  
Author(s):  
Ignasi Mundet i Riera
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Lie Group ◽  
Vector Bundles ◽  
Structure Group ◽  
Higgs Bundles ◽  
Local Systems ◽  
Parabolic Structure ◽  
Reductive Lie Groups ◽  
Reductive Lie Group

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

ON THE STRUCTURE OF SOLUBLE GRADED LIE ALGEBRAS

2011 ◽  
Vol 10 (04) ◽  
pp. 597-604 ◽  
Author(s):  
PAVEL SHUMYATSKY ◽  
CARMELA SICA
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Graded Lie Algebras ◽  
Derived Length ◽  
Elementary Group ◽  
Graded Lie Algebra

Let A be the elementary group of order 2n and L an A-graded Lie algebra with L0 = 0. Assume that L is soluble with derived length k. It is proved that L has a series of ideals of length n all of whose quotients are nilpotent of {k, n}-bounded class.

scholarly journals Maurer–Cartan Elements in the Lie Models of Finite Simplicial Complexes

2017 ◽  
Vol 60 (3) ◽  
pp. 470-477 ◽  
Author(s):  
Urtzi Buijs ◽  
Yves Félix ◽  
Aniceto Murillo ◽  
Daniel Tanré
Keyword(s):  
Simplicial Complex ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simplicial Complexes ◽  
Connected Components ◽  
Homotopy Groups ◽  
Graded Lie Algebras ◽  
Simplicial Sets ◽  
Graded Lie Algebra ◽  
Finite Simplicial Complex

AbstractIn a previous work, we associated a complete diòerential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we also have a realization functor fromthe category of complete diòerential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

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

On certain subgroups of the McCool group

2020 ◽  
Vol 30 (05) ◽  
pp. 1081-1096
Author(s):  
C. E. Kofinas
Keyword(s):  
Automorphism Group ◽  
Positive Integer ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Free Group ◽  
Graded Lie Algebras ◽  
Natural Embedding ◽  
Inner Automorphism ◽  
Graded Lie Algebra

For a positive integer [Formula: see text], with [Formula: see text], let [Formula: see text] be a free group of rank [Formula: see text] and let [Formula: see text] be the subgroup of the automorphism group of [Formula: see text] consisting of all automorphisms which induce the identity on the abelianization of [Formula: see text]. We write [Formula: see text] and [Formula: see text] for the upper McCool group and the partial inner automorphism group, respectively. We show that [Formula: see text] is isomorphic to the quotient of [Formula: see text] by its center and we prove similar results for their associated graded Lie algebras and their Andreadakis–Johnson Lie algebras. Furthermore, we give a presentation of the associated graded Lie algebra over the integers of [Formula: see text] and we prove that it admits a natural embedding into the Andreadakis–Johnson Lie algebra of [Formula: see text]. Although the latter results are known, we present proofs based on different methods.

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