On the classification of abelian extensions
type: Item
instance of: scholarly article
title: On the classification of abelian extensions
zbMATH Open document ID: 1293.17023
author: Paul Bressler
published in: São Paulo Journal of Mathematical Sciences
publication date: 2013-05-22T00:00:00Z
review text: The article sheds some new light on the theory of abelian extensions of Lie algebras. Namely, it is well known that the set \(\text{AbExt}({\mathfrak g},M)\) of equivalence classes of abelian extensions \[ 0\to M\overset{i} \rightarrow E\overset{\pi}\rightarrow{\mathfrak g} 0 \] of a Lie algebra \({\mathfrak g}\) by a \({\mathfrak g}\)-module \(M\) is in bijection to the cohomology space \(H^2({\mathfrak g},M)\). Instead of Lie algebras over a field \(k\), the author places himself in a strict symmetric monoidal closed abelian category linear over a field \(k\) of characteristic zero admitting certain colimits. The goal of the article is twofold: lift the above bijection to groupoid level (\(\text{AbExt}({\mathfrak g},M)\) is a groupoid !), and get rid of the non-canonical choice of a section which usually serves to establish this bijection. For this, the author replaces \[ H^2({\mathfrak g},M)\,=\,\text{Ext}^2_{U{\mathfrak g}}(k,M)\,=\, \text{Ext}^1_{U{\mathfrak g}}(U{\mathfrak g}_+,M), \] (where the last isomorphism follows using the augmentation sequence \[ 0\to U{\mathfrak g}_+\to U{\mathfrak g}\to k\to 0) \] by the groupoid \({\mathbf E}{\mathbf X}{\mathbf T}_{U{\mathfrak g}}(U{\mathfrak g}_+,M)\) of extensions of the type \[ 0\to M\overset{\kappa}\rightarrow E\overset{\sigma}\rightarrow U{\mathfrak g}_{+} 0. \] The main result is then an equivalence of categories of \(\text{AbExt}({\mathfrak g},M)\) and \({\mathbf E}{\mathbf X}{\mathbf T}_{U{\mathfrak g}}(U{\mathfrak g}_+,M)\). The functors \[ {\mathtt E}:\text{AbExt}({\mathfrak g},M){\gets} {\mathbf E}{\mathbf X}{\mathbf T}_{U{\mathfrak g}}(U{\mathfrak g}_+,M) :{\mathtt F} \] inducing this equivalence are not so well-known: the functor \({\mathtt E}\) takes the cone of \(i\) which is a differential graded Lie algebra \({\mathfrak G}\) such that \(U{\mathfrak G}\) is a resolution of \(U{\mathfrak g}\). Chopping it into short exact sequences, and identifying a morphism to \(M\), one pushes out an extension of \…
reviewed by: Friedrich Wagemann
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18G15
Mathematics Subject Classification ID: 18G50
zbMATH DE Number: 6164908
zbMATH Keywords: abelian extensions of Lie algebras
zbMATH Keywords: groupoid of extensions
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: augmentation ideal
zbMATH Keywords: monoidal category
MaRDI profile type: Publication
Wikidata QID: Q121769872
full work available at URL: issn.2316-9028.v6i1p71-80
OpenAlex ID: W2008633698
DOI: 10.11606/ISSN.2316-9028.V6I1P71-80
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description: scientific article; zbMATH DE number 6164908