On the classification of abelian extensions
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 \(U{\mathfrak g}_+\) by \(M\). For \({\mathtt F}\), the author constructs an algebra structure on \(E\) and obtains therefore an underlying Lie bracket on \(E\) and on the pullback to \({\mathfrak g}\subset U{\mathfrak g}_+\). In Section 4, the author shows the compatibility of the functors \({\mathtt E}\) and \({\mathtt F}\) with the monoidal structure induced from the Baer sum on the two groupoids of extensions. The article is well-written and contains interesting new ideas.
scientific article; zbMATH DE number 6164908