The article adapts material which has been introduced in the framework of associative algebras in [H.-J. Baues and E. G. Minian, Homology Homotopy Appl. 4, No. 2(1), 63–82 (2002; Zbl 1004.18012)] to Lie- and Leibniz algebras. Namely, an \(n\)-fold crossed extension of Lie algebras is an exact sequence of vector spaces of \((n+2)\)-terms such that the right-most term is a Lie algebra \({\mathfrak g}\) which is the cokernel of a crossed module (given by the last-but-first map) and all other \((n-1)\)-terms are \({\mathfrak g}\)-modules (and the maps are morphisms of \({\mathfrak g}\)-modules). This kind of exact sequence can be viewed as a splicing together (or Yoneda product) of a crossed module with cokernel \({\mathfrak g}\) and a long exact sequence of \({\mathfrak g}\)-modules. The main theorem is that the set of equivalence classes of such \(n\)-fold crossed extensions carries the structure of an abelian group which is isomorphic to the Chevalley-Eilenberg cohomology space \(H^{n+1}({\mathfrak g},M)\) (where \(M\) is the left-most \({\mathfrak g}\)-module in the long exact sequence). A similar result is shown for Leibniz algebras.

The theorems of Baues-Minian and Das are “versions in degree three” of a theorem of M. Gerstenhaber [Proc. Natl. Acad. Sci. USA 51, 626–629 (1964; Zbl 0166.04205)] stating that the abelian group of equivalence classes of exact sequences obtained from splicing together long exact sequences and square-zero/abelian extensions is isomorphic to usual cohomology. Gerstenhaber shows this (omitting the details) for equationally defined algebras, including associative algebras and Lie algebras. In principle, the set (or abelian group) of equivalence classes of exact sequences where the right-most terms represent \(2\)- or \(3\)-cohomology classes (as for square-zero/abelian extensions or crossed modules) is isomorphic to usual cohomology because the long exact sequence of modules shifts only the cohomological degree. It would be interesting to explore this also for \(1\)- or \(0\)-cohomology classes.

More formally, one shows that the set of equivalence classes of \(n\)-fold crossed extensions constitutes a delta-functor which vanishes on injectives. As on degree \(2\) (resp. degree \(3\)) cohomology classes one already has an isomorphism by construction, one may then use injective presentations and the vanishing of cohomology on injective modules in order to show the theorem by induction. In fact, this kind of theorem can easily be shown (along exactly the same lines as in Baues-Minian or Das) for algebras over an operad and the corresponding operadic cohomology, as soon as operadic cohomology vanishes on injective modules. Thus the main point in a deeper understanding of this kind of theorem is the role of the vanishing of operadic cohomology on injective modules, a property which seems to be equivalent operadic cohomology being an Ext, cf. [J. Millès, Adv. Math. 226, No. 6, 5120–5164 (2011; Zbl 1218.18007)] for related considerations.