Homological aspects of Lie algebra crossed modules

Chevalley-Eilenberg homology of crossed modules of Lie algebras in lower dimensions

“…The present paper provides answers to some questions posed in [4]. In particular, the existence of a five term exact sequence connecting the low-dimensional ChevalleyEilenberg homologies of crossed modules and their cokernel Lie algebras is proved.…”

“…Given a simplicial Lie algebra g * , the Chevalley-Eilenberg homology extends to g * in a natural way (see, e.g., [4]). Namely, applying the Chevalley-Eilenberg complex, C CE (−), dimensional-wise to the simplicial Lie algebra g * , we arrive at the following bicomplex…”

“…A first approach to an internal (low dimensional) homology theory of crossed modules of Lie algebras was done in [5]. In [4], the authors studied the homology theory of crossed modules of Lie algebras in the spirit of [3,10]. In particular, the authors checked that the category of crossed modules of Lie algebras is tripleable and proved that the natural homology theory obtained from this triple (called cotriple homology) can be determined by small complexes formed from the standard Chevalley-Eilenberg complex of Lie algebras.…”

“…For all n 0, one has a natural isomorphism H n (0, g, 0) = H n (g) (see [4]). Moreover, by Proposition 3.1 we have…”

Capability of Crossed Modules of Lie Algebras

“…For instance, the (co)homology theory of crossed modules of Lie algebras was introduced by Casas, Inassaridze, and Ladra in [2], and some results on abelian extensions and perfectness of crossed modules were tied to these topics.…”