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

Abstract

In this paper we prove a five term exact sequence connecting in lower dimensions the Chevalley-Eilenberg homologies of the crossed module of Lie algebras (m, g, µ) and of the Lie algebra g/Im(µ). Moreover, a relationship between the ChevalleyEilenberg homology with coefficients and the homology of a crossed module of Lie algebras is established.

Universal enveloping crossed module of a Lie crossed module

“…Guin [17] has developed low-dimensional non-abelian cohomology of Lie algebras with coefficients in Lie crossed modules, which later has been extended to higher dimensions in [18]. Internal (cotriple) homology and Chevalley-Eilenberg homology theories of Lie crossed modules are investigated in [8,13]. Lie crossed modules also occurred in the "categorification" problem of the theory of Lie algebras [1] as an equivalent formulation of strict Lie 2-algebras (a two-dimensional generalization of the concept of Lie algebra).…”