Chevalley-Eilenberg homology of crossed modules of Lie algebras in lower dimensions (Q388554)

In the present paper the authors answer some of the questions posed in [\textit{J. M. Casas}, \textit{N. Inassaridze} and {the second author}, Manuscr. Math. 131, No. 3--4, 385--401 (2010; Zbl 1239.17014)]. They prove the existence of a five term exact sequence connecting the low dimensional Chevalley-Eilenberg homologies of crossed modules of Lie algebras \((\mathfrak{m}, \mathfrak{g}, \mu)\) and of the Lie algebra \(\mathfrak g/\text{Im}(\mu)\). Finally, a relationship between the Chevalley-Eilenberg homology with coefficients and the homology of a crossed module of Lie algebras is established.NEWLINENEWLINETheir main result reads as follows:NEWLINENEWLINE NEWLINEProposition 3.1. Let \((\mathfrak{m}, \mathfrak{g}, \mu)\) be a crossed module of Lie algebras. Then NEWLINE\[NEWLINE H_0(\mathfrak{m}, \mathfrak{g}, \mu)=\mathfrak k\quad\text{and} \quad H_1(\mathfrak{m}, \mathfrak{g}, \mu)=\text{Coker}(\mu)/[\text{Coker}(\mu),\text{Coker}(\mu)].NEWLINE\]NEWLINE NEWLINEMoreover, if the characteristic of \(\mathfrak k\) is not 2 (i.e. \(1/2\in \mathfrak k\)), then there is an exact sequence of vector spaces NEWLINE\[NEWLINEH_3(\mathfrak{m}, \mathfrak{g}, \mu)\rightarrow H_3(\text{Coker}(\mu))\rightarrow \text{Ker}(\mu)/[\text{Ker}(\mu),\mathfrak g]\rightarrow H_2(\mathfrak{m}, \mathfrak{g}, \mu)\rightarrow H_2(\text{Coker}(\mu)).NEWLINE\]