# nLab Lie algebra homology

The abelian homology of a $k$-Lie algebra $𝔤$ with coefficients in the left $𝔤$-module $M$ is defined as ${H}_{*}^{\mathrm{Lie}}\left(𝔤,M\right)={\mathrm{Tor}}_{*}^{U𝔤}\left(k,M\right)$ where $k$ is the ground field understood as a trivial module over the universal enveloping algebra $U𝔤$. In particular it is a derived functor. It can be computed using Chevalley-Eilenberg chain complex $V\left(𝔤\right)$ as the homology of the chain complex

$M{\otimes }_{U𝔤}V\left(𝔤\right)=M{\otimes }_{U𝔤}U𝔤{\otimes }_{k}{\Lambda }^{*}𝔤=M{\otimes }_{k}{\Lambda }^{*}𝔤.$M \otimes_{U\mathfrak{g}} V(\mathfrak{g}) = M\otimes_{U\mathfrak{g}} U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g} = M\otimes_k \Lambda^* \mathfrak{g}.
• C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, (1948). 85–124.