# nLab Lie algebra homology

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

$M \otimes_{U\mathfrak{g}} V(\mathfrak{g}) = M\otimes_{U\mathfrak{g}} U\mathfrak{g}\otimes_k \Lambda^* \mathfrak{g} = M\otimes_k \Lambda^* \mathfrak{g}.$
• C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, (1948). 85–124.
Revised on July 21, 2010 20:57:05 by Zoran Škoda (161.53.130.104)