# nLab Chevalley-Eilenberg chain complex

Given a $k$-Lie algebra $𝔤$ over a commutative unital ring $k$ which is free as a $k$-module, the Chevalley–Eilenberg chain complex is a particular projective resolution ${V}_{*}\left(𝔤\right)\to k$ of the trivial $𝔤$-module $k$ in the abelian category of $𝔤$-modules (what is the same as $U𝔤$-modules, where $U𝔤$ is the universal enveloping algera of $𝔤$). Graded components of the underlying $k$-module this resolution is given by

${V}_{p}\left(𝔤\right)=U\left(𝔤\right){\otimes }_{k}{\Lambda }^{p}𝔤$V_p(\mathfrak{g}) = U(\mathfrak{g})\otimes_k \Lambda^p{\mathfrak{g}}

and it has the obvious $U𝔤$-module structure by multiplication in the first tensor factor, because ${\Lambda }^{p}𝔤$ is free as a $k$-module.

If $u\in U𝔤$ and ${x}_{i}\in 𝔤$ then the differnetial is given by

$d\left(u\otimes {x}_{1}\wedge \cdots \wedge {x}_{p}\right)=$d(u\otimes x_1 \wedge \cdots \wedge x_p) =
$=\sum _{i=1}^{p}\left(-1{\right)}^{i+1}{\mathrm{ux}}_{i}\otimes {x}_{1}\wedge \cdots \wedge {\stackrel{^}{x}}_{i}\wedge \cdots \wedge {x}_{p}+\sum _{i= \sum_{i = 1}^p (-1)^{i+1} ux_i \otimes x_1 \wedge \cdots \wedge \hat{x}_i\wedge \cdots \wedge x_p + \sum_{i\lt j} (-1)^{i+j} u\otimes [x_i, x_j] \wedge \cdots \wedge \hat{x}_i\cdots \wedge \hat{x}_j\cdots \wedge x_p

Revised on July 28, 2010 21:17:40 by Toby Bartels (173.190.150.114)