nLab Chevalley-Eilenberg chain complex

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

$$V_p(\mathfrak{g}) = U(\mathfrak{g})\otimes_k \Lambda^p{\mathfrak{g}}$$

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

If $u \in U\mathfrak{g}$ and $x_i\in \mathfrak{g}$ then the differential is given by

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

Related entries

• Lie algebra homology,
• Lie algebra cohomology,
• Chevalley-Eilenberg cochain complex

and

• Chevalley-Eilenberg algebra.