Chevalley-Eilenberg complex

Definition
Suppose $$L$$ is a Lie algebra over a base field $$K$$. The Chevalley-Eilenberg complex of $$L$$ is a particular chain complex of $$L$$-modules that can serve as a projective resolution of $$K$$ as a trivial $$L$$-module.

It can be thought of as a kind of Lie algebra analogue of the bar resolution for groups.

The Chevalley-Eilenberg complex can be defined as follows. Denote by $$U$$ the universal enveloping algebra of $$L$$. Denote by $$\bigwedge^n L$$ the $$n$$-fold exterior power of $$L$$ as a $$K$$-vector space. For $$n \in \mathbb{N}_0$$, set $$B_n = U \otimes \bigwedge^nL$$ and define $$d: B_n \to B_{n-1}$$ by:

$$d(u \otimes g_1 \wedge g_2 \wedge \dots \wedge g_n) = \sum_{i=1}^n (-1)^{i+1} ug_i \otimes g_1 \wedge \dots \wedge \hat{g_i} \wedge \dots \wedge g_n + \sum_{i < j} (-1)^{i+j} u \otimes [g_i,g_j] \wedge g_1 \wedge \dots \wedge \hat{g_i} \wedge \dots \wedge \hat{g_j} \wedge \dots \wedge g_n$$

$$d^2 = 0$$, so we get a complex. In fact, this is an exact sequence, and serves as a projective resolution of $$K$$ as a trivial $$L$$-module.

Related notions

 * Bar resolution is the analogous construction for groups.