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 $⋀^{n} L$ the $n$ -fold exterior power of $L$ as a $K$ -vector space. For $n \in N_{0}$ , set $B_{n} = U \otimes ⋀^{n} L$ and define $d : B_{n} \to B_{n - 1}$ by:

$d (u \otimes g_{1} \land g_{2} \land \dots \land g_{n}) = \sum_{i = 1}^{n} (- 1)^{i + 1} u g_{i} \otimes g_{1} \land \dots \land \hat{g_{i}} \land \dots \land g_{n} + \sum_{i < j} (- 1)^{i + j} u \otimes [g_{i}, g_{j}] \land g_{1} \land \dots \land \hat{g_{i}} \land \dots \land \hat{g_{j}} \land \dots \land g_{n}$

$d^{2} = 0$ , 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.