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 $B_{n}=U\otimes \bigwedge ^{n}L$ 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.