DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Chevalley–Eilenberg differential

Definition

For $f \in C^{n} (g, M)$ define $d f \in C^{n + 1} (g, M)$ by

$\begin{aligned} (d f) (x_{0}, \dots, x_{n}) = & \sum_{i = 0}^{n} (- 1)^{i} x_{i} f (x_{0}, \dots, \hat{x_{i}}, \dots, x_{n}) \\ + \sum_{0 \leq i < j \leq n} (- 1)^{i + j} f ([x_{i}, x_{j}], x_{0}, \dots, \hat{x_{i}}, \dots, \hat{x_{j}}, \dots, x_{n}) . \end{aligned}$

This fixes a zero-based sign convention. A hat means omission, and in the second sum the bracket is inserted as the first argument. Empty sums are zero. Thus for $m \in C^{0} = M$ , $(d m) (x) = x m$ . Alternation of the formula makes it factor through $Λ^{n + 1} g$ ; the next theorem proves that successive differentials compose to zero.

Depends on

Chevalley–Eilenberg cochains

Used by

First cohomology is derivations modulo inner derivations Proposition
Zeroth Lie algebra cohomology is invariants Proposition
Second cohomology classifies abelian extensions Theorem
The Chevalley–Eilenberg differential squares to zero Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Weibel, Lie Algebra Homology and Cohomology, §7.7 (standard reference, not scraped)