Difference between Koszul and Chevalley-Eilenberg complexes

Question

Please have a look at these two definitions:

Both are from Wikipedia pages on Lie algebra cohomology, and they look rather similar. Since I'm not very solid on the notation, I'm unsure about what's the difference between them. Furthermore, since both Wiki pages are about Lie algebra cohomology, and no page defines the "other" chain complex, I'm not even sure whether they are different at all.

What's the essential difference between them?

Dietrich Burde · Accepted Answer · 2025-05-22T08:46:54.720

6

They are equivalent complexes. In the Koszul complex the notation $x\wedge y$ is used, which can be identified with an alternating $b(x,y)$ (or more generally the $k$-fold versions). The German Wikipedia version with the Koszul complex explicitly refers to Chevalley-Eilenberg, and furthermore discusses the scalar factor $1/(k+1)$:

"In der angegebenen Arbeit von C. Chevalley und S. Eilenberg wird noch durch $k+1$ dividiert, was im unten angegebenen Lehrbuch von Hilgert und Neeb nicht der Fall ist."

Indeed, the factor $1/(k+1)$ is originally in Chevalley-Eilenberg's paper, but is nowadays skipped, because it causes unnecessary problems over fields of characteristic $p>0$.

edited May 22 '25 at 08:46

answered Aug 05 '16 at 21:28

Dietrich Burde

140,055

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Is there a reason why there are two different names for the same thing? – Andy Miles Aug 15 '16 at 15:18
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Yes, besides Chevalier and Eilenberg also Koszul published a long article on Lie algebra cohomology (J-J. Koszul: Homologie et cohomologie des algebres de Lie. Bull. Soc. Math. France 78 (1950), 65-127.) – Dietrich Burde Aug 15 '16 at 17:51
Koszul's name is also invoked without Lie. nam – jim stasheff May 21 '25 at 20:57

Difference between Koszul and Chevalley-Eilenberg complexes

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