A resolution of length exactly dim g
The Chevalley–Eilenberg resolution is U(g) ⊗ Λng with an explicit differential built from the bracket and the action. It is free, finite, and of length exactly the dimension of g — a Koszul-type complex. Consequently Lie algebra cohomology vanishes above dim g, and the cochain complex it produces is the algebra of alternating multilinear maps, which for a compact Lie group is the complex of invariant differential forms.
Learning objectives
- Write the resolution and its differential.
- Derive the cochain complex computing cohomology.
- Deduce the bound on cohomological dimension.
- Explain the de Rham interpretation.
Section 01The resolution
with differential combining two terms: one moving a generator into the enveloping algebra, one applying the bracket to a pair of generators.
By PBW, U(g) has a filtration whose associated graded is a polynomial algebra, and the resolution becomes the Koszul complex of that polynomial algebra, which is exact. The filtration argument transfers exactness back to the original complex.
Section 02The cochain complex
Applying HomU(g)(−, A) gives the complex of alternating multilinear maps
with the differential dual to the above. In degrees 1 and 2 this reproduces exactly the derivation condition and the extension cocycle condition established independently — a useful consistency check.
| Statement | Reason |
|---|---|
| Hn(g, A) = 0 for n > dim g | The resolution has length dim g |
| gl.dim U(g) = dim g | The resolution is minimal in the graded sense |
| Hdim g(g, K) ≅ K for g unimodular | Top exterior power; a Poincaré duality statement |
| Euler characteristic is 0 for dim g > 0 | Alternating sum of binomial coefficients |
Section 03The de Rham interpretation
For a compact connected Lie group G with Lie algebra g, the complex C*(g, ℝ) is exactly the complex of left-invariant differential forms on G, and averaging over the group shows the inclusion into all forms is a quasi-isomorphism. Hence
This identification means the real cohomology of a compact Lie group is computable purely algebraically from its Lie algebra. It is the cleanest instance in this collection of a homological invariant of an algebraic object coinciding with a topological invariant of a geometric one.
ReferenceFrequently asked questions
Is the Chevalley–Eilenberg complex a Koszul complex?
Essentially yes. After passing to the associated graded of the PBW filtration it becomes the Koszul complex of a polynomial algebra on the basis of g. That is exactly how exactness is proved.
Does the resolution work in characteristic p?
The construction works, but the exterior algebra behaves differently and restricted structures intervene. The resolution remains valid for computing Ext over U(g), but the connection to the representation theory is quite different.
What is the analogue for groups?
There is none of comparable finiteness. The bar resolution is infinite for any non-trivial group, and finite resolutions exist only for special groups — those of finite cohomological dimension, such as free groups and torsion-free arithmetic groups.
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