The Chevalley–Eilenberg Resolution

Executive summary

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

V_n = U(g) ⊗_K Λⁿg, 0 ≤ n ≤ dim g

with differential combining two terms: one moving a generator into the enveloping algebra, one applying the bracket to a pair of generators.

d(u ⊗ x₁ ∧ … ∧ x_n) = ∑_i (−1)ⁱ⁺¹ ux_i ⊗ (… omit x_i …) + ∑_i<j (−1)^i+j u ⊗ [x_i, x_j] ∧ …

Why it is exact

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 Hom_U(g)(−, A) gives the complex of alternating multilinear maps

Cⁿ(g, A) = Hom_K(Λⁿg, A)

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.

Consequences of finiteness
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
H^{dim 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

H^*(g, ℝ) ≅ H^*_dR(G)

Algebra computing topology

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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ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review The Chevalley–Eilenberg Resolution. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat The Chevalley–Eilenberg Resolution as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—resolution, complex, cochain, section, chevalley-eilenberg—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying The Chevalley–Eilenberg Resolution?