Definition of Lie Algebra Cohomology

Executive summary

Invariants, derivations, extensions

Define Hⁿ(g, A) as Extⁿ over U(g) from the trivial module to A. Degree 0 is the invariants; degree 1 is derivations modulo inner derivations; degree 2 classifies extensions. The pattern is identical to group cohomology, and the proofs transfer — but the answers differ sharply, because semisimple Lie algebras in characteristic zero have vanishing cohomology in degrees 1 and 2 where finite groups generally do not.

Learning objectives

Define Lie algebra cohomology and homology.
Identify H⁰ as the invariants.
Describe H¹ via derivations.
State the standard vanishing results.

Section 01The definition

Hⁿ(g, A) = Extⁿ_U(g)(K, A), H_n(g, A) = Tor^U(g)_n(K, A)

Low degrees
Degree	Cohomology	Meaning
0	A^g = { a : x·a = 0 for all x }	The invariants
1	Der(g, A) / Inn(g, A)	Derivations modulo inner ones; with trivial action, Hom(g/[g,g], A)
2	Equivalence classes of extensions	Extensions of g by the abelian ideal A with the given action
3	Obstructions	To realising an outer action by an extension

Section 02Derivations

A derivation d: g → A satisfies

d([x, y]) = x·d(y) − y·d(x)

and is inner when d(x) = x·a for a fixed a. So H¹ measures the derivations that are not inner — the exact analogue of derivations modulo principal derivations for groups.

The classical case

Taking A = g with the adjoint action, H¹(g, g) = Der(g)/Inn(g) is the outer derivation algebra. Its vanishing for semisimple g in characteristic zero — the first Whitehead lemma — says every derivation of a semisimple Lie algebra is inner.

Section 03What differs from the group case

GroupsCohomology often persists

For a finite group in modular characteristic the group algebra has infinite global dimension, and cohomology is non-zero in arbitrarily high degrees.

Lie algebrasCohomology terminates

For g finite-dimensional, U(g) has global dimension dim g, so Hⁿ(g, −) = 0 for n > dim g. The theory is finite-dimensional.

The characteristic matters enormously

In characteristic zero, semisimplicity gives the Whitehead lemmas and complete reducibility. In characteristic p the enveloping algebra behaves differently, restricted Lie algebras enter, and the parallel with modular representation theory of groups becomes the right one.

ReferenceFrequently asked questions

Is there an analogue of Maschke's theorem?

Weyl's theorem on complete reducibility: every finite-dimensional representation of a semisimple Lie algebra in characteristic zero is a direct sum of irreducibles. It is proved from the Whitehead lemmas, exactly as Maschke's theorem gives vanishing cohomology for groups.

Why is the degree bounded by the dimension?

Because the Chevalley–Eilenberg resolution has length equal to dim g, being built from the exterior algebra on g. That resolution is finite, so all higher Ext vanishes.

Does the cohomology have a ring structure?

Yes, by the Yoneda product, and it is graded-commutative. For a compact Lie group the cohomology of its Lie algebra agrees with the de Rham cohomology of the group, so the ring structure has a direct geometric meaning.

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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 Definition of Lie Algebra Cohomology. 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 Definition of Lie Algebra Cohomology as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—algebra, cohomology, invariants, derivations, section—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 Definition of Lie Algebra Cohomology?