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
| Degree | Cohomology | Meaning |
|---|---|---|
| 0 | Ag = { 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
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.
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
For a finite group in modular characteristic the group algebra has infinite global dimension, and cohomology is non-zero in arbitrarily high degrees.
For g finite-dimensional, U(g) has global dimension dim g, so Hn(g, −) = 0 for n > dim g. The theory is finite-dimensional.
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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