H1 = H2 = 0, and everything follows
For a semisimple Lie algebra over a field of characteristic zero, H1(g, A) and H2(g, A) vanish for every finite-dimensional module. The first Whitehead lemma says every derivation is inner and every short exact sequence of modules splits — Weyl's complete reducibility theorem. The second says every extension by an abelian ideal splits, which combined with a further argument gives Levi's decomposition. Both are proved with the Casimir element.
Learning objectives
- State both Whitehead lemmas.
- Explain the role of the Casimir element in the proof.
- Derive Weyl's complete reducibility theorem.
- State Levi's theorem and its relation to H2.
Section 01Semisimplicity and the Casimir
The Killing form is B(x, y) = tr(ad x ∘ ad y). Cartan's criterion: g is semisimple exactly when B is non-degenerate.
- Take a faithful finite-dimensional representation and the associated trace form, which is non-degenerate by semisimplicity.
- Choose a basis xi and the dual basis yi with respect to that form.
- Set c = ∑i xi yi ∈ U(g).
- c is central in U(g), so it acts as a scalar on each irreducible module by Schur's lemma. Centrality is the key computation.
- That scalar is non-zero on any non-trivial irreducible, which is what makes the vanishing arguments work.
Section 02The two lemmas
Every derivation into a finite-dimensional module is inner. Equivalently, every short exact sequence of finite-dimensional modules splits.
Every extension of g by a finite-dimensional abelian ideal splits. This is the input to Levi's theorem.
Decompose the module by the eigenvalues of the Casimir. On the part where it acts invertibly, cohomology vanishes because the Casimir acts both as an invertible scalar and as zero — the latter because it acts trivially on cohomology of the trivial module. On the trivial part, direct computation using semisimplicity of g finishes the argument.
Section 03Consequences
Weyl complete reducibility
Every finite-dimensional representation of a semisimple Lie algebra in characteristic zero is a direct sum of irreducibles. This is the first Whitehead lemma restated.
Levi decomposition
Every finite-dimensional Lie algebra is a semidirect sum of its radical and a semisimple subalgebra. The second Whitehead lemma supplies the splitting.
Rigidity
H²(g, g) = 0 means semisimple Lie algebras admit no non-trivial deformations — they are rigid, which is why the classification by Dynkin diagrams is discrete.
In characteristic p, complete reducibility fails and modular representation theory of Lie algebras resembles that of finite groups. For infinite-dimensional coefficients the lemmas also fail, which is exactly why affine and Virasoro central extensions exist.
ReferenceFrequently asked questions
Why is H²(g, g) = 0 a rigidity statement?
Because infinitesimal deformations of a Lie algebra structure are classified by H² with adjoint coefficients, and obstructions to extending them lie in H³. Vanishing of H² means no first-order deformation exists, so the structure cannot be deformed at all.
Is there a group-theoretic analogue?
Maschke's theorem is the analogue of the first Whitehead lemma, and the Schur–Zassenhaus theorem plays a role like the second. In both settings the mechanism is a form of averaging that requires the relevant order or characteristic condition.
What replaces the Casimir in characteristic p?
Nothing with the same force. The trace form can degenerate and the Casimir need not act invertibly, which is precisely why complete reducibility fails and why restricted Lie algebra cohomology is a separate subject.
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