# Whitehead's lemma (Lie algebra)
**Whitehead's lemma** is a pair of vanishing theorems in the cohomology of Lie algebras, due to [J. H. C. Whitehead](https://www.edgepedia.org/j-h-c-whitehead), stating that if \( \mathfrak{g} \) is a finite-dimensional semisimple [Lie algebra](https://www.edgepedia.org/lie-algebra) over a field of characteristic zero and \( M \) is a finite-dimensional \( \mathfrak{g} \)-module, then the first and second cohomology groups vanish: \( H^{1}(\mathfrak{g}, M) = 0 \) and \( H^{2}(\mathfrak{g}, M) = 0 \)[1](https://encyclopediaofmath.org/wiki/Cohomology_of_Lie_algebras) • [2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf). The two statements are known as the first and second Whitehead lemmas, and together they account for much of the special simplicity of the cohomology of semisimple Lie algebras[3](https://indico.ictp.it/event/a04198/session/4/contribution/2/material/0/0.pdf).
| Key fact | Detail |
|---|---|
| First lemma | \( H^{1}(\mathfrak{g}, M) = 0 \) for \( \mathfrak{g} \) finite-dimensional semisimple over a field of characteristic zero and \( M \) a finite-dimensional \( \mathfrak{g} \)-module[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf) |
| Second lemma | \( H^{2}(\mathfrak{g}, A) = 0 \) under the same hypotheses[4](https://www.math.ubc.ca/~reichst/Lie-Algebra-Cohomology.pdf) |
| Equivalent structure theorems | Vanishing of \( H^{1} \) yields Weyl's complete reducibility theorem; vanishing of \( H^{2} \) yields Levi's theorem[5](https://planetmath.org/CohomologyOfSemiSimpleLieAlgebras) |
| Converse | A finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if \( H^{1}(\mathfrak{g}, M) = 0 \) for all finite-dimensional modules \( M \)[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf) |
| Characteristic restriction | The first lemma fails in characteristic \( p > 0 \); simple Lie algebras in characteristic \( p \) can have non-trivial outer derivations, that is, \( H^{1}(\mathfrak{g}, \mathfrak{g}) \neq 0 \)[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf) |
| Origin | Proved algebraically by J. H. C. Whitehead in 1936–37; recast as cohomology vanishing by Chevalley and Eilenberg in 1948[6](https://sites.math.rutgers.edu/~weibel/HA-history.pdf) |
## Statement and meaning
[Lie algebra cohomology](https://www.edgepedia.org/lie-algebra-cohomology) \( H^{n}(\mathfrak{g}, M) \), in the form introduced by [Claude Chevalley](https://www.edgepedia.org/claude-chevalley) and [Samuel Eilenberg](https://www.edgepedia.org/samuel-eilenberg) in 1948, measures obstructions in the structure of a Lie algebra and its representations[6](https://sites.math.rutgers.edu/~weibel/HA-history.pdf). The first Whitehead lemma says that for a finite-dimensional semisimple \( \mathfrak{g} \) over a field \( K \) of characteristic zero, \( H^{1}(\mathfrak{g}, M) = 0 \) for every finite-dimensional \( \mathfrak{g} \)-module \( M \)[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf). The second lemma makes the parallel statement in degree two: \( H^{2}(\mathfrak{g}, A) = 0 \) for every finite-dimensional module \( A \)[4](https://www.math.ubc.ca/~reichst/Lie-Algebra-Cohomology.pdf).
In degree one, cohomology has concrete interpretations. With coefficients in the adjoint module \( \mathfrak{g} \) itself, \( H^{1}(\mathfrak{g}, \mathfrak{g}) = \mathrm{Der}(\mathfrak{g})/\mathrm{ad}(\mathfrak{g}) \), the outer derivations modulo inner ones; the first lemma therefore implies that a semisimple Lie algebra in characteristic zero has only inner derivations, \( \mathrm{Der}(\mathfrak{g}) = \mathrm{ad}(\mathfrak{g}) \)[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf). In general, \( H^{1} \) classifies module extensions, and \( H^{2} \) classifies extensions of the Lie algebra \( \mathfrak{g} \) by the module \( M \) with a specified action[5](https://planetmath.org/CohomologyOfSemiSimpleLieAlgebras).
## Consequences
**Weyl's theorem.** Extensions of a module \( M \) by a module \( N \) are classified by \( H^{1}(\mathfrak{g}, \mathrm{Hom}(M, N)) \), so the vanishing of the first cohomology implies that every such extension splits. This is Weyl's theorem: finite-dimensional representations of a semisimple Lie algebra in characteristic zero are completely reducible, that is, direct sums of irreducible modules[5](https://planetmath.org/CohomologyOfSemiSimpleLieAlgebras) • [4](https://www.math.ubc.ca/~reichst/Lie-Algebra-Cohomology.pdf). The implication runs in both directions: Weyl's theorem gives a short proof of the first lemma, since if every module is semisimple all extensions are trivial and \( H^{1} \) vanishes[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf).
**Levi's theorem.** The vanishing of \( H^{2} \) is equivalent to Levi's theorem, which states that every finite-dimensional Lie algebra in characteristic zero is a split extension of a semisimple algebra by a solvable algebra, that is, the radical is complemented by a semisimple subalgebra[5](https://planetmath.org/CohomologyOfSemiSimpleLieAlgebras) • [1](https://encyclopediaofmath.org/wiki/Cohomology_of_Lie_algebras). Whitehead's original algebraic lemma was used precisely to prove Levi's theorems[7](https://www.ams.org/tran/1948-063-01/S0002-9947-1948-0024908-8/S0002-9947-1948-0024908-8.pdf).
**Rigidity.** By the Nijenhuis–Richardson criterion, a finite-dimensional Lie algebra with \( H^{2}(\mathfrak{g}, \mathfrak{g}) = 0 \) is rigid, meaning that nearby Lie algebra structures are isomorphic to it. Semisimple Lie algebras are therefore rigid by Whitehead's lemma[3](https://indico.ictp.it/event/a04198/session/4/contribution/2/material/0/0.pdf) • [8](https://www.sciencedirect.com/science/article/pii/S0022404922002146).
## Higher degrees
The vanishing pattern breaks down at degree three. For a simple Lie algebra \( \mathfrak{g} \), \( \dim H^{3}(\mathfrak{g}, K) = 1 \), with the basic cocycle constructed from the [Killing form](https://www.edgepedia.org/killing-form)[3](https://indico.ictp.it/event/a04198/session/4/contribution/2/material/0/0.pdf) • [9](https://arxiv.org/abs/0808.0212). A related result, sometimes called the Whitehead theorem, holds in all higher degrees: for a finite-dimensional semisimple \( L \) and any non-trivial finite-dimensional irreducible module \( V \), \( H^{n}(L, V) = 0 \) for all \( n \geq 3 \)[9](https://arxiv.org/abs/0808.0212). A stronger form due to Whitehead states that if \( \mathfrak{g} \) is semisimple and \( W \) is a finite-dimensional module with no invariant elements, \( W^{\mathfrak{g}} = 0 \), then \( H^{k}(\mathfrak{g}, W) = 0 \) for all \( k \geq 0 \)[3](https://indico.ictp.it/event/a04198/session/4/contribution/2/material/0/0.pdf). Combining the two lemmas with complete reducibility gives the general formula \( H^{*}(\mathfrak{g}, W) = H^{*}(\mathfrak{g}, K) \otimes W^{\mathfrak{g}} \), so for semisimple \( \mathfrak{g} \) only the cohomology with trivial coefficients carries information[3](https://indico.ictp.it/event/a04198/session/4/contribution/2/material/0/0.pdf).
## Method of proof
The standard proof rests on the [Casimir element](https://www.edgepedia.org/casimir-element), a central element of the universal enveloping algebra built from an invariant bilinear form on \( \mathfrak{g} \), such as the Killing form[10](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec18.pdf). A central element that acts by zero on the trivial representation and by a non-zero scalar on a module \( E \) forces \( H^{1}(\mathfrak{g}, E) = 0 \), because it produces an invariant complement to any extension[10](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec18.pdf). For a non-trivial irreducible module \( V \) of a semisimple \( \mathfrak{g} \) in characteristic zero, the trace form \( B_{V}(x, y) = \mathrm{Tr}_{V}(xy) \) is non-degenerate, and the corresponding Casimir element acts on \( V \) by the scalar \( \dim \mathfrak{g} / \dim V \), which is non-zero[10](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec18.pdf). The general case then follows by induction on module dimension using the long exact cohomology sequence, reducing to irreducible modules and finally to the trivial module \( K \), where \( H^{1}(\mathfrak{g}, K) = (\mathfrak{g}/[\mathfrak{g}, \mathfrak{g}])^{*} = 0 \) because semisimple algebras are perfect[4](https://www.math.ubc.ca/~reichst/Lie-Algebra-Cohomology.pdf) • [10](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec18.pdf). The same Casimir argument is the principal step in Whitehead's lemma in the modular setting studied by Askar Dzhumadil'daev, a mathematician known for work on the cohomology of modular Lie algebras[11](https://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=2841&what=fullteng).
## Limits and converses
The characteristic-zero hypothesis is essential. The first Whitehead lemma does not hold in characteristic \( p > 0 \): there are even simple Lie algebras in characteristic \( p \) with non-trivial outer derivations, so \( H^{1}(\mathfrak{g}, \mathfrak{g}) \neq 0 \)[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf). For \( \mathfrak{sl}_{2}(K) \), \( \dim H^{1}(\mathfrak{g}, \mathfrak{g}) = 0 \) when the characteristic is not 2, but 4 in characteristic 2, where the algebra is no longer semisimple[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf).
The lemmas also characterize semisimplicity. A finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if \( H^{1}(\mathfrak{g}, M) = 0 \) for all finite-dimensional modules \( M \)[2](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf). A converse to the second lemma was established by Pasha Zusmanovich, a researcher in Lie theory: finite-dimensional Lie algebras over a field of characteristic zero whose second cohomology group vanishes in every finite-dimensional module are, essentially, semisimple[12](https://arxiv.org/abs/0704.3864). A converse to the higher Whitehead theorem shows that algebras whose high-degree cohomology in every non-trivial irreducible module vanishes are essentially direct sums of semisimple and nilpotent algebras[9](https://arxiv.org/abs/0808.0212).
## Extensions
The lemmas have been generalized in several directions. Arezoo Zohrabi and Pasha Zusmanovich established a δ-version of the first Whitehead lemma: for a simple finite-dimensional Lie algebra defined over a field of characteristic zero, every δ-derivation taking values in a finite-dimensional module is either inner, or—when the module is the adjoint one—a scalar multiplication, apart from certain exceptional cases tied to \( \mathfrak{sl}(2) \)[13](https://www.sciencedirect.com/science/article/pii/S0021869321000193). A Leibniz analog of Whitehead's vanishing theorem yields a second Whitehead lemma for Leibniz algebras[14](https://arxiv.org/abs/1902.06128). Analogues for semisimple Filippov (n-Lie) algebras hold as well: they admit no non-trivial central extensions and are rigid[15](https://arxiv.org/abs/0905.3083).
## References
1. [Cohomology of Lie algebras, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Cohomology_of_Lie_algebras)
2. [Dietrich Burde, Cohomology of Groups and Algebras (lecture notes), University of Vienna](https://homepage.univie.ac.at/dietrich.burde/papers/burde_76_coho.pdf)
3. [Lectures on Poisson cohomology, ICTP notes, §2.3 Cohomology of Lie algebras](https://indico.ictp.it/event/a04198/session/4/contribution/2/material/0/0.pdf)
4. [M. Reichstein, Lie Algebra Cohomology (lecture notes), University of British Columbia](https://www.math.ubc.ca/~reichst/Lie-Algebra-Cohomology.pdf)
5. [Cohomology of semi-simple Lie algebras, PlanetMath](https://planetmath.org/CohomologyOfSemiSimpleLieAlgebras)
6. [Charles Weibel, History of Homological Algebra](https://sites.math.rutgers.edu/~weibel/HA-history.pdf)
7. [Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Transactions of the American Mathematical Society 63 (1948), 85–124](https://www.ams.org/tran/1948-063-01/S0002-9947-1948-0024908-8/S0002-9947-1948-0024908-8.pdf)
8. [Deformations and rigidity in varieties of Lie algebras, Journal of Algebra (2022)](https://www.sciencedirect.com/science/article/pii/S0022404922002146)
9. [Pasha Zusmanovich, A converse to the Whitehead Theorem, arXiv:0808.0212](https://arxiv.org/abs/0808.0212)
10. [MIT OpenCourseWare 18.745, Lecture 18: Extensions of Representations, Whitehead's Theorem, and Complete Reducibility](https://ocw.mit.edu/courses/18-745-lie-groups-and-lie-algebras-i-fall-2020/mit18_745_f20_lec18.pdf)
11. [A. S. Dzhumadil'daev, On the cohomology of Lie algebras, Mathematics of the USSR-Sbornik](https://www.mathnet.ru/php/getFT.phtml?jrnid=sm&paperid=2841&what=fullteng)
12. [Pasha Zusmanovich, A converse to the Second Whitehead Lemma, arXiv:0704.3864](https://arxiv.org/abs/0704.3864)
13. [Arezoo Zohrabi and Pasha Zusmanovich, A δ-first Whitehead Lemma, Journal of Algebra 573 (2021), 476–491](https://www.sciencedirect.com/science/article/pii/S0021869321000193)
14. [On Leibniz cohomology, arXiv:1902.06128](https://arxiv.org/abs/1902.06128)
15. [Cohomology of Filippov algebras and an analogue of Whitehead's lemma, arXiv:0905.3083](https://arxiv.org/abs/0905.3083)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Lie representations and modules › Lie algebra cohomology*
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