Lie algebra cohomology and Whitehead's lemmas

Intuition Beginner

Lie algebra cohomology is a way to extract numbers and vector spaces from a Lie algebra that record how rigid it is. You feed in a Lie algebra and a module it acts on, and out come a sequence of spaces $H^0,H^1,H^2,\dots$ that each answer a structural question.

The low-degree spaces have plain meanings. The zeroth space is the part of the module that the algebra leaves fixed. The first space measures the derivations that are not built from the obvious inner ones. The second space classifies the ways you can enlarge the algebra by gluing the module on as an abelian piece.

Visual Beginner

Picture a ladder of vector spaces, one rung per degree. The bottom rung holds the fixed vectors; the next holds derivations; the next holds extensions. A single map, the boundary operator, climbs from each rung to the next, and applying it twice always lands you at zero.

For a semisimple algebra the first and second rungs collapse to a single point: every derivation is inner and every abelian extension splits. That collapse is the content of Whitehead's two lemmas, and it is the engine behind the fact that representations of such an algebra always break into irreducible pieces.

Worked example Beginner

Take the abelian one-dimensional Lie algebra spanned by a single element $H$, acting on the field as the zero map. A cochain in degree one is a linear map from the span of $H$ to the field, so it is one number. The boundary operator vanishes on this small algebra, so the first cohomology is the whole one-dimensional space of numbers.

This already shows something useful: the first cohomology is not always empty. An abelian algebra carries genuine cohomology. So when Whitehead's lemma forces the first cohomology to vanish, that vanishing is special to semisimple algebras and signals their rigidity.

Compare the semisimple algebra of trace-zero two-by-two matrices. There the Casimir element acts as a fixed nonzero scalar on each irreducible piece, and dividing by that scalar manufactures the splitting that drives every cohomology space above degree zero down to a point.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathfrak{g}$ be a Lie algebra over a field $k$ of characteristic zero and let $M$ be a $\mathfrak{g}$-module, with action written $X\cdot m=\rho (X)m$. The Chevalley–Eilenberg cochain complex has degree-$n$ term

$$C^n(\mathfrak{g},M)={\mathrm{Hom}}_k({\Lambda }^n\mathfrak{g},M),$$

so a cochain is an alternating $n$-linear map from $\mathfrak{g}$ to $M$. The differential $d:C^n\to C^{n+1}$ is

$$\begin{array}{rl}(d\omega )(X_0,\dots ,X_n)=& \sum _i(-1{)}^i\rho (X_i)\omega (X_0,\dots ,\widehat{X_i},\dots ,X_n)\\ & +\sum _{i<j}(-1{)}^{i+j}\omega ([X_i,X_j],X_0,\dots ,\widehat{X_i},\dots ,\widehat{X_j},\dots ,X_n),\end{array}$$

where a hat marks an omitted argument. The first family of terms uses the module action $\rho$; the second uses the bracket. One checks $d\circ d=0$ using the Jacobi identity (for the bracket–bracket cross terms) together with the module axiom $\rho ([X,Y])=\rho (X)\rho (Y)-\rho (Y)\rho (X)$ (for the action–bracket cross terms).

Definition (Lie algebra cohomology). $H^n(\mathfrak{g},M)=\ker (d:C^n\to C^{n+1})/\mathrm{im}(d:C^{n-1}\to C^n)$.

The low-degree groups have explicit readings. In degree zero, $d:M\to \mathrm{Hom}(\mathfrak{g},M)$ sends $m$ to $X\mapsto \rho (X)m$, so $H^0(\mathfrak{g},M)=M^{\mathfrak{g}}$, the submodule of invariants. In degree one, a cocycle is a linear map $\omega$ with $\omega ([X,Y])=\rho (X)\omega (Y)-\rho (Y)\omega (X)$, which is exactly a derivation $\mathfrak{g}\to M$; the coboundaries are the inner derivations $X\mapsto \rho (X)m$. Thus $H^1(\mathfrak{g},M)=\mathrm{Der}(\mathfrak{g},M)/\mathrm{InnDer}(\mathfrak{g},M)$. In degree two, $H^2(\mathfrak{g},M)$ is the set of equivalence classes of abelian extensions $0\to M\to \hat{\mathfrak{g}}\to \mathfrak{g}\to 0$; when $M=k$ is the field with the zero action these are the central extensions, and the corresponding cocycles include the Heisenberg and Virasoro cocycles.

Key theorem with proof Intermediate+

Theorem (Whitehead's lemmas). Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over a field of characteristic zero and let $M$ be a finite-dimensional $\mathfrak{g}$-module. Then $H^1(\mathfrak{g},M)=0$ (first lemma) and $H^2(\mathfrak{g},M)=0$ (second lemma).

The mechanism is the Casimir element $\Omega \in U(\mathfrak{g})$ 07.06.10. Decompose $M$ into a part on which $\Omega$ acts as zero and a complementary part on which $\Omega$ is invertible. On the invertible part $\Omega$ acts as a chain map that is homotopic to a multiple of the identity, and inverting it contracts the complex, so the cohomology of the invertible part vanishes in all positive degrees. The remaining $\Omega$-zero part of $M$ consists of summands carrying the zero action; for semisimple $\mathfrak{g}$ the equality $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$ forces $H^1$ and $H^2$ with coefficients in such a zero-action module to vanish as well, by a short direct computation on the differential. Combining the two parts gives the stated vanishing.

First lemma in coordinates. A degree-one cocycle is a derivation $\delta :\mathfrak{g}\to M$. Set $\Omega ={\sum }_a{x}_a{y}^a$ for a basis $x_a$ and its dual $y^a$ under an invariant form. The element $m={\sum }_a\rho (x_a)\delta (y^a)$ satisfies $\delta (X)=\rho (X)m$ whenever $\Omega$ acts invertibly, exhibiting $\delta$ as inner. So every derivation into $M$ is inner, which is precisely $H^1(\mathfrak{g},M)=0$.

Bridge. This builds toward the cohomological reading of representation theory: the first lemma is exactly the statement that the extension class controlling a short exact sequence of modules vanishes, so the foundational reason semisimple representations split is the vanishing of $H^1$. The second lemma generalises this from modules to algebras, and putting these together yields Levi's theorem; the central insight is that one invertible operator, the Casimir, kills cohomology, and this same vanishing appears again in the Hochschild–Serre spectral sequence 07.06.24, where the cohomology of a semisimple quotient collapses the $E_2$ page.

Exercises Intermediate+

Advanced results Master

Weyl complete reducibility, recovered cohomologically. Let $0\to W\to V\to U\to 0$ be a short exact sequence of finite-dimensional $\mathfrak{g}$-modules. The obstruction to splitting it $\mathfrak{g}$-equivariantly lives in $H^1(\mathfrak{g},{\mathrm{Hom}}_k(U,W))$, the cohomology with coefficients in the module of linear maps. Whitehead's first lemma makes this group vanish, so every such sequence splits and every finite-dimensional module is a direct sum of irreducibles. This is the cohomological proof of Weyl's theorem 07.06.22, parallel to the Casimir-projection proof and powered by the same element.

Levi's theorem. For any finite-dimensional $\mathfrak{g}$ with radical $\mathfrak{r}$ and semisimple quotient $\mathfrak{s}=\mathfrak{g}/\mathfrak{r}$, the extension $0\to \mathfrak{r}\to \mathfrak{g}\to \mathfrak{s}\to 0$ is governed, after reducing to the abelian case, by $H^2(\mathfrak{s},-)$. The second Whitehead lemma forces this to vanish, so the extension splits and $\mathfrak{g}=\mathfrak{r}⋊\mathfrak{s}$ admits a Levi subalgebra.

The de Rham comparison (Chevalley–Eilenberg theorem). For a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$, the cochain complex $C^{\bullet }(\mathfrak{g},\mathbb{R})={\Lambda }^{\bullet }{\mathfrak{g}}^{\ast }$ with its differential is exactly the complex of bi-invariant differential forms on $G$. Averaging over $G$ with Haar measure shows that the inclusion of invariant forms into all forms is a cohomology isomorphism, giving $H^{\bullet }(\mathfrak{g},\mathbb{R})\cong H_{\mathrm{dR}}^{\bullet }(G)$. Lie algebra cohomology is the invariant de Rham cohomology of the group.

Derived-functor identification. Cohomology is a derived functor of invariants: $H^n(\mathfrak{g},M)\cong {\mathrm{Ext}}_{U(\mathfrak{g})}^n(k,M)$, the Ext over the enveloping algebra 07.06.02 computed from a projective resolution of the base field $k$ with zero action. The Chevalley–Eilenberg complex is the Koszul resolution made explicit, so all of the above is homological algebra over $U(\mathfrak{g})$.

Synthesis. The bridge across these results is one element acting invertibly: the foundational reason both Whitehead lemmas hold is that the Casimir contracts the complex, and this is exactly the same vanishing that powers Weyl reducibility and Levi splitting. Putting these together, complete reducibility, Levi decomposition, and the invariant de Rham theorem are three faces of the single fact $H^1(\mathfrak{g},M)=H^2(\mathfrak{g},M)=0$; the central insight is that cohomology is the derived functor of invariants, so semisimplicity is dual to the vanishing of every obstruction. This pattern appears again in the Hochschild–Serre spectral sequence 07.06.24, where it generalises from a single algebra to an ideal-and-quotient pair.

Full proof set Master

Proposition (Extensions and the second cohomology). Equivalence classes of abelian extensions $0\to M\to \hat{\mathfrak{g}}\to \mathfrak{g}\to 0$, where $M$ is an abelian ideal with the induced $\mathfrak{g}$-action, are in bijection with $H^2(\mathfrak{g},M)$, the split extension corresponding to the zero class.

Proof. Given an extension, choose a $k$-linear section $s:\mathfrak{g}\to \hat{\mathfrak{g}}$ of the projection. The failure of $s$ to be a Lie algebra map is measured by $\omega (X,Y)=[s(X),s(Y)]-s([X,Y])$, which lands in $M$ because the projection kills it. Alternation is immediate, and the Jacobi identity in $\hat{\mathfrak{g}}$ translates into the two-cocycle condition ${\sum }_{\mathrm{cyc}}(\rho (X)\omega (Y,Z)-\omega ([X,Y],Z))=0$, so $\omega$ is a Chevalley–Eilenberg two-cocycle. Changing the section $s$ by a linear map $t:\mathfrak{g}\to M$ changes $\omega$ by the coboundary $dt$, so the class $[\omega ]\in H^2(\mathfrak{g},M)$ is independent of the section. Conversely, any two-cocycle $\omega$ defines a bracket on $M\oplus \mathfrak{g}$ by $[(m,X),(n,Y)]=(\rho (X)n-\rho (Y)m+\omega (X,Y),[X,Y])$; the cocycle condition is exactly the Jacobi identity for this bracket, producing an extension whose class is $[\omega ]$. The split extension, the semidirect product, has section a Lie map and hence cocycle zero, so it corresponds to $0\in H^2(\mathfrak{g},M)$. $\square$

Proposition (First lemma forces inner derivations). If $\mathfrak{g}$ is semisimple and $M$ is a finite-dimensional module on which the Casimir $\Omega$ acts invertibly, every derivation $\delta :\mathfrak{g}\to M$ is inner.

Proof. Write $\Omega ={\sum }_a{x}_a{y}^a$ for dual bases of $\mathfrak{g}$ under the Killing form. Because $\Omega$ is central, $\rho (\Omega )$ commutes with every $\rho (X)$. Set $m=\rho (\Omega {)}^{-1}{\sum }_a\rho (x_a)\delta (y^a)$. A computation using the cocycle identity $\delta ([X,Y])=\rho (X)\delta (Y)-\rho (Y)\delta (X)$ and the invariance of the form under the bracket shows ${\sum }_a\rho (x_a)\delta (y^a)$ transforms as $\rho (\Omega )$ applied to the inner derivation generated by $\delta$; dividing by the invertible $\rho (\Omega )$ gives $\delta (X)=\rho (X)m$ for all $X$. Hence $\delta$ is inner and $H^1(\mathfrak{g},M)=0$ on the invertible part. The $\Omega$-zero part contributes only zero-action summands, killed by $\mathfrak{g}=[\mathfrak{g},\mathfrak{g}]$. $\square$

Connections Master

The enveloping algebra 07.06.02 supplies the home of the Casimir element and the derived-functor description: $H^n(\mathfrak{g},M)={\mathrm{Ext}}_{U(\mathfrak{g})}^n(k,M)$, so this whole theory is homological algebra over $U(\mathfrak{g})$ with the Chevalley–Eilenberg complex as the Koszul resolution of the base field with zero action.

The Lie algebra itself 03.04.01 provides the bracket and the Jacobi identity that make $d\circ d=0$ hold; the cochain spaces $\mathrm{Hom}({\Lambda }^n\mathfrak{g},M)$ are built directly from its underlying vector space.

The Casimir element and Weyl complete reducibility 07.06.22 are re-proved here cohomologically: the Casimir contracts the complex, making $H^1$ vanish, and that vanishing is exactly the statement that every short exact sequence of finite-dimensional modules splits.

The Lie algebroid Chevalley–Eilenberg differential 03.04.22 is the geometric generalisation: a Lie algebroid over a point is an ordinary Lie algebra, and its algebroid cohomology recovers the complex defined here as a special case.

Forward, the Hochschild–Serre spectral sequence 07.06.24 organises the cohomology of $\mathfrak{g}$ from that of an ideal $\mathfrak{h}$ and the quotient $\mathfrak{g}/\mathfrak{h}$, with the five-term inflation–restriction sequence as its low-degree shadow.

Historical & philosophical context Master

Claude Chevalley and Samuel Eilenberg introduced the cochain complex now bearing their names in 1948 ^{[Chevalley-Eilenberg 1948]}, with the explicit aim of giving an algebraic account of the real cohomology of a compact Lie group: their paper proves that the cohomology of the group equals the cohomology of the complex of invariant forms, which is an entirely algebraic object built from the Lie algebra. This was one of the founding moments of homological algebra, predating the general theory of derived functors that later subsumed it.

The vanishing theorems are older still. J. H. C. Whitehead established the vanishing of the relevant low-degree cohomology for semisimple infinitesimal groups in 1937 ^{[Whitehead 1937]}, working in the language of solving certain equations in the universal algebra rather than cohomology, which had not yet been named. Serre's lectures ^{[Serre 1965]} later placed these lemmas at the centre of Part I, deriving complete reducibility and the Levi decomposition from them, and Weibel ^{[Weibel 1994]} recast the entire subject as Ext over the enveloping algebra. The philosophical lesson is that rigidity of an algebraic structure is most cleanly expressed as the vanishing of an obstruction group, a viewpoint that now pervades deformation theory and representation theory alike.

Bibliography Master

@article{chevalley-eilenberg1948,
  author = {Chevalley, Claude and Eilenberg, Samuel},
  title = {Cohomology theory of {Lie} groups and {Lie} algebras},
  journal = {Trans. Amer. Math. Soc.},
  volume = {63},
  pages = {85--124},
  year = {1948}
}

@article{whitehead1937,
  author = {Whitehead, J. H. C.},
  title = {Certain equations in the algebra of a semi-simple infinitesimal group},
  journal = {Quart. J. Math.},
  volume = {8},
  pages = {220--237},
  year = {1937}
}

@book{serre-lalg,
  author = {Serre, Jean-Pierre},
  title = {Lie Algebras and Lie Groups},
  series = {Lecture Notes in Mathematics},
  volume = {1500},
  publisher = {Springer},
  year = {2006}
}

@book{weibel1994,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  year = {1994}
}

@book{knapp1988,
  author = {Knapp, Anthony W.},
  title = {Lie Groups, Lie Algebras, and Cohomology},
  publisher = {Princeton University Press},
  year = {1988}
}