Whitehead's Lemmas and the Rigidity of Semisimple Algebras

Derives Lemma A37.2. In degree zero, Equations (A37.2) and (A37.3) give

which is precisely the statement that \(\rho\) is a homomorphism.

In degree one, write \(g = \dd f\) and expand \(\left(\dd g\right)(X,Y,Z)\) by Equation (A37.4), substituting Equation (A37.3) in each of the six terms. Suppressing \(\rho\) and writing \(Xu\) for \(\rho(X)u\), the six terms are

Add them and collect. The terms carrying \(f(Z)\) are \(XY - YX - \comm{X}{Y}\), which vanish because \(\rho\) is a homomorphism; likewise those carrying \(f(Y)\), namely \(-XZ + ZX + \comm{X}{Z}\), and those carrying \(f(X)\), namely \(YZ - ZY - \comm{Y}{Z}\). The terms \(\pm Xf\left(\comm{Y}{Z}\right)\), \(\pm Yf\left(\comm{X}{Z}\right)\) and \(\pm Zf\left(\comm{X}{Y}\right)\) cancel in pairs. What remains is

because \(-\comm{\comm{X}{Z}}{Y} = \comm{\comm{Z}{X}}{Y}\) and the argument is then the Jacobi identity.

Derives Lemma A37.5. Expand \(\comm{T_{a}}{Y} = \mu_{a}{}^{b}T_{b}\) and \(\comm{T^{a}}{Y} = \nu^{a}{}_{b}T^{b}\), so that \(\mu_{a}{}^{b} = \beta\left(\comm{T_{a}}{Y},T^{b}\right)\) and \(\nu^{a}{}_{b} = \beta\left(\comm{T^{a}}{Y},T_{b}\right)\). By Equation (A37.9) and the symmetry of \(\beta\),

The left-hand side of Equation (A37.11) is therefore \(\sum_{a,b}\left(\nu^{a}{}_{b} + \mu_{b}{}^{a}\right) T^{b}\otimes T_{a}\), after renaming the summation indices in the second term, and each coefficient vanishes by Equation (A37.12).

For the last statement apply the bilinear map \((u,w) \mapsto \rho(u)\rho(w)\) to Equation (A37.11):

which is the image of Equation (A37.11) and hence zero.

Derives Lemma A37.6. Symmetry is cyclicity of the trace, and invariance is the computation Equation (A31.15): \(\tr\left(\rho\left(\comm{X}{Y}\right) \rho(Z)\right) = \tr\left(\rho(X)\rho\left(\comm{Y}{Z}\right)\right)\). Let \(\mathfrak{s}\) be the radical of \(\beta_{\mathbb{V}}\); it is an ideal by the argument of Equation (A31.16) with \(\beta_{\mathbb{V}}\) in place of \(\kappa\). For \(W \in \comm{\mathfrak{s}}{\mathfrak{s}}\) and \(Y \in \mathfrak{s}\), \(\tr\left(\rho(W)\rho(Y)\right) = \beta_{\mathbb{V}}(W,Y) = 0\), so Theorem A31.11 makes \(\rho(\mathfrak{s})\) solvable; as \(\rho\) is faithful, \(\mathfrak{s}\) is a solvable ideal of a semisimple algebra, hence zero (Remark A31.4). Finally

Derives Theorem A37.7. Degree one. Let \(f\) be a \(1\)-cocycle, so that by Equation (A37.3)

Put \(X = T_{a}\), apply \(\rho\left(T^{a}\right)\) and sum over \(a\). The first term gives \(\Gamma f(Y)\). In the second, commute \(\rho\left(T^{a}\right)\) past \(\rho(Y)\),

and set

The result is

The last two terms are the image of the invariance identity Equation (A37.11) under the bilinear map \((u,w) \mapsto \rho(u)f(w)\), hence vanish. So \(\Gamma f(Y) = \rho(Y)v\) for every \(Y\). Since \(\Gamma\) commutes with every \(\rho(Y)\) (Lemma A37.5) so does \(\Gamma^{-1}\), and therefore

by Equation (A37.2): every \(1\)-cocycle is a coboundary.

Degree two. Let \(c\) be a \(2\)-cocycle. Put \(X = T_{a}\) in \(\left(\dd c\right)(X,Y,Z) = 0\), apply \(\rho\left(T^{a}\right)\), sum over \(a\), and set

The six terms of Equation (A37.4) contribute as follows. The first gives \(\Gamma c(Y,Z)\) outright. In the second and the third, Equation (A37.18) moves \(\rho\left(T^{a}\right)\) past \(\rho(Y)\) and \(\rho(Z)\), producing \(-\rho(Y)b(Z)\) and \(+\rho(Z)b(Y)\) together with one commutator term each. The fourth and fifth are unchanged. The sixth is \(b\left(\comm{Y}{Z}\right)\), since \(-c\left(\comm{Y}{Z},T_{a}\right) = +c\left(T_{a},\comm{Y}{Z}\right)\). Collecting,

where the leftover is

Both lines of Equation (A37.24) vanish: applying the bilinear map \((u,w) \mapsto \rho(u)c(w,Z)\) to Equation (A37.11) gives

and the same computation with \(Y\) and \(Z\) exchanged kills the second line. So Equation (A37.23) leaves

that is, by Equation (A37.3), \(\Gamma c = \dd b\). As in degree one, \(\Gamma^{-1}\) commutes with every \(\rho(Y)\) and therefore with \(\dd\), so \(c = \dd\left(\Gamma^{-1}b\right)\): every \(2\)-cocycle is a coboundary.

Derives Lemma A37.9. The argument is that of Theorem 9.158, with the family of operators \(\set{U(a)}\) replaced by \(\set{\rho(X)}\): the characteristic polynomial of \(A\) has a root \(\lambda\) because \(\mathbb{F}\) is algebraically closed and \(\dim\mathbb{V} \ge 1\), so \(\ker\left(A - \lambda\identity\right) \neq 0\); that kernel is a submodule, because \(A - \lambda\identity\) commutes with the action; and irreducibility forces it to be all of \(\mathbb{V}\).

Derives Lemma A37.10. First, \(\rho(\mathfrak{g})\mathbb{V} \subseteq \mathbb{W}\): the quotient \(\mathbb{V}/\mathbb{W}\) is a one-dimensional module, so the action on it is a homomorphism of \(\mathfrak{g}\) into an abelian algebra, which annihilates \(\comm{\mathfrak{g}}{\mathfrak{g}} = \mathfrak{g}\) by Corollary A31.13.

Next, we may assume \(\rho\) faithful. Otherwise put \(\mathfrak{k} = \ker\rho\), an ideal; by Corollary A31.13, \(\mathfrak{g}/\mathfrak{k} \cong \mathfrak{k}^{\perp}\) is again semisimple, and \(\mathbb{V}\) is a module over it with the same submodules. If \(\mathfrak{g}/\mathfrak{k} = 0\) the action is trivial, every subspace is a submodule and any complementary line serves.

Induct on \(\dim\mathbb{W}\).

Case 1: \(\mathbb{W}\) has a submodule \(\mathbb{W}'\) with \(0 \neq \mathbb{W}' \neq \mathbb{W}\). In \(\mathbb{V}/\mathbb{W}'\) the submodule \(\mathbb{W}/\mathbb{W}'\) has codimension one and smaller dimension, so by induction there is a one-dimensional submodule \(\widetilde{\mathbb{X}}/\mathbb{W}'\) with \(\mathbb{V}/\mathbb{W}' = \mathbb{W}/\mathbb{W}' \oplus \widetilde{\mathbb{X}}/\mathbb{W}'\). Inside \(\widetilde{\mathbb{X}}\) the submodule \(\mathbb{W}'\) has codimension one and \(\dim\mathbb{W}' < \dim\mathbb{W}\), so by induction again \(\widetilde{\mathbb{X}} = \mathbb{W}' \oplus \mathbb{X}\) with \(\mathbb{X}\) a one-dimensional submodule. Then \(\mathbb{X} \cap \mathbb{W} \subseteq \widetilde{\mathbb{X}} \cap \mathbb{W} = \mathbb{W}'\) and \(\mathbb{X} \cap \mathbb{W}' = 0\), so \(\mathbb{X} \cap \mathbb{W} = 0\) and dimensions give \(\mathbb{V} = \mathbb{W}\oplus\mathbb{X}\).

Case 2: \(\mathbb{W}\) is irreducible. If \(\rho(\mathfrak{g})\mathbb{W} = 0\) then \(\mathbb{W}\) is one-dimensional, \(\dim\mathbb{V} = 2\), and every product of two elements of \(\rho(\mathfrak{g})\) vanishes because \(\rho(\mathfrak{g})\mathbb{V}\subseteq\mathbb{W}\); so \(\rho(\mathfrak{g})\) is abelian and \(\rho(\mathfrak{g}) = \rho\left(\comm{\mathfrak{g}}{\mathfrak{g}}\right) = 0\), and any complementary line serves. So assume \(\rho(\mathfrak{g})\mathbb{W} \neq 0\), and let \(\Gamma\) be the Casimir built from the trace form \(\beta_{\mathbb{V}}\) of Lemma A37.6, which is available because \(\rho\) is faithful. By Lemma A37.5 \(\Gamma\) commutes with the action; \(\Gamma\mathbb{V} \subseteq \rho(\mathfrak{g})\mathbb{V} \subseteq \mathbb{W}\), since \(\Gamma\) is a sum of products of two operators \(\rho(\cdot)\); hence \(\Gamma\) acts as zero on \(\mathbb{V}/\mathbb{W}\) and

by Lemma A37.6. On the irreducible \(\mathbb{W}\), Lemma A37.9 gives \(\Gamma|_{\mathbb{W}} = \lambda\identity\) with \(\lambda\dim\mathbb{W} = \dim\mathfrak{g}\), so \(\lambda \neq 0\) in characteristic zero and \(\Gamma|_{\mathbb{W}}\) is invertible. Therefore \(\Gamma\mathbb{V} = \mathbb{W}\), \(\dim\ker\Gamma = \dim\mathbb{V} - \dim\mathbb{W} = 1\), and \(\ker\Gamma \cap \mathbb{W} = 0\). As \(\Gamma\) commutes with the action, \(\ker\Gamma\) is a submodule, and \(\mathbb{V} = \mathbb{W}\oplus\ker\Gamma\).

Derives Theorem A37.11. It suffices to show that every submodule \(\mathbb{W} \subseteq \mathbb{V}\) has a complementary submodule; the decomposition then follows by induction on \(\dim\mathbb{V}\). Assume \(0 \neq \mathbb{W} \neq \mathbb{V}\) and let \(\mathcal{H} = \operatorname{Hom}(\mathbb{V},\mathbb{W})\), a \(\mathfrak{g}\)-module under

Let \(\mathcal{V} \subseteq \mathcal{H}\) be the set of \(\phi\) whose restriction to \(\mathbb{W}\) is a scalar multiple of the identity, and \(\mathcal{W} \subseteq \mathcal{V}\) the set of \(\phi\) vanishing on \(\mathbb{W}\). Both are subspaces, and the map \(\mathcal{V} \to \mathbb{F}\) sending \(\phi\) to its scalar is linear, surjective — any projection of \(\mathbb{V}\) onto \(\mathbb{W}\) lies in \(\mathcal{V}\) with scalar \(1\) — and has kernel \(\mathcal{W}\), so \(\mathcal{W}\) has codimension one in \(\mathcal{V}\).

They are submodules: if \(\phi|_{\mathbb{W}} = \lambda\identity\) then for \(w \in \mathbb{W}\),

using that \(\mathbb{W}\) is a submodule; so \(X\cdot\mathcal{V} \subseteq \mathcal{W} \subseteq \mathcal{V}\).

Apply Lemma A37.10 to \(\mathcal{W} \subseteq \mathcal{V}\): there is a one-dimensional submodule \(\mathbb{F}\phi_{0}\) complementary to \(\mathcal{W}\), and \(\phi_{0}\) may be normalized so that \(\phi_{0}|_{\mathbb{W}} = \identity\). Being a one-dimensional submodule with \(X\cdot\mathcal{V}\subseteq\mathcal{W}\), it satisfies \(X\cdot\phi_{0} \in \mathbb{F}\phi_{0}\cap\mathcal{W} = 0\): that is, \(\phi_{0}\) is a homomorphism of modules. Its kernel is therefore a submodule; it meets \(\mathbb{W}\) in \(0\) and has dimension \(\dim\mathbb{V} - \dim\mathbb{W}\) because \(\phi_{0}\) is onto \(\mathbb{W}\). Hence \(\mathbb{V} = \mathbb{W}\oplus\ker\phi_{0}\).

Derives Proposition A37.12. Degree one. With \(\rho = 0\), Equation (A37.3) makes a \(1\)-cocycle a linear map \(f\) with \(f\left(\comm{X}{Y}\right) = 0\), that is, a linear map vanishing on \(\comm{\mathfrak{g}}{\mathfrak{g}} = \mathfrak{g}\) (Corollary A31.13); so \(f = 0\). The coboundaries are zero as well, and \(H^{1} = 0\).

Degree two. Let \(c\) be a \(2\)-cocycle. By the construction in the proof of Proposition 18.76 — “every class is realized” — the vector space \(\tilde{\mathfrak{g}} = \mathfrak{g}\oplus\mathbb{F}Z\) with \(Z\) central and the bracket Equation (18.120) is a Lie algebra, the Jacobi identity being Equation (18.121), which by Remark A37.3 is \(\dd c = 0\).

Make \(\tilde{\mathfrak{g}}\) a \(\mathfrak{g}\)-module. For \(\xi \in \mathfrak{g}\) and \(u \in \tilde{\mathfrak{g}}\) set \(\xi \cdot u = \comm{\tilde{\xi}}{u}\), where \(\tilde{\xi}\) is any preimage of \(\xi\) under the projection \(\pi : \tilde{\mathfrak{g}} \to \mathfrak{g}\). This is well defined, because two preimages differ by a multiple of the central \(Z\); and it is a module structure, because \(\comm{\widetilde{\comm{\xi}{\eta}}}{u} = \comm{\comm{\tilde{\xi}}{\tilde{\eta}}}{u}\) — the two lifts differ by a central element — and the right-hand side expands by the Jacobi identity of \(\tilde{\mathfrak{g}}\) into \(\comm{\tilde{\xi}}{\comm{\tilde{\eta}}{u}} - \comm{\tilde{\eta}}{\comm{\tilde{\xi}}{u}}\).

The line \(\mathbb{F}Z\) is a submodule, \(Z\) being central. By Theorem A37.11 it has a complementary submodule \(\mathfrak{m}\), so \(\tilde{\mathfrak{g}} = \mathbb{F}Z\oplus\mathfrak{m}\) with \(\comm{\tilde{\mathfrak{g}}}{\mathfrak{m}} \subseteq \mathfrak{m}\); in particular \(\mathfrak{m}\) is a subalgebra, and \(\pi|_{\mathfrak{m}} : \mathfrak{m} \to \mathfrak{g}\) is a bijective homomorphism of Lie algebras. Its inverse \(s = \left(\pi|_{\mathfrak{m}}\right)^{-1}\) is a linear splitting of \(\pi\) which is also a homomorphism, so the cocycle it defines through Equation (18.120) is \(\comm{s(\xi)}{s(\eta)} - s\left(\comm{\xi}{\eta}\right) = 0\).

Finally, two linear splittings of \(\pi\) differ by a linear map \(\mathfrak{g} \to \mathbb{F}\), and by the first part of the proof of Proposition 18.76 the cocycles they define differ by a coboundary. The splitting \(\xi \mapsto (\xi,0)\) defines \(c\) and the splitting \(s\) defines \(0\); hence \(c\) is a coboundary and \(H^{2}\left(\mathfrak{g},\mathbb{F}\right) = 0\).

Derives Lemma A37.13. Write \(\mathfrak{g} = \mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{s}\) as a direct sum of simple ideals (Corollary A31.13) and let \(\kappa_{j}\) be the Killing form of \(\mathfrak{g}_{j}\), extended to \(\mathfrak{g}\) by declaring the other summands orthogonal to \(\mathfrak{g}_{j}\) and to each other. Each \(\kappa_{j}\) is invariant and, by Corollary A31.14, \(\kappa = \sum_{j}\kappa_{j}\). For nonzero scalars \(t_{1},\ldots,t_{s}\) put \(\beta_{t} = \sum_{j}t_{j}\kappa_{j}\), an invariant symmetric form, nondegenerate because each \(\kappa_{j}\) is nondegenerate on \(\mathfrak{g}_{j}\). Choosing a basis adapted to the decomposition, the \(\beta_{t}\)-dual basis is \(t_{j}^{-1}\) times the \(\kappa\)-dual basis in block \(j\), so

the inner sum running over a basis of \(\mathfrak{g}_{j}\) with its \(\kappa_{j}\)-dual. Each \(\Gamma_{j}\) commutes with \(\rho\left(\mathfrak{g}_{j}\right)\) by Lemma A37.5 applied inside \(\mathfrak{g}_{j}\), and with \(\rho\left(\mathfrak{g}_{i}\right)\) for \(i \neq j\) because \(\comm{\mathfrak{g}_{i}}{\mathfrak{g}_{j}} = 0\); so it commutes with the whole action, and Lemma A37.9 gives \(\Gamma_{j} = \lambda_{j}\identity\) on the irreducible \(\mathbb{V}\).

The \(\lambda_{j}\) are not all zero. Let \(\beta_{\mathbb{V}}\) be the trace form Equation (A37.14), which is invariant but need not be nondegenerate. Its restriction to the simple ideal \(\mathfrak{g}_{j}\) is a multiple of \(\kappa_{j}\): writing \(\beta_{\mathbb{V}}(X,Y) = \kappa_{j}(AX,Y)\) for \(X,Y \in \mathfrak{g}_{j}\) with \(A \in \operatorname{End} \left(\mathfrak{g}_{j}\right)\), invariance of both forms gives

for all \(Z \in \mathfrak{g}_{j}\), so \(A\) commutes with every \(\ad_{Z}\). The adjoint representation of a simple algebra is irreducible — its submodules are its ideals — so Lemma A37.9 gives \(A = c_{j}\identity\) and \(\beta_{\mathbb{V}}|_{\mathfrak{g}_{j}} = c_{j}\kappa_{j}\). Now

If \(c_{j} = 0\) then \(\beta_{\mathbb{V}}\) vanishes on \(\mathfrak{g}_{j}\), so \(\tr\left(\rho(X)\rho(Y)\right) = 0\) for all \(X \in \comm{\mathfrak{g}_{j}}{\mathfrak{g}_{j}} = \mathfrak{g}_{j}\) and \(Y \in \mathfrak{g}_{j}\), and Theorem A31.11 makes \(\rho\left(\mathfrak{g}_{j}\right)\) solvable; being a homomorphic image of the simple \(\mathfrak{g}_{j}\) it is either isomorphic to \(\mathfrak{g}_{j}\), which is not solvable, or zero. So \(c_{j} = 0\) forces \(\rho\left(\mathfrak{g}_{j}\right) = 0\). Since \(\rho \neq 0\), some \(\mathfrak{g}_{j_{0}}\) acts nontrivially and \(\lambda_{j_{0}} \neq 0\).

Choice of \(t\). The eigenvalue of \(\Gamma_{t}\) on \(\mathbb{V}\) is \(\sum_{j}t_{j}^{-1}\lambda_{j}\). If \(\sum_{j}\lambda_{j} \neq 0\) take every \(t_{j} = 1\). Otherwise take \(t_{j_{0}} = 2\) and \(t_{j} = 1\) for \(j \neq j_{0}\); the eigenvalue is then \(\sum_{j}\lambda_{j} - \tfrac{1}{2}\lambda_{j_{0}} = -\tfrac{1}{2}\lambda_{j_{0}} \neq 0\). In either case \(\Gamma_{t}\) is a nonzero scalar on \(\mathbb{V}\), hence invertible.

Derives Theorem A37.14. Reduction to an algebraically closed field. Each \(C^{n}(\mathfrak{g},\mathbb{V})\) is a finite-dimensional \(\mathbb{K}\)-vector space and \(\dd\) is \(\mathbb{K}\)-linear; extending scalars to \(\mathbb{F}\) gives \(C^{n}(\mathfrak{g},\mathbb{V})\otimes_{\mathbb{K}}\mathbb{F} = C^{n}\left(\mathfrak{g}_{\mathbb{F}}, \mathbb{V}_{\mathbb{F}}\right)\), with the same differential, because a multilinear map is fixed by its values on a \(\mathbb{K}\)-basis. The ranks of \(\dd\) in each degree are unchanged by field extension, so \(H^{n}(\mathfrak{g},\mathbb{V})\otimes_{\mathbb{K}}\mathbb{F} = H^{n}\left(\mathfrak{g}_{\mathbb{F}}, \mathbb{V}_{\mathbb{F}}\right)\), and a space vanishes if and only if its extension does. Moreover \(\mathfrak{g}_{\mathbb{F}}\) is again semisimple by Remark A31.12. It therefore suffices to prove the theorem over \(\mathbb{F}\).

Reduction to irreducible coefficients. By Theorem A37.11, \(\mathbb{V} = \mathbb{V}_{1}\oplus\cdots\oplus\mathbb{V}_{r}\) with each \(\mathbb{V}_{i}\) irreducible. A cochain with values in a direct sum is the same thing as a tuple of cochains with values in the summands, and \(\dd\) acts componentwise, so

The two cases. If \(\mathfrak{g}\) acts as zero on the irreducible \(\mathbb{V}_{i}\), then every subspace is a submodule, so \(\dim\mathbb{V}_{i} = 1\) and \(\mathbb{V}_{i} \cong \mathbb{F}\) with the trivial action; Proposition A37.12 gives \(H^{1} = H^{2} = 0\). Otherwise Lemma A37.13 supplies an invariant nondegenerate form whose Casimir is invertible on \(\mathbb{V}_{i}\), and Theorem A37.7 gives \(H^{1} = H^{2} = 0\). Summing over \(i\) in Equation (A37.34) completes the proof.

Derives Corollary A37.15. Take \(\mathbb{V} = \R\) with the trivial action in Theorem A37.14; by Remark A37.3 the group \(H^{2}\) so computed is the group \(H^{2}(\mathfrak{g},\R)\) of Definition 18.75, and by Proposition 18.76 its vanishing says that every central extension of \(\mathfrak{g}\) by \(\R\) is trivial, that is, isomorphic to \(\mathfrak{g}\oplus\R\) as a Lie algebra. That \(\mathfrak{so}(p,q)\) is semisimple for \(D \ge 3\) is Corollary A31.15, proved there by computing its Killing form and applying Theorem A31.1.