Differential Geometry—MTG 6257—Spring 2000 Problem Set 4

Notation for all the problems below. Unless otherwise specified, G and H are Lie groups with Lie algebras 𝔤 and 𝔥 respectively. Identity elements of groups are denoted e. Summation notation for repeated indices is used.

Definition. Let {X_{i}} be a basis of 𝔤. Then [X_{i}, X_{j}] = c^{k}_{ij}X_{k} for some scalars c^{k}_{ij} called the structure constants of 𝔤 relative to this basis. The Jacobi identity and antisymmetry of the Lie bracket impose constraints that the structure constants must satisfy.

1. Let {θ^{i}} be left-invariant 1-forms which pointwise are a basis of T^{∗}G. Let the vector fields {X_{i}} be the dual basis of T G.

(a) Show that the vector fields are {X_{i}} left-invariant.

(b) Let {c^{k}_{ij}} be the structure constants of 𝔤 relative to the basis {X_{i}}. Show that dθ^{i} = −\frac{1}{2}c^{k}_{ij}θ^{i} ∧ θ^{j}.

(c) Let ω ∈ Ω^{p}(G) be left-invariant. Show that if p = 0 and G is connected, then ω is constant, while if p > 0 then there exist constants \{a_{I}\} such that ω = \frac{1}{p!} ∑_{I} a_{I}θ^{I}, where I runs over all multi-indices (i_{1}, …, i_{p}) with and where θ^{I} means θ^{i_{1}} ∧ … ∧ θ^{i_{p}}.

2. Let V be a finite-dimensional vector space, V^{∗} its dual, A : V → V an isomorphism. Recall that A extends to an isomorphism Ã of the tensor algebra of V be declaring Ã(v_{1} ⊗ … ⊗ v_{p}) = A(v_{1}) ⊗ … ⊗ A(v_{p}). Recall that the adjoint A^{∗} : V^{∗} → V^{∗} is defined by A^{∗}(θ) = θ ∘ A; this extends similarly to the tensor algebra of V^{∗}.

(a) Show that Ã maps ⋀^{p}(V) to itself.

(b) Let {A_{t}} be a 1-parameter family of isomorphisms of V, where t ranges over an open interval containing 0, and let a = \frac{d}{dt}A_{t}|_{t=0} ∈ End(V) (=Hom(V, V); “End” stands for “endomorphism” = linear transformation of a vector space to itself). Let a also stand for the derivative of Ã_{t}. Show that a is a derivation of the tensor algebra:

a(v_{1}⊗v_{2}⊗…⊗v_{p}) = a(v_{1})⊗v_{2}⊗…⊗v_{p}+v_{1}⊗a(v_{2})⊗…⊗v_{p}+…+v_{1}⊗v_{2}⊗…⊗a(v_{p}).

(c) From (a), (b), and the definition of wedge product, deduce that a acts as a derivation on the exterior algebra of V:

a(v_{1} ∧v_{2} ∧…∧v_{p}) = a(v_{1})∧v_{2} ∧…∧v_{p} +v_{1} ∧a(v_{2})∧…∧v_{p} +…+v_{1} ∧v_{2} ∧…∧a(v_{p}).

(d) If we replace V by V^{∗} and a by a^{∗}, the result in (c) still holds. Show that this result is equivalent to the statement that for ω ∈ ⋀^{p}(V^{∗}) and v_{1}, …, v_{p} in V we have

(a^{∗}ω)(v_{1}, v_{2}, …, v_{p}) = ω(a(v_{1}), v_{2}, …, v_{p}) + ω(v_{1}, a(v_{2}), …, v_{p}) + ω(v_{1}, v_{2}, …, a(v_{p})).

3. Apply problems 2cd to V = 𝔤, A_{t} = Ad_{exp(tX)} for some X ∈ 𝔤, and w ∈ ⋀^{p} 𝔤^{∗} for some p > 0; from class, we have a = ad_{X}. Specifically, show the following.

(a) (ad_{X})^{∗}ω(Y_{1}, …, Y_{p}) = ω([X, Y_{1}], Y_{2}, …, Y_{p}) + ω(Y_{1}, [X, Y_{2}], …, Y_{p}) + … + ω(Y_{1}, Y_{2}, …, [X, Y_{p}]).

(b) If {X_{i}} and {θ^{i}} dual bases of 𝔤 and 𝔤^{∗}, respectively, and {c^{k}_{ij}} are the structure constants relative to {X_{i}}, then ad_{X_{i}}^{∗}θ^{k} = c^{k}_{ij}θ^{j}.

(c) (Notation as in (b).) If we write ω = \frac{1}{p!} ∑_{I} a_{I}θ^{I} as in problem 1, and we choose {a_{i_{1}i_{2}...i_{p}}} to be antisymmetric in every pair of indices (as we always can do), then

ad_{X_{i}}^{∗}ω = \frac{1}{(p − 1)!}a_{k_{1}k_{2}...k_{p}}c^{k_{1}}_{ij}θ^{j} ∧ θ^{k_{2}} ∧ … ∧ θ^{k_{p}}.

4. (a) Let {X_{i}}, {θ^{i}} be dual bases of 𝔤, 𝔤^{∗} respectively, and let {c^{k}_{ij}} be the associated structure constants. Regarding each θ^{i} as a left-invariant form on G, use problems 1bc and 3c to show that if ω ∈ Ω^{p}(G) is left-invariant, then

dω = −θ^{i} ∧ ((ad_{X_{i}})^{∗}ω).

(b) As a corollary, deduce that if ω is bi-invariant then ω is closed. (You will have to handle the case deg(ω) = 0 separately, since in the preceding we always assumed deg(ω) > 0.) Hence, as stated in class, if G is compact then H^{∗}(G) ≅ Ω_{bi}^{∗}(G). (This is an isomorphism not only of vector spaces, but of algebras, where the product on Ω^{∗}(G) is wedge-product and the product on H^{∗}(G) is induced by wedge product.)

5. The direct sum of Lie algebras 𝔤_{1}, 𝔤_{2} is the vector space 𝔤_{1} ⊕ 𝔤_{2} with the obvious bracket: for X_{1}, Y_{1} ∈ 𝔤_{1} and X_{2}, Y_{2} ∈ 𝔤_{2}, define [(X_{1}, X_{2}), (Y_{1}, Y_{2})]_{𝔤_{1}⊕𝔤_{2}} = ([X_{1}, Y_{1}]_{𝔤_{1}}, [X_{2}, Y_{2}]_{𝔤_{2}}). (The direct sum of any finite number of Lie algebras is defined analogously.) The direct product of Lie groups G_{1}, G_{2} is the manifold G_{1} × G_{2} with the group operation (g_{1}, g_{2}) · (h_{1}, h_{2}) = (g_{1}h_{1}, g_{2}h_{2}). Show the following.

(a) 𝔤1 ⊕ {0} and {0} ⊕ 𝔤_{2} are ideals in 𝔤_{1} ⊕ 𝔤_{2}.

(b) If 𝔤_{1}, 𝔤_{2} are the Lie algebras of G_{1}, G_{2} respectively, then the Lie algebra of G_{1} × G_{2} is 𝔤_{1} ⊕ 𝔤_{2}.

(Note: the converse of (b) is not quite true; if G has Lie algebra 𝔤_{1} ⊕ 𝔤_{2}, then G need not be isomorphic to G_{1} × G_{2}—one can only conclude that the universal covers of G and G_{1} × G_{2} must be isomorphic.)

6. A Lie algebra 𝔤 is called simple if (i) dim(𝔤) > 1 and (ii) 𝔤 has no nontrivial ideals (i.e. the only ideals are {0} and 𝔤). We call 𝔤 semisimple if it is the direct sum of simple ideals. We call a Lie group simple (respectively, semisimple) if its Lie algebra is simple (respectively, semisimple).

Remark. “Simple” in the context of Lie groups thus has a slightly different meaning from its conventional use in group theory, where “simple” means “no nontrivial normal subgroups”, and in particular implies that the center of the group is trivial.

A simple Lie group can have a discrete, nontrivial center, but cannot have a center of dimension greater than zero.

(a) Let 𝔤 be an arbitrary Lie algebra. Show that [𝔤, 𝔤] is an ideal in 𝔤. (Here, for any vector subspaces 𝔞, 𝔟 of 𝔤, the notation [𝔞, 𝔟] stands for the space of linear combinations of elements of the form [a, b], where a ∈ 𝔞, b ∈ 𝔟.)

(b) Show that the center 𝔷 of any Lie algebra 𝔤 is an ideal in 𝔤.

(c) Using (a) and (b), show that if 𝔤 is simple, then [𝔤, 𝔤] = 𝔤.

(d) Using (b) and (c), show that if 𝔤 is semisimple, then its center is trivial and [𝔤, 𝔤] = 𝔤.

7. Let G be a semisimple Lie group and let Ω_{bi}^{∗}(G) be the complex of bi-invariant differential forms.

(a) Show that if ω ∈ Ω^{p}(G), p > 0, is bi-invariant iff ad_{X}^{∗}ω ≡ 0 for all X ∈ 𝔤.

(b) Show that if ω ∈ Ω_{bi}^{1}(G), then ω ≡ 0. (Hint: part (a), plus 3b and 6c.)

(c) Show that if ω ∈ Ω_{bi}^{2}(G), then ω ≡ 0. (Same hint.)

(d) Deduce as a corollary of (b) and (c) that H^{1}(G) = 0 and H^{2}(G) = 0 for any compact semisimple Lie group G. (Problem 9 below will address H^{3}(G).)

8. The Killing form κ on a Lie algebra 𝔤 is the symmetric bilinear form defined by κ(X, Y) = tr(ad_{X} ∘ ad_{Y}). (The Killing form does not kill anything; it’s named after Mr. Killing.)

(a) Show that κ is indeed symmetric.

(b) Show that κ is Ad-invariant.

(c) Show that κ cannot be positive-definite. (Hint: ad_{X} would be skew-adjoint with respect to an Ad-invariant inner product.) Thus κ must either be indefinite, negative-definite, or degenerate. In (d) and (e) you will see that all of these possibilities are realized.

(d) Show that if X is in the center of 𝔤 then κ(X, Y) = 0 for all Y ∈ 𝔤. Hence if 𝔤 has a nontrivial center then κ is degenerate.

(e) Compute the Killing form on (i) 𝔰𝔬(3) and (ii) 𝔰𝔩(2, R). Remark. More generally it can be shown that the Killing form is nondegenerate iff 𝔤 is semisimple. (This is a theorem of Elie Cartan.)

9. (a) Show that if G is compact and semisimple, then H^{3}(G) ≠ 0. (Hint: 6d above, plus Problem Set 3 #8c.)

(b) Without using Cartan’s theorem, show that if the Killing form of 𝔤 is negativedefinite, then G is compact and H^{3}(G) ≠ 0. (Hint: 6d above, plus Problem Set 3 #10f.)

Remark. With more work, one can show that if G is compact and simple, then H^{3}(G) = R. More generally, if 𝔤 is semisimple then its decomposition as a direct sum of simple ideals is unique up to ordering, and if additionally G is compact then H^{3}(G) = R^{m}, where m is the number of simple ideals in the decomposition of 𝔤.

10. Let Θ ∈ Ω^{1}(G; 𝔤) be the Maurer-Cartan form. Show the following.

(a) [Θ, [Θ, Θ]] = 0.

(b) dΘ + \frac{1}{2}[Θ, Θ] = 0. (One way to do this is to use problem 1b, but there are others.)