Problems

3.4. PROBLEMS 51

3 Letgbe a compact simple Lie algebra. Prove that thead-invariant inner product ongis unique, up to a multiplicative constant. (Hint: Problem 12 in Chapter 1).

4 LetG be a Lie group equal to one ofSO(n)(n ≥ 3) orSU(n) (n ≥ 2), and denote its Lie algebra byg. Prove that for anyc >0

hX, Yi=−ctrace(XY),

whereX,Y ∈g, defines aAd-invariant positive definite inner product on g. Conclude that the Cartan-Killing form ofgis of this form for somec <0.

5 Explain whysl(2,R)is not a compact Lie algebra.

6 Prove that a complex Lie algebra whose realification is a compact Lie algebra must be Abelian.

7 Prove that a real Lie algebra with a positive-definite Cartan-Killing form must be zero-dimensional.

8 A Lie algebragis callednilpotentif thelower central seriesof ideals ofg C⁰g⊃ C¹g⊃ · · · ⊃C^qg⊃ · · · ,

defined byC⁰g =gandC^qg = [g,C^q−1g]forq ≥ 1, terminates at zero, that is,C^pg= 0for somep≥1.

a. Show that the space of strictly upper triangular matrices ingl(n,R)is a nilpotent Lie algebra.

b. Check that ifgis nilpotent then, for allX ∈ g,ad_X is nilpotent as an endomorphism ofg(that is,ad^m_x = 0for somem >0); we say thatgis ad-nilpotent(Engel’s Theorem is the statement that everyad-nilpotent Lie algebra is nilpotent).

c. Prove that the Cartan-Killing form of a nilpotent Lie algebra is null.

9 Compute the Killing form ofgl(n,F) for F = R orC directly from the formulaad_XY =XY −Y X.

10 Obtain the following expressions for the Cartan-Killing forms of the classical complex Lie algebras:

sl(n,C) :β(X, Y) = 2ntrace(XY);

so(n,C) :β(X, Y) = (n−2) trace(XY);

sp(n,C) :β(X, Y) = 2(n+ 1) trace(XY).

11 Let Gbe a compact connected Lie group of dimension at least3with Lie algebrag, and denote the Cartan-Killing form ofgbyβ.

3.A. EXISTENCE OF COMPACT REAL FORMS 53 a. Letωbe the left-invariant3-form onGwhose value at the identity is

ω1(X, Y, Z) =β([X, Y], Z)

forX,Y,Z ∈g. Prove thatωis skew-symmetric and right-invariant, so it defines a bi-invariant differential form of degree3onG.

b. In caseG=S³, show that ₁₆¹ ωcoincides with the volume form, with respect to some orientation. Deduce that the bi-invariant integral on S³is given byR

S³f(g)dµ(g) =±_32π¹² R

S³f ωforf ∈C(S³).

3.A Existence of compact real forms

Proof of Theorem 3.3.7. (Sketch) A complex Lie algebra of dimensionncan be thought ofCⁿwith a skew-symmetric multiplication satisfying the Ja-cobi identity. In other words, the Lie bracket belongs to the space V_n = Λ²(C^n∗)⊗CCⁿand its coordinates satisfy quadratic polynomial equations corresponding to the Jacobi equation. Fix a basis(e1, . . . , en)ofCⁿ. Then

µ(e_i, e_j) = Xn k=1

µ^k_ije_k for someµ^k_ij ∈C. The Jacobi condition is

Xn m=1

(µ^m_ijµ^ℓ_mk+µ^m_jkµ^ℓ_mi+µ^m_kiµ^ℓ_mj) = 0

for all i, j, k, ℓ = 1, . . . , n. Now there is a closed subvariety Ln of V_n parametrizing all complexn-dimensional Lie algebras.

Consider the natural action ofG:=GL(n,C)onV_n, namely, g·µ(x, y) :=g(µ(g⁻¹x, g⁻¹y))

forg ∈ G, µ ∈ V_n, x, y ∈ Cⁿ. It amounts to “change of basis” in the Lie algebra. The idea of this proof is to find a suitable basis whose real span will be a compact real form. Thus we identify the given complex semisimple Lie algebra with (Cⁿ, µ)and need to find g ∈ Gsuch that ν := g·µhas coordinatesν_ij^k all real and a negative definite Cartan-Killing form on the real span of(e₁, . . . , e_n).

Denote the Cartan-Killing form of ν ∈ Ln by B_ν. By semisimplicity, may assume the basis ofCⁿhas been chosen so that Bµ(ei, ej) = −δij for all i, j. We shall restrict to changes of basis that preserve B_µ. This will ensure that a potential real form is compact. So let

H={g∈G|B_gµ =B_µ} and consider the orbitH(µ) =:Y.

CLAIM. Y is a closed subvariety ofLn. In order to prove the claim, let X:=G(µ)andZ ={ν∈ Ln|B_ν =B_µ}. Note thatZis a closed subvariety ofLn. Plainly,Y =X∩Z. For allν ∈Z,(Cⁿ, ν)is semisimple and

dimH(ν) = dimH−dimH_ν = dimH−n

is independent ofν, asH_ν = Aut(Cⁿ, ν) = Der(Cⁿ, ν)⁰ ∼= (Cⁿ, ν)has di-mensionn. Now all orbits of H in Z have the same dimension. It is an elementary result of algebraic actions that the lowest dimensional orbit of HinZ is closed inZ. It follows that allH-orbits inZ are closed inZ, and hence inLⁿ, includingY, as we wished.

EndowCⁿwith the Hermitian inner product such that(e₁, . . . , e_n)is a unitary basis. This specifies a subgroupKofGisomorphic toU(n). Recall that its Lie algebrakis a real form ofg.

Consider ρ_µ : H → Rgiven byρ_µ(h) = ||hµ||². Since H(µ) is closed inLn, there exists a point of minimum ofρ_µ, which we may assume to be 1∈H. Therefore

(3.A.1) 0 = (dρµ)1(A) = 2ℜ(Aµ, µ) for allA∈h. Note that

h∼=so(n,C) =so(n) +√

−1so(n)⊂u(n) +√

−1u(n) =g,

so thathis invariant under taking the transpose conjugate matrix. Now we can apply (3.A.1) to[A, A^∗]and obtain

||A^∗µ||²− ||Aµ||²=ℜ([A, A^∗]µ, µ) = 0.

This equation shows thatAµ = 0if and only ifA^∗µ= 0, that is, alsoh_µis invariant under taking the transpose conjugate matrix. This means

h_µ = h_µ∩so(n) +h_µ∩(√

−1so(n))

= h_µ∩so(n) +√

−1(h_µ∩so(n)).

We have shown that h_µ ∩so(n) is a compact real form of h_µ. But h_µ = Der(Cⁿ, µ)is isomorphic to(Cⁿ, µ).

It remains to prove the uniqueness, up to conjugacy. We first observe that a real form of(Cⁿ, µ)is equivalent to a critical point ofρ_µ. Indeed, we first compute:

µ^k_ij =−B_µ(µ(e_i, e_j), e_k) =−B_µ(e_i, µ(e_j, e_k)) =µⁱ_jk and

δ_ij =−B_µ(e_i, e_j) =−X

kℓ

µ^ℓ_jkµ^k_iℓ=−X

kℓ

µ^k_iℓµ^k_ℓj =X

kℓ

µ^k_iℓµ^k_jℓ.

3.A. EXISTENCE OF COMPACT REAL FORMS 55 In particular,P

kℓ(µⁱ_kℓ)²=P

kℓ(µ^k_iℓ)²= 1for alli, and therefore X

ijk

(ν_ij^k)² =nfor allν ∈H(µ).

Hence

||ν||²=X

ijk

|ν_ij^k|² ≥X

ijk

ℜ(ν_ij^k)² =n, and equality holds if and only if allν_ij^k are real.

Now suppose(e^′₁, . . . , e^′_n)is another basis ofCⁿwhose real span yields a compact real form ofµ. We may choose this basis so thatBµ(e^′_i, e^′_j) =−δij

for ali,j. Leth ∈ Gbe such thathei = e^′_i for all i. Then ν := h⁻¹ ·µhas real coefficients in the basis(e₁, . . . , e_k)andB_h⁻1µ(e_i, e_j) = −1for alli,j.

It follows that h ∈ H. We have shown that a real form of(Cⁿ, µ) indeed corresponds to a point of minimum ofρµonH.

Next, suppose1andh⁻¹are two points of minimumρ_µinH=SO(n,C).

Writeh⁻¹=kexpAfork∈SO(n)andA∈√

−1so(n).⁴Put f(t) =||h⁻¹·µ||²=||exp(tA)·µ||² =X

i

e^2cit||µ_i||²,

whereµ_i are the eigenvectors of Aandc_i the corresponding eigenvalues.

Then f is a strictly convex function unless, for each i, we have µ_i = 0 or c_i = 0, that is, unless h⁻¹µ = kµ. Since f(0) and f(1) are points of minimum, we must haveh⁻¹µ=kµ∈(K∩H)(µ), whereK∩H=SO(n).

We have shown that the set of minima is theK∩H-orbit ofµ.

Finally,hkµ=µ, sohk∈H_µ= Aut(µ); moreover,k·span_R(e₁, . . . , e_n) = span_R(e₁, . . . , e_n), so

hk·span_R(e₁, . . . , e_n) =h·span_R(e₁, . . . , e_n) = span_R(e^′₁, . . . , e^′_n),

as wished.

4Need to explain this Cartan decomposition.

C H A P T E R 4

Root theory

One says an endomorphismAof a real or complex vector spaceV is semisim-pleif every invariant subspace admits an invariant complement. It is easy to see that, overC, this is equivalent toAbeing diagonalizable. In this case, we can writeA=A₁⊕ · · · ⊕A_randV =V₁⊕ · · · ⊕V_r, whereA_iis a scalar operator onV_ifor eachi.

In our case, for a given complex semisimple Lie algebrag(say the com-plexification of a compact Lie algebrau), we want to understand its fine structure, namely, describe its multiplication table in conceptual terms. It is very natural to look atad[g], the algebra of endomorphisms ofg gener-ated byad_X, for allX∈g. We first observe thatad_X :g→gis semisimple (not for allX ∈g, but) for allX ∈u, hence diagonalizable. This is already very good, but to be really useful we need a notion of simultaneous diag-onalization. A commuting family of semisimple endomorphisms can be diagonalized in the same basis. Sincead_[X,Y]= [ad_X,ad_Y]for allX,Y ∈g, we need to consider operatorsad_X :g→gwhereXbelongs to an Abelian subalgebrah. Of course, the bigger theh, the better. This brings us to the notion of a Cartan subalgebra (CSA) of a semisimple Lie algebra.

The counterpart of a CSA on the group level is the notion of a maximal torus. IfU is a compact Lie group, there is an inner product on its Lie alge-brauwith respect to which all transformationsAd_g :u→uare orthogonal and hence diagonalizable over C. In this chapter, we introduce maximal tori and CSA, but we will focus on the adjoint representation of a CSA. In later chapters, when we talk about the Weyl formulae and the Peter-Weyl theorem, we will come back to maximal tori.

4.1 Maximal tori

LetGbe a compact connected Lie group, letgdenote its Lie algebra and as-sumeGis not Abelian. We claimGcontains proper subgroups isomorphic to a torusTⁿ = S¹× · · · ×S¹, withn ≥1factors. In fact, for any nonzero X ∈ g, the closure of the image of the one-parameter subgroup defined

57

byX,

{exptX |t∈R},

is a compact connected Abelian subgroup, hence isomorphic to a torus. A maximal torusofGis a torus subgroup which is not properly contained in a bigger torus. By dimensional reasons, maximal tori exist.

4.1.1 Examples









 e^it1

...

e^itn





t₁, . . . , t_n∈R







is a torusT in U(n). This is a maximal torus ofU(n) because an element g ∈ U(n) that commutes with all elements ofT must lie in T. Indeed g commutes with a diagonal matrix with all entries distinct, which implies thatgpreserves its eigenspaces inCⁿ. Hencegis diagonal.

Similarly,























cost₁ −sint₁ sint₁ cost₁

...

cost_n −sint_n sintn costn









: t₁, . . . , t_n∈R













 is a torus inSO(2n),





























cost1 −sint1

sint₁ cost₁ ...

cost_n −sint_n sintn costn

1











: t₁, . . . , t_n∈R



















is a torus inSO(2n+ 1), and one checks these are also maximal tori.

4.1.2 Lemma LetTⁿ =Rⁿ/Zⁿ be ann-torus. ThenTⁿ ismonogenic, that is, there isg∈Tⁿsuch that the cyclic grouphgigenerated bygis dense inTⁿ.

Proof. Fix a countable basis{U_i}i∈Nof open sets ofTⁿ. Given any cube C₀inTⁿ(i.e. projection of a product of closed intervals inRⁿ), we shall con-struct a descending chain of cubesC0 ⊃C1 ⊃C2 ⊃ · · · whose intersection contains a generatorgas desired.

We proceed by induction. Suppose we have already definedC₀⊃C₁ ⊃

· · · ⊃C_i−1 andC_i−1has sideǫ. Take an integerN_i >1/ǫ. ThenC_i^N₋ⁱ₁ =Tⁿ.

4.1. MAXIMAL TORI 59 By continuity, we can find a cubeC_i contained inC_i−1such thatC_i^Ni ⊂U_i. Letg∈ ∩^∞i=0C_i. Theng^Ni ∈U_ifor alli, sohgiis dense inTⁿ. LetT be a maximal torus ofG, and denote its Lie algebra byt. Thent is an Abelian Lie subalgebra ofg. Indeed it is amaximal Abelian subalgebra, for ifsis an Abelian subalgebra containingt, then the associated connected Lie subgroupSofGis Abelian, so its closureS¯is a torus containingT. By maximality ofT,S¯=T and thuss⊂t.

For later use, we introduce the following terminology. Let g be a Lie algebra and letΣbe a subset ofg. ThecentralizerofΣingis

Zg(Σ) ={X ∈g : [X, Y] = 0for allY ∈Σ}. ThenormalizerofΣingis

Ng(Σ) ={X ∈g : [X, Y]∈Σfor allY ∈Σ}. Note thatZ_g(Σ)andN_g(Σ)are subalgebras ofg, by Jacobi.

Similarly, ifGis a Lie group with Lie algebrag, we can define the cen-tralizerand thenormalizerofΣinGrespectively as

Z_G(Σ) ={g∈G|Ad_gY =Y for allY ∈Σ} and

N_G(Σ) ={g∈G|Ad_gY ∈Σfor allY ∈Σ}. These are subgroups ofG(closed, ifΣis a closed subset ofg).

4.1.3 Theorem (É. Cartan’s Maximal Torus Theorem) Any two maximal tori in a compact connected Lie groupU are conjugate under an inner automorphism.

In particular, given a maximal torus T of G, every element of U is conjugate under an inner automorphism to an element inT; equivalently,Tintersects every conjugacy class ofG.

Proof. LetT andT^′ be maximal tori inU. It suffices to show that their Lie algebrastandt^′are conjugate under the adjoint representation. Choose H ∈ t that generates a dense one-parameter subgroup ofT Thent is the centralizer ofHinu.

LetH^′ ∈ube arbitrary and use the compactness ofU to choose a critical pointu₀∈U of the smooth functionf(u) =β(H,Ad_uH^′), whereβdenotes the Cartan-Killing form ofu. Then, forX∈u,

0 = d

dt

t=0β(H,Ad_(exptX)u0H^′)

= β(H,[X,Ad_u0H^′])

= β(X,[Ad_u0H^′, H]).

Since X is arbitrary and β is negative definite, [Ad_k0H^′, H] = 0. Now Ad_u0H^′ ∈t, and this proves the last assertion in the statement of the Theo-rem.

In particular, we apply the above reasoning to the caseH^′ generates a dense one-parameter subgroup oft^′. Then Ad_u0H^′ ∈ t implies that t ⊂ Zu(Ad_u0H^′) = Ad_u0Zu(H^′) = Ad_u0t^′. Nowdimt≤t^′and, by symmetry, we

obtain equality. Hencet= Ad_u0t^′.

We define the rank of a compact Lie group to be the dimension of a maximal torus. Since any two maximal tori are conjugate, they all have the same dimension.

4.1.4 Remark It follows from Theorem 4.1.3 that the exponential map of a compact connected Lie groupGis surjective. Indeed, it is explicitly surjec-tive in the case of a torus; any element ofGsits inside a maximal torusT;

sinceexp^T is the restriction ofexp^G, we are done (compare Remark 3.2.14).

No documento Lecture Notes on Compact Lie Groups and Their Representations (páginas 55-64)