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 LetGbe a Lie group equal to one ofSO(n)orSU(n), 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 compact Lie algebra is trivial.
7 Prove that a real Lie algebra with a positive-definite Cartan-Killing form must be zero.
8 A Lie algebragis callednilpotentif thelower central seriesof ideals ofg C0g⊃ C1g⊃ · · · ⊃Cqg⊃ · · · ,
defined byC0g =gandCqg = [g,Cq−1g]forq ≥ 1, terminates at zero, that is,Cpg= 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,adX is nilpotent as an endomorphism ofg(that is,admx = 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 formulaadXY =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 14ω coincides with the volume form. De-duce that the bi-invariant integral onS3 is given byR
S3f(g)dµ(g) =
1 8π2
R
S3f ωforf ∈C(S3).
3.A Existence of compact real forms
Proof of Theorem 3.3.7.A complex Lie algebra of dimensionncan be thought ofCnwith a skew-symmetric multiplication satisfying the Jacobi identity.
In other words, the Lie bracket belongs to the spaceVn = Λ2(Cn∗)⊗CCn and its coordinates satisfy quadratic polynomial equations corresponding to the Jacobi equation. Fix a basis(e1, . . . , en)ofCn. Then
µ(ei, ej) = Xn k=1
µkijek
for someµkij ∈C. The Jacobi condition is Xn
m=1
(µmijµℓmk+µmjkµℓmi+µmkiµℓmj) = 0
for all i, j, k, ℓ = 1, . . . , n. Now there is a closed subvariety Ln of Vn parametrizing all complexn-dimensional Lie algebras.
Consider the natural action ofG:=GL(n,C)onVn, namely, g·µ(x, y) :=g(µ(g−1x, g−1y))
forg ∈ G, µ ∈ Vn, 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 coordinatesνijk all real and a negative definite Cartan-Killing form.
Denote the Cartan-Killing form of ν ∈ Ln by Bν. 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 Bµ. This will ensure that a potencial real form is compact. So let
H={g∈G|Bgµ =Bµ} 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|Bν =Bµ}. Note thatZis a closed subvariety ofLn. Plainly,Y =X∩Z. For allν ∈Z,(Cn, ν)is semisimple and
dimH(ν) = dimH−dimHν = dimH−n
is independent ofν, asHν = 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, . . . , en)is a unitary basis. This specifies a subgroupKofGisomorphic toU(n). Recall that its Lie algebrakis a real form ofg.
Consider ρµ : H → Rgiven byρµ(h) = ||hµ||2. Since H(µ) is closed inLn, there exists a point of minimum ofρµ, 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
||Aµ||2− ||Aµ||2 =ℜ([A, A∗]µ, µ) = 0.
This equation shows thatAµ = 0if and only ifA∗µ= 0, that is, alsohµis invariant under taking the transpose conjugate matrix. This means
hµ = hµ∩so(n) +hµ∩(√
−1so(n))
= hµ∩so(n) +√
−1(hµ∩so(n)).
We have shown that hµ ∩so(n) is a compact real form of hµ. But hµ =
Der(Cn, µ)is isomorphic to(Cn, µ).
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⊕ · · · ⊕ArandV =V1⊕ · · · ⊕Vr, whereAiis a scalar operator onVifor 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 byadX, for allX∈g. We first observe thatadX :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]= [adX,adY]for allX,Y ∈g, we need to consider operatorsadX :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 ia an inner product on its Lie alge-brauwith respect to which all transformationsAdg :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
55
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
eit1
...
eitn
t1, . . . , tn∈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 ...
costn −sintn
sintn costn
: t1, . . . , tn∈R
is a torus inSO(2n),
cost1 −sint1
sint1 cost1 ...
costn −sintn
sintn costn 1
: t1, . . . , tn∈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{Ui}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 ⊃
· · · ⊃Ci−1 andCi−1has sideǫ. Take an integerNi >1/ǫ. ThenCiN−i1 =Tn.
4.2. CARTAN SUBALGEBRAS 57