Lattices

Roots and Associated Structures

6.3 Lattices

130 6 Roots and Associated Structures

P^∨ = {H ∈it|α(H)∈Zforα∈(g_C)}.

(c)Let kerE=kerE(T)be the lattice initgiven by kerE = {H ∈it|exp(2πi H)=I}.

(d)In general, if1is a lattice in(it)^∗that spans(it)^∗and if2is a lattice initthat spansit, define thedual lattices,^∗₁and^∗₂initand(it)^∗, respectively, by

^∗₁ = {H ∈it|λ(H)∈Zforλ∈1} ^∗₂ = {λ∈(it)^∗|λ(H)∈ZforH∈2}.

It is well known that^∗₁ and^∗₂ are lattices and that they satisfy^∗∗i = i

(Exercise 6.24). Notice kerEis a lattice by the proof of Theorem 5.2.

6.3.2 Relations

Lemma 6.25.Let G be a compact connected Lie group with Cartan subalgebra t.

For H ∈t,expH∈ Z(G)if and only ifα(H)∈2πiZfor allα∈(g_C).

Proof. Letg = expH and recall from Lemma 5.11 thatg ∈ Z(G)if and only if Ad(g)X = X for all X ∈ g. Now forα ∈ (g_C)∪ {0}and X ∈ g_α, Ad(g)X = e^adHX =e^α(H)X. The root decomposition finishes the proof.

Definition 6.26.LetGbe a compact Lie group andT a maximal torus. Writeχ(T) for thecharacter groupon T, i.e.,χ(T)is the set of all Lie homomorphismsξ : T →C\{0}.

Theorem 6.27.Let G be a compact Lie group with a maximal torus T . (a) R⊆A⊆P.

(b)Givenλ∈(it)^∗,λ∈ A if and only if there existsξ_λ∈χ(T)satisfying ξλ(expH)=e^λ(H)

(6.28)

for H ∈ t, whereλ ∈ (it)^∗ is extended to an element of(t_C)^∗ byC-linearity. The mapλ→ξλestablishes a bijection

A←→χ(T). (c)For semisimpleg,|P/R|is finite.

Proof. Letα∈(g_C)and supposeH ∈twith expH =e. Lemma 6.25 shows that α(H)∈ 2πiZ, so thatR ⊆ A. Next choose a standardsl(2,C)-triple{Eα,h_α,F_α} associated toα. As in the proof of Corollary 6.22, exp 2πi h_α = I. Thus ifλ ∈ A, λ(2πi h_α)∈2πiZ, so thatA⊆P which finishes part (a).

132 6 Roots and Associated Structures

For part (b), start withλ ∈ A. Using the fact that expt= T and using Lemma 6.25, Equation 6.28 uniquely defines a well-defined functionξλonT. It is a homo- morphism by Theorem 5.1. Conversely, if there is aξ_λ ∈χ(T)satisfying Equation 6.28, then clearlyλ(H) ∈ 2πiZwhenever expH = I, so thatλ ∈ A. Finally, to see that there is a bijection from Atoχ(T), it remains to see that the mapλ → ξ_λ is surjective. However, this requirement follows immediately by taking the differ- ential of an element ofχ(T)and extending viaC-linearity. Theorem 6.9 shows the differential can be viewed as an element of(it)^∗.

Next, Theorem 6.11 shows thatR spans(it)^∗ for semisimpleg. Part (c) there- fore follows immediately from elementary lattice theory (e.g., see [3]). In fact, it is straightforward to show|P/R|is equal to the determinant of the so-calledCartan

matrix(Exercise 6.42).

Theorem 6.29.Let G be a compact Lie group with a semisimple Lie algebragand let T be a maximal torus of G with corresponding Cartan subalgebrat.

(a) R^∗=P^∨. (b) P^∗=R^∨. (c)A^∗=kerE.

(d) P^∗⊆ A^∗ ⊆R^∗, i.e., R^∨⊆kerE⊆ P^∨. Proof. The equalitiesR^∗ =P^∨,

R^∨_∗

= P, and(kerE)^∗ = Afollow immediately from the definitions. This proves parts (a), (b), and (c) (Exercise 6.24). Part (d) fol-

lows from Theorem 6.27 (Exercise 6.24).

6.3.3 Center and Fundamental Group

The proof of part (b) of the following theorem will be given in §7.3.6. However, for the sake of comparison, part (b) is stated now.

Theorem 6.30.Let G be a connected compact Lie group with a semisimple Lie al- gebra and maximal torus T .

(a) Z(G)∼=P^∨/kerE ∼=A/R.

(b)π1(G)∼=kerE/R^∨∼= P/A.

Proof (part (a) only).By Theorem 5.1, Corollary 5.13, and Lemma 6.25, the expo- nential map induces an isomorphism

Z(G)∼= {H∈t|α(H)∈2πiZforα∈(g_C)}/{H ∈t|expH=I}

=(2πi P^∨) / (2πi kerE),

so that Z(G) ∼= P^∨/kerE. Basic lattice theory shows R^∗/A^∗ ∼= A/R (Exercise

6.24) which finishes the proof.

While the proof of part (b) of Theorem 6.30 is postponed until §7.3.6, in this section we at least prove the simply connected covering of a compact semisimple Lie group is still compact.

Let G be a compact connected Lie group and letG be the simply connected covering ofG. A priori, it is not known thatGis alinear groupand thus our devel- opment of the theory of Lie algebras and, in particular, the exponential map is not directly applicable toG. Indeed for more general groups, Gmay not be linear. As usual though, compact groups are nicely behaved. Instead of redoing our theory in the context of arbitrary Lie groups, we instead use the lifting property of covering spaces. Write exp_G:g→Gfor the standard exponential map and let

exp_G:g→G

be the unique smooth lift of exp_Gsatisfying exp_G(0)=eand exp_G =π◦exp_G. Lemma 6.31.Let G be a compact connected Lie group, T a maximal torus of G,G the simply connected covering of G,π :G→G the associated covering homomor- phism, andT =

π⁻¹(T)₀ .

(a)Restricted tot,exp_Ginduces an isomorphism of Lie groupsT ∼=t/

t∩ker exp_G . (b)Ifgis semisimple, thenT is compact.

Proof. Elementary covering theory shows thatT is a covering of T. From this it follows that T is Abelian on a neighborhood ofeand, sinceT is connected, T is Abelian everywhere. Sinceπexp_Gt =exp_Gt = T and since exp_Gtis connected, exp_Gt ⊆T. In particular, exp_G : t→ T is the unique lift of exp_G : t→ T satis- fying exp_G(0) =e. In turn, uniqueness of the lifting easily shows exp_G(t0+t) = exp_G(t0)exp_G(t). To finish part (a), it suffices to show expGtcontains a neighbor- hoodeinT. For this, it suffices to show the differential of exp_Gat 0 is invertible. But sinceπ is a local diffeomorphism and since exp_Gis a local diffeomorphism near 0, we are done.

For part (b), it suffices to show thatT is a finite cover ofT whengis semisim- ple. For this, first observe that ker exp_G ⊆ ker exp_G = 2πi kerE since exp_G = π◦exp_G. AsT ∼=t/ (2πi kerE), it follows that the kerπrestricted toT is isomor- phic to(2πi kerE) /

t∩ker expG

. By Theorems 6.27 and 6.29, it therefore suffices to show that 2πi R^∨⊆t∩ker exp_G.

Givenα∈ (g_C), let{Iα,Jα,Kα}be a standardsu(2)-triple ingassociated to α. Writeϕα : SU(2)→ Gfor the corresponding homomorphism. Since SU(2)is simply connected, writeϕα : SU(2)→ Gfor the unique lift ofϕαmapping I toe.

Using the uniqueness of lifting fromsu(2)toG, if follows easily thatϕ_α◦exp_SU(2)= exp_G◦dϕ_α. Therefore by construction,

e=ϕ_α(I)=ϕ_α(expSU(2)2πi H)=exp_G(2πi dϕ_αH)=exp_G(2πi h_α) ,

which finishes the proof.

Lemma 6.32.Let G be a compact connected Lie group, T a maximal torus of G, G the simply connected covering of of G,π : G → G the associated covering homomorphism, andT =

π⁻¹(T)₀ . (a)G=0

g∈G

cgT . (b)G=exp_G(g).

134 6 Roots and Associated Structures

Proof. The proof of this lemma is a straightforward generalization of the proof of the Maximal Torus theorem, Theorem 5.12 (Exercise 6.26).

Corollary 6.33.Let G be a compact connected Lie group with semisimple Lie alge- brag, T a maximal torus of G,G the simply connected covering of of G,π:G→G the associated covering homomorphism, andT =

π⁻¹(T)0

. (a)G is compact.

(b)gmay be identified with the Lie algebra ofG, so thatexp_Gis the corresponding exponential map.

(c)T =π⁻¹(T)andT is a maximal torus ofG.

(d)kerπ⊆Z(G)⊆T .

Proof. For part (a), observe thatG =0

g∈G

cgT

by Lemma 6.32. ThusG is the continuous image of the compact setG/Z(G)×T ∼=G/Z(G)×T (Exercise 6.26).

For part (b), recall from Corollary 4.9 that there is a one-to-one correspondence between one-parameter subgroups ofGand the Lie algebra ofG. By the uniqueness of lifting, exp_G(t X)exp_G(s X) =exp_G((t +s)X)for X ∈ gandt,s ∈ R, so that t →exp_G(t X)is a one-parameter subgroup ofG. On the other hand, ifγ :R→Gis a one-parameter subgroup, then so isπ◦γ :R→G. Thus there is a unique X ∈g, so thatπ(γ (t))=exp_G(t X). As usual, the uniqueness property of lifting fromRto Gshows thatγ (t)=exp_G(t X), which finishes part (b).

For parts (c) and (d), we already know from Lemma 6.31 thatT = exp_G(t).

Since t is a Cartan subalgebra, Theorem 5.4 shows thatT is a maximal torus of G. By Lemma 1.21 and Corollary 5.13, kerπ ⊆ Z(G) ⊆ T so thatπ⁻¹(T) =

T(kerπ)=T is, in fact, connected.

6.3.4 Exercises

Exercise 6.24 Supposeiis a lattice in(it)^∗that spans(it)^∗. (1)Show that^∗i is a lattice init.

(2)Show that^∗∗i =i.

(3)If1⊆2, show that^∗₂ ⊆^∗₁. (4)If1⊆2, show that2/1∼=^∗₁/^∗₂.

Exercise 6.25 (1)Use the standard root system notation from §6.1.5. In the follow- ing table, write (θi)for the element diag(θ1, . . . , θn)in the case ofG = SU(n), for the element diag(θ1, . . . , θn,−θ1, . . . ,−θn) in the cases of G = Sp(n) or S O(E2n), and for the element diag(θ1, . . . , θn,−θ1, . . . ,−θn,0) in the case of G=S O(E2n+1). Verify that the following table is correct.

G R^∨ kerE P^∨ P^∨/R^∨ SU(n) {(θin)|θi∈Z,

i=1θi =0} R^∨ {(θi+n^θ_n⁰)|θi ∈Z,

i=0θi =0} Zn

Sp(n) {(θi)|θi ∈Z} R^∨ {(θi+^θ₂⁰)|θi∈Z} Z2

S O(E2n) {(θni)|θi ∈Z,

i=1θi ∈2Z} {(θi)|θi ∈Z} {(θi+^θ₂⁰)|θi∈Z} Z2×Z2 neven Z4 nodd S O(E2n+1) {(θni)|θi ∈Z,

i=1θi ∈2Z} P^∨ {(θi)|θi∈Z} Z2.

(2)In the following table, write(λi)for the element

iλii. Verify that the follow- ing table is correct.

G R A P P/R

SU(n) {(λni)|λi ∈Z,

i=1λi =0} P {(λi+n^λ_n⁰)|λi∈Z,

i=0λi =0} Zn

Sp(n) {(λni)|λi ∈Z,

i=1λi ∈2Z} P {(λi)|λi ∈Z} Z2

S O(E2n) {(λni)|λi ∈Z,

i=1λi ∈2Z} {(λi)|λi∈Z} {(λi+^λ₂⁰)|λi ∈Z} Z2×Z2 neven Z4 nodd S O(E2n+1){(λi)|λi ∈Z} R {(λi+^λ₂⁰)|λi ∈Z} Z2. Exercise 6.26 LetGbe a compact connected Lie group,T a maximal torus ofG,G the simply connected covering of ofG,π : G→ Gthe associated covering homo- morphism, andT =

π⁻¹(T)0

. This exercise generalizes the proof of the Maximal Torus theorem, Theorem 5.12, to show thatG=0

g∈G

cgT

andG=expG(g).

(1) Make use of Lemma 5.11 and the fact that kerπ is discrete to show that ker(Ad◦π)=Z(G).

(2)Supposeϕ : g→ G is lift of a mapϕ : g→ G. Use the fact thatπ is a local diffeomorphism to show thatϕis a local diffeomorphism if and only ifϕis a local diffeomorphism.

(3)Use the uniqueness property of lifting to show that exp_G◦Ad(πg)=cg◦exp_G forg∈G.

(4)Show that0

g∈GcgT =exp_G(g).

(5)If dimg=1, show thatG ∼= S¹andg∼= G ∼=Rwith exp_Gbeing the identity map. Conclude thatG=exp_G(g).

(6) Assume dimg > 1 and use induction on dimgto show that G = exp_G(g)as outlined in the remaining steps. First, in the case where dimg < dimg, show that G∼=

G×T^k

/F, whereFis a finite Abelian group. Conclude thatG∼=Gss×R^k. Use the fact that the exponential map fromR^ktoT^k is surjective and the inductive hypothesis to showG=exp_G(g).

(7) For the remainder, assumegis semisimple, so that T is compact. Use Lemma 1.21 to show that kerπ⊆Z(G). Conclude thatG/Z(G)∼=G/Z(G)and use this to show that expG(g)is compact and therefore closed.

(8)It remains to show that exp_G(g)is open. FixX₀ ∈gand writeg₀ =exp_G(X₀). Use Theorem 4.6 to show that it suffices to considerX₀=0.

136 6 Roots and Associated Structures

(9)As in the proof of Theorem 5.12, leta=zg(πg₀)andb=a^⊥. Consider the map ϕ :a⊕b→Ggiven byϕ(X,Y)=g₀⁻¹exp_G(Y)g₀exp_G(X)exp_G(−Y). Show that ϕis a local diffeomorphism near 0. Conclude that{expG(Y)g0exp_G(X)exp_G(−Y)| X ∈a,Y ∈b}contains a neighborhood ofg₀inG.

(10)Let A = π⁻¹A0

, a covering of the compact Lie subgroupA =ZG(πg0)⁰of G. Show that exp_G(a)⊆A. Conclude that0

g∈Gg⁻¹Agcontains a neighborhood of g0inG.

(11) If dima < dimg, use the inductive hypothese to show that A = exp_G(a).

Conclude that0

g∈Gg⁻¹Ag = 0

g∈Gexp_G(Ad(πg)a), so that expG(g)contains a neighborhood ofg0.

(12)Finally, if dima = dimg, show that g0 ∈ Z(G). Lett be a Cartan subalge- bra containing X0 so thatg= 0

g∈GAd(πg)t. Show thatg0expG(g) ⊆expG(g).

Conclude that exp_G(g)contains a neighborhood ofg₀.