Introduction to Compact Groups

Introduction to Compact Groups

An Introductory Course

from the Fourth Semester up Qualification Module

Wahlpflichtbereich und Haupstudium

Karl Heinrich Hofmann

Summer 2006

Chapter 1

Compact Groups: Basics

In the first chapter we introduce basic concepts, look at elementary examples and constructions, and provide the essential tools of the trade.

Definitions and Examples

Definition 1.1.Acompact groupGis a compact Hausdorff space whose underlying set has a group structure such that the function

(∗) (x, y)7→xy⁻¹:G×G→G

is continuous. ut

For the concept of compactness and Hausdorff separation of a space see the set of Lecture Notes “Introductction to Topology.” Definition 1.1 is a special case of the definition of atopological group, which is a topologcial space and a group such that (∗) is continuous.

Our principal source of reference for compact groups is

[1] Hofmann, K. H., and S. A. Morris, The Structure of Compact Groups, de Gruyter Verlag, Berlin, 1998, xvii + 834pp.

Second Completely Revised, Corrected and Augmented Edition 2006, xviii + 860pp. To appear at de Gruyter Verlag, Berlin.

Exercise E1.1. (i) Let G be a group and a topological space Show that the following conditions are equivalent:

(1) The function (x, y)7→xy⁻¹:G×G→Gis continuous.

(2) Multiplication and inversion are continuous functions.

Here we recall that multiplication is the function (x, y)7→xy:G×G→G.

Examples 1.2.(i) All finite groups with the discrete topology are compact groups.

(ii) The multiplicative groups

S⁰={r∈R:|r|= 1}, S¹={z∈C:|z|= 1}, S³={q∈H:|q|= 1}

are compact groups on the unit spheres of the fields of real and complex numbers and of the skew field of quaternions.

The skew field ofquaternionsis isomorphict to a real 4-dimensional subalgebra of the algebra of 2×2 complex matrices

M(u, v)^def=

u v

−v u

∈M2(C), u, v∈C. Using the basis 1, i, j, kofH, we find the isomorphism via

r·1 +x·i+y·j+z·k7→M(r+x·i, y+z·i).

Accordingly,S³∼= SU(2)^def= {M(u, v) :uu+vv= 1}.

If G is a compact group on an n-sphere Sⁿ, then n = 0,1,3. But his is a nontrivial result: see [1] 9.59(iv).

(iii) Each of the groups O(n) ofn×n-orthogonal matrices forms a closed and bounded subset in the vector spaceMn(R) of alln×nreal matrices and therefore is a compact group, since matric multiplication, being polynomial in each coefficient, is continuous and inversion agrees with transposition and is, therefore, continuous.

By a similar argument, each of the groups U(n) of n×n-unitary matrices is a compact group; alternatively, one may identify U(n) with a closed subgroup of O(2n).

(iv) Everyclosed subgroupof a compact group is a compact group.

(v) If{Gj:j∈J}is an arbitrary family of compact groups, then their cartesian productG^def= Q

j∈JGjwith componentwise multiplication and the product topol- ogy is a compact group. The compactness ofGis a consequence of theTychonoff product theorem; the continuity of (x, y)7→ xy⁻¹ :G×G→ Gfollows from the natural homeomorphy ofG×GandQ

j∈JGj×Gj and the continuity of (xj, yj)7→xjy_j⁻¹:Gj×Gj →Gj

for allj∈J.

(va) Letp∈N. Definef:Z→P^def= Q

n∈NZ/pⁿZbyf(x) = (x+pⁿZ)_n∈N. Then Zp

def= f(Z) is a compact abelian group. If pr_n:P→Z/pⁿZdenotes the projection given by pr_n((xm)_m∈N =xn, then fn

def= pr_n|Zp is a morphism Zp →Z/pⁿZ. It is an exercise to show that it is surjective. Thus kerfn is a subgroupIn such that Zp/In ∼=Z/pⁿZ. All morphisms in sight also preserve multiplication, so Zp is a ring and theIn are ideals, andIn turns out to agree exactly withpⁿZp. One calls Zp thering ofp-adic integers, and its elements are calledp-adic integers.

(vb) Every productQ

j∈JO(n_j) or any productQ

j∈JU(n_j) of a family of or- thogonal, respectively, unitary groups is a compact group, as is any closed subgroup

of these. ut

It is remarkable and is a first goal of this course to prove that every compact group is isomorphic as a topological group to a closed subgroup of one of the groups exhibited inExample 1.2(vb).

Exercise E1.2. Verify the details of the propositions that a product of an arbi- trary family{Gj:j ∈J}of compact groups is a compact group and that a closed subgroup of a compact group is a compact group.

Exercise E1.3. Prove the following assertion:

If G is a compact group and N a closed normal subgroup, then G/N is a compact group with respect to the quotient topology.

We must know that a subset V ∈ G/N is open iff q⁻¹V is open inG where q:G→G/N is the quotient map given byq(g) =gN=N g.

Exercise E1.4. (p-adic integers). LetL denote the subset of all sequences (xn+ pⁿZ)_n∈N∈P ^def= Q

n∈NZ/pⁿZ,x_n∈Zsuch thatx_n+1∈x_n+pⁿZ. Show thatLis a compact subring ofP and that it contains the subringZ^0def= {(x+pⁿZ)_n∈N∈L: x∈Z}. Prove thatZ⁰ ∼=Zand that every open subset of L contains an element ofZ⁰, that is,Z⁰ is dense inLandL=Z⁰. Conclude thatL=Zp.

Applications to Abelian Groups

An important example arises out of the preceding proposition. For two setsX and Y the set of all functionsf:X→Y will be denoted byY^X.

Definition 1.3. If A is an abelian group (which we prefer to write additively) then the group

Hom(A,T)⊆T^A

of all morphisms of abelian groups into the underlying abelian group of the circle group (no continuity involved!) given the induced group structure and topology of the product groupT^A(that is, pointwise operations and the topology of pointwise convergence) is calledthe character group ofAand is writtenA. Its elements areb

calledcharactersof A. ut

Proposition 1.4. The character group Ab of any abelian group A is a compact abelian group.

Proof. By Exercise 1.2(v), the productT^A is a compact abelian group. For any pair (a, b)∈A×Athe set M(a, b) ={χ∈T^A |χ(a+b) =χ(a) +χ(b)} is closed sinceχ7→χ(c):T^A →T is continuous by the definition of the product topology.

But thenAb=T

(a,b)∈A×AM(a, b) is closed inT^A and therefore compact. ut Let us look at a few examples: In order to recognizeZbwe note that the function f 7→f(1): Hom(Z,T)→ Tis an algebraic isomorphism and is continuous by the definition of the topology of pointwise convergence. Since Zb is compact and T

Hausdorff, it is an isomorphism of compact groups. Hence

(1) Zb ∼=T.

If Z(n) = Z/nZ is the cyclic group of order n, then the function z+nZ 7→

1

nz+Zgives an injectionj:Z(n)→Twhich induces an isomorphism Hom(Z(n), j):

Hom Z(n),Z(n)

→ Hom(Z(n),T) = Zd(n). Since the function f 7→ f(1 +nZ):

Hom Z(n),Z(n)

→Z(n) is an isomorphism, we have

(2) Zd(n)∼=Z(n).

IfX is a set, and{Ax|x∈X}a family of abelian groups, let us denote with L

x∈XAxthe direct sum of theAx, that is, the subgroup of the cartesian product Q

x∈XAx consisting of all elements (ax)_x∈X with ax = 0 for all xoutside some finite subset ofX. A special case isZ^(X)=L

x∈XAx withAx=Zfor allx∈X. This is thefree abelian group onX

Proposition 1.5.The function Φ: Y

x∈X

Hom(Ax,T)→Hom(M

x∈X

Ax,T)

which associates with a family(f_x)_x∈X of morphismsf_x:A_x→Tthe morphism (a_x)_x∈X7→ X

x∈X

f_x(a_x):M

x∈X

A_x→T

is an isomorphism of compact groups. Notably,

(3) (M

x∈X

A_x)b∼= Y

x∈X

Ac_x.

In particular

(4) Z^(X)b

∼=Zb^X ∼=T^X.

Proof. Abbreviate L

x∈XAx by A. We notice that Φ is well defined, since the fx(ax) vanish with only finitely many exceptions for (ax)_x∈X. Clearly Φ is a mor- phism of abelian groups. Further (fx)_x∈X∈ker Φ if and only ifP

x∈Xfx(ax) = 0 for all (ax)_x∈X ∈A. Choosing for a given y ∈X the family (ax) so thatax = 0 for x6=y and ay =a we obtain fy(a) = 0 for any a ∈ Ay. Thus fy = 0 for all y ∈ X. Hence Φ is injective. If f:A → T is a morphism, define fy:Ay → T by fy=f◦copr_ywhere copry:Ay→Ais the natural inclusion. Then Φ (fx)x∈X

=f follows readily. Thus Φ is surjective, too, and thus is an isomorphism of abelian groups. Next we show that Φ is continuous. By the definition of the topology on Hom(A,T) ⊆ T^A, it suffices to show that for each (a_x)_x∈X ∈ A, the function (f_x)_x∈X 7→Φ (f_x)_x∈X

(a_x)_x∈X

=P

x∈Xf_x(a_x) : Hom(A_x,T)^X →T is contin- uous. Since only finitely manya_xare nonzero, this is the case if (f_x)_x∈X 7→f_y(a_y) is continuous for each fixedy, and this holds if f_y 7→f_y(a_y): Hom(A_y,T)→Tis

continuous. However, by definition of the topology of pointwise convergence, this is indeed the case. Since the domain of Φ is compact by the theorem of Tychonoff and the range is Hausdorff, this suffices for Φ to be a homeomorphism.

The last assertion of the proposition is a special case. This remark concludes

the proof of the proposition. ut

The compact abelian groupsT^X are calledtorus groups. The finite dimensional toriTⁿ are special cases.

We cite from the basic theory of abelian groups the fact that a finitely generated abelian group is a direct sum of cyclic groups. Thus (1), (2) and (3) imply the following remark:

Remark 1.6.IfE is a finite abelian group, thenEb is isomorphic toE (although not necessarily in any natural fashion!). IfF is a finitely generated abelian group of rankn, that is,F =E⊕Zⁿ with a finite abelian groupE, thenFb∼=Eb×Tⁿ.ut In particular, the character groups of finitely generated abelian groups are compact manifolds. (We shall not make any use of this fact right now.

There are examples of compact abelian groups whose topological nature is quite different.

Example 1.7. Let {Gj | j ∈ J} be any family of finite discrete nonsingleton groups. Then G =Q

j∈JGj is a compact group. All connected components are singleton, andGis discrete if and only ifJ is finite. ut A topological space in which all connected components are singletons is called totally disconnected. Arbitrary products of totally disconnected spaces are totally disconnected, and all discrete spaces are totally disconnected. The standard Cantor middle third setCis a compact metric totally disconnected space. In fact it may be realized as the set of all real numbersrin the closed unit interval, whose expansion r =P∞

n=1an3⁻ⁿ with respect to base 3 has all coefficients an in the set {0,2}.

Then the map f:{−1,1}^N → C given by f (rn)n∈N

= P∞

n=1(rn+ 1)3⁻ⁿ is a homeomorphism. The set S⁰ = {−1,1} is a finite group, and thus, by Exercise 1.2(v), the domain off is a compact group.

Hence the Cantor set can be given the structure of a compact abelian group.

In this group, every element has order 2, so that in fact, algebraically, it is a vector space over the field GF(2) of 2 elements, and by (2) and (3) above, it is the character group ofZ(2)^(N).

One can show that all compact metric totally disconnected spaces without isolated points are homeomorphic toC. In particular, all metric compact totally disconnected infinite groups are homeomorphic toC.

Definition 1.8. LetX andY be sets and F ⊆Y^X a set of functions from X to Y. We say thatF separates the points of X if for any two different pointsx₁ and x₂ inX, there is anf ∈F such thatf(x₁)6=f(x₂). ut

IfGandH are groups, then a setF of homomorphisms fromGto H is easily seen to separate the points ofGif and only if for eachg6= 1 inGthere is anf ∈F withf(g)6= 1.

For any abelian group A there is always a large supply of characters. In fact there are enough of them to separate the points. In order to see this we resort to some basic facts on abelian groups:

An abelian group A is called divisible if for each a ∈ A and each natural number nthere is an x∈A such thatn·x=a. Examples of divisible groups are QandR. Every homomorphic image of a divisible group is divisible, whenceTis divisible. The crucial property of divisible groups is that for every subgroupS of an abelian groupAand a homomorphismf:S →Iinto a divisible group there is a homomorphic extensionF:A→I off, as we shall argue now.

Definition 1.9. An abelian groupI is calledinjective if for every injective mor- phismi:A→Band every morphismj:A→Ithere is a morphismf:B →Iwith j=f◦i.

I −−−−−−−−−→^idI I

j

x





x



^f A −−−−−−−−−→

i B

u t

One may rephrase injectivity in the following convenient fashion:An abelian group I is injective if and only if any homomorphism j:A → I of a subgroup A of a groupB extends to a homomorphismf:B→I on the whole group.

Proposition 1.10.For an abelian groupGthe following conditions are equivalent:

(1) Gis divisible.

(2) Gis injective.

Proof. (1)⇒(2). (AC) Assume thatAis a subgroup ofBand that a homomorphism j:A→Gis given. We must extend j to a morphism f:B →G. We consider the set of all morphismsϕ:C→GwithA⊆C⊆Bandϕ|A=j. This set is partially ordered by inclusion of domains and extension of mappings (i.e.ϕ≤ϕ⁰ ifC⊆C⁰ and ϕ⁰|C = ϕ). One verifies quickly that this set is inductive, hence by Zorn’s Lemma contains a maximal element µ:M → G. We must show M = B. Let b∈B. ThenM ∩Z·bis a cyclic group, say nZ·b. Since Gis divisible, there is an elementd∈Gsuch thatn·d=µ(n·b). Assume now that m1+z1·b =m2+z2·b.

Then m2−m1 = (z1−z2)·b ∈ M ∩Z·b = nZ·b. In particular, the kernel of m7→m·b:Z→Gis contained innZ. Thus there is az∈Zwith (z1−z2−zn)·b= 0, and thusz1−z2−zn=z⁰nfor somez⁰∈Z. Henceµ(m2−m1) =µ (z1−z2)·b

= µ (z+z⁰)n·b

= (z+z⁰)n·d= (z₁−z₂)·dand thusµ(m₁) +z₁·d=µ(m₂) +z₁·d.

Therefore we define unambiguously a function µ⁰:M⁰ → I, M⁰ = M +Z·b by µ⁰(m+z·b) = µ(m) +z·d satisfying µ⁰|M = µ. It is easy to verify that µ is a morphism. Hence µ ≤ µ⁰. By the maximality of µ we have µ⁰ = µ and thus M⁰=M. Henceb∈M. ThusM =B.

(2)⇒(1). There is a setX and a surjective homomorphism p:Z^(X) → G. We may assume that G= Z^(X)/K with K = kerpand that cardX = cardG. Now K⊆Z^(X)⊆Q^(X). ThenG=Z^(X)/K ⊆Q^(X)/K, andD =Q^(X)/K is divisible.

Hence there is a divisible group D with G ⊆ D. Since G is injective there is a morphismf:D→Gsuch thatf|G= id_G. HenceGis a homomorphic image of a

divisible group and is, therefore, divisible. ut

T −−−−−−−−−→⁼ T

f

x





x





χ

S −−−−−−−−−→

incl A

Lemma 1.11. The characters of an abelian groupA separate the points.

Proof. Assume that 0 6=a∈ A. We must find a morphism χ:A →T such that χ(a)6= 0. LetS be the cyclic subgroupZ·a ofA generated by a. IfS is infinite, then S is free and for any nonzero element t in T (e.g. t = ¹₂ +Z) there is an f:S→Twithf(a) =t6= 0. If Shas ordern, thenS is isomorphic to ¹_nZ/Z⊆T, and thus there is an injectionf:S →T. If we letχ:A→Tbe an extension off which exists by the divisibility ofT, thenχ(a) =f(a)6= 0. ut Definitions 1.12.For a compact abelian groupGa morphism of compact groups χ:G → T is called a character of G. The set Hom(G,T) of all characters is an abelian group under pointwise addition, called thecharacter group ofGand written G. Notice that we do not consider any topology onb G.b ut

Now we can of course iterate the formation of character groups and oscillate be- tween abelian groups and compact abelian groups. This deserves some inspection;

the formalism is quite general and is familiar from the duality of finite-dimensional vector spaces.

Lemma 1.13. (i)If Ais an abelian group, then the function ηA:A→ b

A,b ηA(a)(χ) =χ(a) is an injective morphism of abelian groups.

(ii)If Gis a compact abelian group, then the function η_G:G→G,bb η_G(g)(χ) =χ(g) is a morphism of compact abelian groups.

Proof. (i) The morphism property follows readily from the definition of pointwise addition inA. An elementb g is in the kernel ofηA ifχ(g) = 0 for all characters.

Since these separate the points by Lemma 1.11, we concludeg = 0. HenceηA is injective.

(ii) Again it is immediate thatη_Gis a morphism of abelian groups. We must ob- serve its continuity: The functiong7→χ(g):G→Tis continuous for every charac- terχby the continuity of characters. Hence the functiong7→ χ(g)

χ∈Gb:G→T^Gb is continuous by the definition of the product topology. Since b

Gb= Hom(G,b T)⊆T^Gb inherits its structure from the product,ηG is continuous. ut Exercise E1.5.For a discrete group Aand a compact groupGthe members of

bb

Aand GGbb separate the points ofA, respectively,b G. Equivalently, the evaluationb morphismsη

Ab:Ab→ bb Abandη

Gb:Gb→ bb

Gbare injective.

[Hint. Observe that alreadyηA(A) separates the points ofA.]b ut

Let us look at our basic examples: If A is a finite abelian group, then Ab is isomorphic to A by Remark 1.6. Hence b

Abis isomorphic to A and ηA:A → b Abis injective by Lemma 1.23. Henceη_Ais an isomorphism.

Every character χ:T→ Tyields a morphism of topological groups f:R→T viaf(r) =χ(r+Z). Let q:R→T be the quotient homomorphism. We setV = ]− ¹₃,¹₃[⊆ R and W =q(V). Then q|V:V → W is a homeomorphism. Assume thatxandy are elements of W such thatx+y ∈W, too. Thenr= (q|V)⁻¹(x), s= (q|V)⁻¹(y) andt= (q|V)⁻¹(x+y) are elements ofV such thatq(r+s−t) = q(t) +q(s)−q(t) =x+y−(x+y) = 0 in T. Hence r+s−t ∈ kerq =Z. But also|r+s−t| ≤ |r|+|s|+|t|<3·¹₃ = 1. Hencer+s−t= 0 and (q|V)⁻¹(x) + (q|V)⁻¹(y) = r+s = t = (q|V)⁻¹(x+y). Now let U denote an open interval around 0 inRsuch that f(U)⊆W. If we setϕ= (q|V)⁻¹◦f|U:U →Rthen for allx, y, x+y ∈U we haveϕ(x+y) =ϕ(x) +ϕ(y). Under these circumstancesϕ extends uniquely to a morphism F:R→ R of abelian groups (see Exercise E1.6 below). Nowq◦F =f =χ◦qsinceF extendsϕandU generates the abelian group R. ThenZ= kerq⊆ker(q◦F), that is,F(Z)⊆kerq=Z. Thus if we setn=F(1), thenn∈Z. Sinceϕis continuous, thenF is continuous at 0. As a morphism, F is continuous everywhere (see Exercise E1.7 below). As a morphism of abelian groups,F is quickly seen to beQ-linear, and from its continuity it follows that it is R-linear. Thus F(t) = nt and χ(t+Z) =nt+Z follows. Thus the characters of T are exactly the endomorphisms µn = (g 7→ ng) and n 7→ µn:Z → Tb is an isomorphism.

Exercise E1.6. Prove the following proposition:

The Extension Lemma. Let U be an arbitrary interval inRcontaining0 and assume thatϕ:U →Gis a function into a group such thatx, y, x+y∈U implies

ϕ(x+y) =ϕ(x)ϕ(y). Then there is a morphismF:R→Gof groups extendingϕ.

IfU contains more than one point thenF is unique.

[Hint. Step 1. Show by induction that for anyu∈U such thatu,2u, . . . nu∈U we haveϕ(ku) =ϕ(u)^k,k= 1,2, . . . , n. Step 2. Ifu, mu∈U for a natural numberm, then for anyn∈N,ϕ(u)^mn=ϕ(mu)ⁿ. Step 3. Forr∈Rand two integersmandn withr/m, r/n∈U showϕ(r/m)^m=ϕ(r/n)ⁿ. Indeed assume first thatm, n≥1;

letu=r/mn, then ϕ(u)^mn =ϕ(r/m)^m by Step 2; likewiseϕ(u)^mn =ϕ(r/n)ⁿ. Reduce the casem, n <0 to this case. Step 4. DefineF(r) to be the unique element ofGfor which there is an integermsuch thatr/m∈U andF(r) =ϕ(r/m)^mand

show thatF is a morphism.] ut

Exercise E1.7.A homomorphism between topological groups is continuous iff it is continuous at the identity.

Now that we have determined Tb we look at η_Z. We haveη_Z(n)(χ) = χ(n) = nχ(1) =µ_n χ(1)

for any characterχ ofZ. Sinceχ7→χ(1):Zb→Tis an isomor- phism by (1) above and since every character of Tis of the form µ_n, this shows thatη_Zis an isomorphism.

Now we show thatη_Tis an isomorphism, too. We recall thatTbis infinite cyclic and is generated by the identity map ε:T → T. In other words, any character χ:T → T of bT is of the form χ = n·ε = µ_n(ε). Now we observe η_T(g)(n·ε) = n·ε(g) =n·g for alln∈Z. Takingn= 1 we note that the kernel ofη_Tis singleton and thusη_T is injective. In order to show surjectivity we assume that Ω:Tb →T is a character ofTb ∼=Z. Then Ω(ε) is an element g ∈T and we see η_T(g)(n·ε) = n·g = n·Ω(ε) = Ω(n·ε). Thus η_T(g) = Ω. This shows that η_T is surjective, too.

Thusη_T is an isomorphism.

Remark 1.24.(i) Assume thatAandB are abelian groups such thatηA andηB

are isomorphisms. Thenη_A⊕B is an isomorphism.

(ii) IfGandH are compact abelian groups andηG andηH are isomorphisms, thenη_G×H is an isomorphism.

(iii) For any finitely generated abelian group A, the map ηA:A → b Ab is an isomorphism.

(iv) IfG∼=Tⁿ×E for a natural numbernand a finite abelian group E then ηG:G→ b

Gb is an isomorphism.

(v) Every torus groupTⁿcontains an element such that the subgroup generated by it is dense.

Proof. Exercise E1.8. ut

Exercise E1.8. Prove Remarks 1.24(i)–(v).

[Hint. For (iii) and (iv) recall that the evaluation morphism is an isomorphism for cyclic groups, forZand forT. Also recall the Fundamental Theorem for Finitely Generated Abelian Groups (cf.[1], Appendix A1.11).

For a proof of (v) setT =Tⁿ. Every quotient group ofT modulo some closed subgroup is a compact group which is a quotient group ofRⁿ and is, therefore, a torus by [1] Appendix 1, Theorem 1.12(ii). Now letx∈T; thenT /hZ·xiis a torus, and by (iv) above, its characters separate the points. Thus, Z·xis dense in T iff all characters ofT vanish onZ·x, i.e. onx, iff

(∀χ∈Tb) [χ(Z·x) ={0}]⇒[χ= 0]

iff the mapχ 7→ n7→χ(n·x)

:Tb→Zb is injective iff the map χ7→χ(x):Tb→T is injective (via the natural isomorphism Zb ∼= T). But since η:T → Tbb is an isomorphism by (iv) above, any homomorphism α:Tb → T is an evaluation, i.e.

there is a uniquex∈T such that for any χ∈Tb we haveχ(x) =α(χ). Thus, in conclusion, we have an elementx∈T such thatZ·xis dense inT iff we have an injective morphismZⁿ ∼=Tb→T. But the injective morphismsZⁿ →R/Zabound (cf. Appendix A1.43).

Provide a direct proof of Remark 1.24(v) as follows:Let rj ∈R,j = 1, . . . , n, benreal numbers such that{1, r1, . . . , rn}is a set of linearly independent elements of theQ-vector space R. Then the element x+Z∈Rⁿ/Zⁿ, x= (r1, . . . , rn) has

the property that Z·(x+Z)is dense.] ut

Exercise E1.9. Prove the following universal property of the evaluation mor- phism:

Lemma A. For every morphism f:A → Gb from an abelian group A to the character group of a compact abelian groupGthere is a unique morphismf⁰:G→ Absuch that f =fb⁰◦η_A.

Lemma B. For every morphismf:G→Abfrom a compact abelian group Gto the character group of an abelian groupA there is a unique morphism f⁰:A→Gb such thatf =fb⁰◦ηG.

[Hint. Lemma B is proved in the same way as Lemma A. In the case of Lemma A, definef⁰ byf⁰(g)(a) =f(a)(g) fora∈Aundg∈G. Verify the asserted property and uniqueness by using the definitions off⁰ andηAandfb⁰.]

Apply this to show

Lemma C. For each abelian group A we have ηcA◦η

Ab = idA and for each compact abelian group wie haveηcG◦η

Gb= idG.

Exercise E1.10. Prove the following observation on morephisms of abelian groups:

Lemma. If f:A → B is a morephism of abelian groups and fb:Bb → Ab its adjoint, then a character β of B is in kerfbiff it annihilates f(A), that is, β(f(A) ={0}.

Projective Limits

Definition 1.25.LetJ be a directed set, that is, a set with a reflexive, transitive and antisymmetric relation≤such that every finite nonempty subset has an upper bound. Aprojective system of topological groups overJ is a family of morphisms {fjk:Gk →Gj | (j, k)∈J ×J, j ≤k}, where Gj, j ∈J are topological groups, satisfying the following conditions:

(i) f_jj= id_Gj for allj∈J

(ii) f_jk◦f_kl=f_jl for allj, k, l∈J withj≤k≤l. ut Lemma 1.26. (i)For a projective system of topological groups, define the topo- logical groupP byP =Q

j∈JG_j. Set

G={(g_j)_j∈J ∈P |(∀j, k∈J)j≤k⇒f_jk(g_k) =g_j}.

Then G is a closed subgroup of P. If incl:G → P denotes the inclusion and pr_j:P → Gj the projection, then the function fj = pr_j◦incl:G → Gj is a morphism of topological groups for all j ∈ J, and for j ≤ k in J the relation fj=fjk◦fk is satisfied.

(ii) If all groups Gj in the projective system are compact, then P and G are compact groups.

Proof. (i) Assume that j ≤k in J. Define Gjk ={(gl)l∈J ∈P | fjk(gk) =gj}.

Since fjk is a morphism of groups, this set is a subgroup of P, and since fjk is continuous, it is a closed subgroup. But G = T

(j,k)∈J×J, j≤kGjk. Hence G is a closed subgroup. The remainder is straightforward.

(ii) If allG_j are compact, thenPis compact by Tychonoff’s Theorem, and thus

Gas a closed subgroup ofP is compact, too. ut

Definitions 1.27.IfP ={f_jk:G_k→G_j |(j, k)∈J×J, j ≤k}is a projective sys- tem of topological groups, then the groupGof Lemma 1.26 is called itsprojective limitand is writtenG= limP. As a rule it suffices to remind oneself of the entire projective system by recording the family of groupsGjinvolved in it; therefore the notationG= lim_j∈JGj is also customary. The morphismsfj:G→Gj are called limit mapsand the morphismsfjk:Gk→Gj are calledbonding maps. ut Example 1.28. Assume that we have a sequence ϕn:Gn+1 → Gn, n ∈ N of morphisms of compact groups:

G₁←^ϕ1 G₂^ϕ←² G₃←^ϕ3G₄← · · ·^ϕ4

Then we obtain a projective system of compact groups by defining, for natural numbersj≤k, the morphisms

fjk=ϕj◦ϕj+1◦ · · · ◦ϕ_k−1:Gk →Gj.

ThenG= lim_n∈NGn is simply given by{(gn)_n∈N|(∀n∈N)ϕn(gn+1) =gn}.

(i) Choose a natural number p and set Gn = Z(pⁿ) = Z/pⁿZ. Define ϕn:Z(pⁿ⁺¹)→Z(pⁿ) byϕn(z+pⁿ⁺¹Z) =z+pⁿZ:

Z(p)←^ϕ1Z(p²)←^ϕ2Z(p³)←^ϕ3Z(p⁴)← · · ·^ϕ4

The projective limit of this system is none other than our group Zp of p-adic integers.

(ii) SetGn =Tfor alln∈Nand defineϕn(g) =p·g for all n∈Nandg∈T. (It is customary, however, to writepin place ofϕp):

T

←p T

←p T

←p T

← · · ·p

The projective limit of this system is called thep-adic solenoidTp. ut

Proposition 1.29. Assume that G = lim_j∈JGj for a projective system fjk:Gk → Gj of compact groups, j ≤ k in J, and denote with fj:G → Gj the limit maps. Then the following statements are equivalent:

(1) All bonding mapsfjk are surjective.

(2) All limit mapsfj are surjective.

Proof. (1)⇒(2) Fixi∈J. Leth∈G_i; we must find an elementg= (g_j)_j∈J∈G withg_i=f_i(g) =h. For allk∈J withi≤kwe defineC_k ⊆Q

j∈JG_j by {(xj)_j∈J|(∀j≤k)xj=fjk(xk) andxi =h}.

Since f_ik is surjective, C_k 6= Ø. If i ≤ k ≤ k⁰ then we claim C_k⁰ ⊆ C_k. Indeed (x_j)_j∈J ∈ C_k⁰ implies f_jk(x_k) = f_jkf_kk⁰(x_k⁰) = f_jk⁰(x_k⁰) = x_j and x_i = h.

Thus (x_j)_j∈J ∈ C_k and the claim is established. Now {C_k | k ∈ J, i ≤ k} is a filter basis of compact sets in Q

j∈JG_j and thus has nonempty intersection.

Assume that g= (g_m)_m∈J is in this intersection. Then, firstly, g_i =h. Secondly, letj ≤k. SinceJ is directed, there is ak⁰ withi, k ≤k⁰. Then (gm)_m∈J ∈Ck⁰. Hence gj = fjk⁰(gk⁰) = fjkfkk⁰(gk⁰) = fjk(gk) by the definition of Ck⁰. Hence g∈lim_j∈JGj. Thusg is one of the elements we looked for.

(2)⇒(1) Letj≤k. Thenfj =fjkfk. Thus the surjectivity offj implies that

offjk. ut

Definition 1.30.A projective system of topological groups in which all bonding maps and all limit maps are surjective is called astrict projective systemand its

limit is called astrict projective limit. ut

Proposition 1.31.(i)Let G= lim_j∈JGj be a projective limit of compact groups.

Let Uj denote the filter of identity neighborhoods of Gj, U the filter of identity neighborhoods ofG, andN the set{kerfj |j∈J}. Then

(a) U has a basis of identity neighborhoods {f_k⁻¹(U)|k∈J, U ∈ Uk}.

(b) N is a filter basis of compact normal subgroups converging to1.(That is, given a neighborhood U of1, there is an N∈ N such that N ⊆U.)

(ii) Conversely, assume that G is a compact group with a filter basis N of compact normal subgroups withT

N ={1}. ForM ⊆N in N letf_{N M}:G/M → G/N denote the natural morphism given by f_{N M}(gM) = gN. Then the f_{N M} constitute a strict projective system whose limit is isomorphic toGunder the map

g 7→ (gN)N∈N:G → lim_N∈NG/N. With this isomorphism, the limit maps are equivalent to the quotient mapsG→G/N.

Proof. (i)(a) LetV ∈ U. Then by the definition of the projective limit there is an identity neighborhood of Q

j∈JG_j of the form W = Q

j∈JW_j with W_j ∈ U_j for which there is a finite subset F of J such that j ∈ J \F implies Wj = Gj

such thatW∩lim_j∈JGj ⊆V. SinceJ is directed, there is an upper boundk∈J of F. There is a U ∈ Uk such that fjk(U)⊆ Wj for all j ∈J. Then f_k⁻¹(U) ⊆ W∩lim_j∈JGj ⊆V.

(i)(b) Evidently, each kerfj is a compact normal subgroup. Since i, j ≤ k implies kerfk ⊆ kerfi∩kerfj and J is directed, N is a filter basis. For each j∈J we have kerfj=f_j⁻¹(1)⊆f_j⁻¹(U) for anyU ∈ Uj. Sincef_j⁻¹(U) is a basic neighborhood of the identity by (a), we are done.

(ii) It is readily verified that the family of all morphismsfN M:G/M →G/N for M ⊆ N in N constitutes a strict projective system of compact groups. An element (g_NN)_N∈N ∈Q

N∈NG/N withg_N ∈Gis in its limitLif and only if for each pairM ⊇N in N we havef_{M N}(g_NN) =g_MM, that is,g_M⁻¹g_N ∈M. Thus for each g ∈ G certainly (gN)_N∈N ∈L. The kernel of the morphism ϕ = (g 7→

(gN)_N∈N):G→LisTN ={1}. Henceϕis injective. Assumeγ= (g_NN)_N∈N ∈ L. Then {g_NN | N ∈ N } is a filter basis of compact sets in G, for if M ⊇ N theng⁻¹_MgN ∈M, and thusgN ∈gMM∩gNN. Hence its intersection contains an element g and then g ∈ gNN is equivalent to gN = gNN. Thus ϕ(g) = γ. We have shown thatϕis also surjective and thus is an isomorphism of compact groups (see Remark 1.8). IfqN:G→G/N is the quotient map, and iffN:L →G/N is the limit map defined byfN (gNN)_N∈N

=gNN, then clearlyqN =fN ◦ϕ. The

proof of the proposition is now complete. ut

The significance of the preceding proposition is that we can think of a strict projective limit G as a compact group which is approximated by factor groups G/N modulo smaller and smaller normal subgroups N. This is not a bad image.

The groupG is decomposed into cosets gN whose size can be made as small as we wish using the normal subgroups in the filter basisN.

More Duality Theory

LetA be an arbitrary abelian group. LetF denote the family of all finitely gen- erated subgroups. This family is directed, for if F, E ∈ F then F +E ∈ F.

Also, A = S

F∈FF. If E, F ∈ F and E ⊆ F then the inclusion E → F in- duces a morphism fEF:Fb → Eb via fEF(χ) = χ|E for χ:F → T. The family {fEF:Fb → Eb | E, F ∈ F, E ⊆ F} is a projective system of compact abelian groups. By the divisibility ofT, each character onE⊆F extends to one onF and so this system is strict. The inclusionF →A induces a morphismf_F:Ab→Fb by f_F(χ) =χ|F for each characterχ:A→T.

Proposition 1.32. The map χ7→(χ|F)_F∈F:Ab→ limF∈FFb is an isomorphism of compact abelian groups.

Proof. Define ϕ: Hom(A,T) → lim_F∈FHom(F,T) by ϕ(χ) = (χ|F)_F∈F. This definition yields a morphism of compact groups. A characterχofAis in its kernel if and only if χ|F = 0 for all F ∈ F. But since A =S

F∈FF this is the case if and only ifχ= 0. Thusϕis injective. Now letγ= (χF)_F∈F ∈lim_F∈FFb. By the definition of the bonding maps, this means that for every pair of finitely generated subgroupsE⊆F in Awe haveχF|E=χE. Now we can unambiguously define a functionχ:A→Tas follows. We pick for eacha∈A anF ∈ F with a∈F (for instance,F =Z·a). By the preceding, the elementχF(a) inTdoes not depend on the choice ofF. Hence we define a functionχ:A→Tbyχ(a) =χF(a). Ifa, b∈A, takeF =Z·a+Z·band observeχ(a+b) =χF(a+b) =χF(a)+χF(b) =χ(a)+χ(b).

Thus χ ∈ Hom(A,T) and χ|F = χF. Hence ϕ(χ) =γ. Thus ϕ is bijective and hence an isomorphism of compact groups (see Remark 1.8). ut In short:The character groupAbof any abelian groupA is the strict projective limit of the character groups Fb of its finitely generated subgroups F. We know that Fb is a direct product of a finite group and a finite-dimensional torus group (see Remark 1.18). In particular, every character group of an abelian group is approximated by compact abelian groups on manifolds.

Assume that G = lim_j∈JG_j is a strict projective limit of compact abelian groups with limit mapsfj:G→Gj. Every characterχ:Gj→Tgives a character χ◦fj:G→TofG. Sincefj is surjective, χ7→χ◦fj:Gcj →Gb is injective. Under this map, we identifyGcj with a subgroup ofG.b

Proposition 1.33. IfGis a strict projective limit limj∈JGj thenGb=S

j∈JGcj. Proof. With our identification ofGb_jas a subgroup ofG, the right side is containedb in the left one. Now assume thatχ:G→Tis a character ofG. If we denote withV the image of ]−¹₃,¹₃[ inT, then{0}is the only subgroup ofTwhich is contained inV. NowU =χ⁻¹(V) is an open neighborhood of 0 inG. Hence by Proposition 1.31(i) there is a j ∈ J such that kerfj ⊆ U. Hence χ(kerfj) is a subgroup of Tcontained in V and therefore is{0}. Thus kerfj ⊆kerχand there is a unique morphism χj:Gj → T such that χ = χj ◦fj. With our convention, this means exactlyχ∈Gcj. ThusGb⊆S

j∈JGcj. ut

The next theorem is one half of the famous Pontryagin Duality Theorem for compact abelian groups.

Theorem 1.34.For any abelian groupAthe morphismη_A:A→Abbis an isomor- phism.

Proof. We know thatAbis the strict projective limit limF∈FFb with the directed familyF of finitely generated subgroups of A. (See Proposition 1.32.) The limit maps fF:Ab → Fb are given by fF(χ) = χ|F, and these surjective maps induce injective morphisms Hom(f_F,T): Hom(F ,b T)→Hom(A,b T) with Hom(f_F,T)(Σ) = Σ◦fF. By Proposition 1.33, Hom(A,b T) is the union of the images of the injective morphisms Hom(fF,T). Thus for any Ω ∈ Hom(A,b T) there is an F ∈ F such that Ω is in the image of Hom(fF,T). Hence there is a Σ ∈ Hom(F ,b T) such that Ω = Hom(fF,T)(Σ) = Σ◦fF. ButηF:F →Hom(F ,b T) is an isomorphism by Remark 1.24(i). Hence there is ana∈F such that Σ =η_F(a). Thus Ω =η_f(a)◦fF. Therefore, for any character χ:A → T of A we have Ω(χ) = η_F(a) f_F(χ)

= η_F(a)(χ|F) = (χ|F)(a) =χ(a) =η_A(a)(χ). Thus η_A is surjective. The injectivity

was established in Lemma 1.23. ut

It is helpful to visualize our argument by diagram chasing:

F −−−−−−−−−→^ηF Hom(F ,b T)

inc



 y





y^Hom(dinc,T) A −−−−−−−−−→

ηA

Hom(A,b T).

The other half of the Pontryagin Duality Theorem claims that η_G:G→ Gbb is an isomorphism for any compact abelian groupG, too. We cannot prove this at the present level of information. However, in practicing the concept of a projective limit we can take one additional step.

Let us, at least temporarily, use the parlance that a compact abelian groupG is said to have duality if η_G:G → Gbb is an isomorphism. We propose the follow- ing exercise whose proof we indicate rather completely since it is of independent interest.

Lemma 1.35. If a compact abelian group G is the limit lim_j∈JG_j of a strict projective system of compact abelian groups Gj which have duality, then G has duality.

Proof. After Lemma 1.23, we have to show thatη_G:G→Gbbis bijective. We attack the harder part first and show thatηG is surjective. Assume that Ω∈ b

G; that is,b Ω is a morphism of abelian groups Gb → T. By Proposition 1.36,Gb =S

j∈JGcj. If we denote with Ω_j the restriction Ω|cG_j, then Ω_j:Gc_j →Tis an element of Gcc_j. SinceG_j has duality by hypothesis,η_Gj is surjective and thus there is ag_j ∈G_j such thatη_Gj(g_j) = Ω_j. We claim thatg ^def= (g_j)_j∈J ∈Q

j∈JG_j is an element of lim_j∈JG_j=G. For this purpose assume thatj≤kinJ. We have a commutative

diagram

G_k −−−−−−−−−→^ηGk Gcc_k

f_jk



 y



 y^fddjk Gj −−−−−−−−−→

η_Gj

cc Gj.

(We shall consider this claim in a separate exercise below.) We notice that cc

f_jk:Gcc_k →Gcc_j

is the restriction map sending Ωk to Ωk|Gcj= Ωj. Thus ηG_j fjk(gk)

= c

fcjk ηG_k(gk)

= c

fcjk(Ωk) = Ωj=ηG_j(gj).

But sinceGj has duality,ηG_j isinjective, and thus fjk(gk) =gj,

which establishes the claim g ∈ lim_j∈JG_j. For each limit map f_j:G → G_j, as before, we have a commutative diagram:

G −−−−−−−−−→^ηG Gbb

fj



 y



 y^fccj Gj −−−−−−−−−→

η_Gj

cc Gj. Thus fbb_j η_G(g)

=η_Gj f_j(g)

=η_Gj(g_j) = Ω_j for all j ∈ J. Now we observe that b

fbj:Gbb→ c

Gcjis the restriction Σ→Σ|cGj. Thus the restriction of the morphism ηG(g):Gb→Tto eachGcjis Ωj, and therefore this morphism is none other than the given map Ω. HenceηG(g) = Ω and the claim of the surjectivity ofηG is proved.

As a second step we show thatηG is injective. We have observed before that this statement is equivalent to the assertion that the characters of G separate the points. Hence we assume that 0 6=g ∈ G. Set N ={kerf_j | j ∈ J}. From Proposition 1.33(i) we know that TN ={0}. Hence there is a j ∈J such that g /∈kerf_j, that is,f_j(g)6= 0. Since the groupG_jhas duality, its characters separate its points. Hence there is aχ∈Gcj such thatχ fj(g)

6= 0. Hence χ◦fj∈Gb is a character ofGwhich does not annihilateg. The assertion is now proved. ut

Let us now assume the following result

Proposition 1.36.The characters of a compact abelian groups separate the points.

u t The proof is a consequence of the basic theorem for compact groups saying that every compact group has enough finite dimensional continuous lienar representa- tions to separate the points. [See e.g. Lecture Notes “Introduction to Topological Groups”, WS 2005-06,topgr.pdf.]

Now we can prove the following result.

Theorem 1.37. For any compact abelian group G the morphism η_G:G → Gbb is an isomorphism.

Proof. From Proposition 1.36, it follows at once thatηG:G→ b

Gbis injective. Hence the corestriction g 7→ η_G → Γ ^def= η_G(G) is an isomorphism onto the subgroup Γ ⊆ b

G. We claim that Γ =b b

G; a proof of this claim will finish the proof. Byb Proposition 1.36 once again, the claim is proved if every character ofG/Γ is zero,bb that is, if every character ofGbb which vanishes on Γ is zero. By Theorem 1.34 we may identifyGb with the character group ofGbb under the evaluation isomorphism.

Thus a characterf of b

Gb vanishing on Γ is given by an element χ∈Gb such that f(Ω) = Ω(χ). But we have 0 = f ηG(g)

= ηG(g)(χ) for all g ∈ G since f annihilates Γ. By the definition of ηG we then note χ(g) = ηG(g)(χ) = 0 for all g∈G, that is,χ= 0 and thusf = 0. ut Theorems 1.34 and 1.37 constitute the object portion of thePontryagin Duality Theorem for discrete and compact abelian groups. Up to natural isomorphism it sets up a bijection between the class of discrete and that of compact abelian groups. It shall reveal its true power when it is complemented by the morphism part which sets up a similar bijection between morphisms. However, this belongs to the domain of generalities and does, in fact, not require more work in depth.

The nontrivial portion of the duality is accomplished.

The following consequence of the duality theorem turns out to be very useful.

Corollary 1.38. (i) Let G be a compact abelian group and A a subgroup of the character groupG. The following two conditions are equivalent:b

(1) A separates the points ofG.

(2) A=G.b

(ii) (The Extension Theorem for Characters) If H is a closed subgroup of G, then every character ofH extends to a character ofG.

Proof. (i) Proposition 1.36 says that (2) implies (1), and so we have to prove that (1) implies (2). Since the characters of the discrete groupG/Ab separate the points by Lemma 1.11, in order to prove (2) it suffices to show that every character ofGb vanishing onAmust be zero. Thus let Ω be a character ofGb vanishing onA. By Theorem 1.37, there is ag ∈GwithηG(g) = Ω. Thusχ∈Aimplies 0 = Ω(χ) = ηG(g)(χ) =χ(g). From (1) we now concludeg= 0. Hence Ω =ηG(g) = 0.

(ii) The collection of all restrictionsχ|H of characters ofGtoH separates the points ofH since the characters ofGseparate the points ofGby Proposition 1.36.

Then (i) above shows that the function χ7→χ|H:Gb →Hb is surjective, and this

proves the assertion. ut

Corollary 1.39. For every compact abelian group G there is a filter basis N of compact subgroups such thatGis the strict projective limitlimN∈NG/N of factor groups each of which is a character group of a finitely generated abelian group.

Proof. LetA =Gb denote the character group of Gand F the family of finitely generated subgroups. If F ∈ F, let N_F = F^⊥ denote the annihilator {g ∈ G | χ(g) = 0 for allχ ∈ F}. Since F ⊆ F⁰ in F implies N_F⁰ ⊆ N_F, the family N ={N_F |F ∈ F }is a filter basis of closed subgroups. An elementgis inTN if and only if it is in the annihilator of every finitely generated subgroup ofA, hence if and only if it is annihilated by all ofA, sinceAis the union of all of its finitely generated subgroups. Thus g = 0 by Proposition 1.36. By Proposition 1.31(ii), therefore,Gis the strict projective limitG= lim_F∈FG/NF.

Now we claim that the character group ofG/NF may be identified withF. This will finish the proof of the corollary. IfqF:G→G/NF denotes the quotient map, then the functionϕ7→ ϕ◦qF: (G/NF)b → Gb is injective as qF is surjective. Its image is precisely the groupF^⊥⊥ of all characters vanishing onNF. Since every characterχ∈F vanishes onNF, we haveF ⊆F^⊥⊥. We shall now show equality and thereby prove the claim. But when F^⊥⊥ is identified with the Extension character group ofG/N_F then the subgroupF separates the points ofG/N_F since the only cosetg+N_F ∈G/N_F annihilated by all ofFisN_F by the definition ofN_F. Now the Extension Theorem for Characters, Corollary 1.38(ii) showsF =F^⊥⊥.ut

Corollary 1.39 yields the following remark:

Corollary 1.40. Every compact abelian group is the strict projective limit of a projective system of groupsG/N isomorphic toT^n(N)×EN with suitable numbers n(N) = 0,1, . . . , and finite abelian groupsEN. ut Thus every compact abelian groups is the strict projective limit of compact abelian groups defined on compact manifolds. Such groups are called compact abelianLie groups

Index

L

x∈XAx, 4 G,b 7

p-adic integers, 2, 12 p-adic solenoid, 12 Q, 6

R, 6 T^A, 3 Tp, 12 Y^X, 3 Z(n), 4 Zp, 12

Abelian group, divisible, 6 abelian group, injective, 6 AC, 6

Axiom of Choice, 6 Bonding map, 11 Cantor space, 5 character, 3, 7 character group, 3 closed subgroup, 2

compact abelian group, 18 compact group, 1

compact manifold, 5 Direct sum, 4 directed set, 11 divisible, 6 divisible group, 6 duality, 15

Evaluation morphism, 8f Extension Lemma, 8 extension of characters, 17

Extension Theorem for Characters, 17

Filter basis, 12

finitely generated abelian group, 5, 18

free abelian group, 4 GF(2), 5

group, character, 3, 7

group, compact, 1 group, divisible, 6

group, divisible abelian, 6

group, finitely generated abelian, 5 group, free abelian, 4

group, injective abelian, 6 group, topological, 1 group, torus, 5, 9 groups on spheres, 1 Has duality group, 15 Hom(A,T), 3

Injective, 6 Lie groups, 18 limit, 15 limit map, 11

Manifold, compact, 5 matrices, orthogonal, 2 matrices, unitary, 2

Ofp-adic integers group, 12 orthogonal matrices, 2

Pontryagin Duality Theorem, 14 projective limit, 11

projective system, 11f Quaternions, 2

Ring ofp-adic integers, 2 Separates the points, 5, 17 space, Cantor, 5

space, totally disconnected, 5 sphere groups, 1

strict projective limit, 12, 14, 18 strict projective system, 12, 15 subgroup, closed, 2

Topological group, 1 topology, of pointwise convergence, 3 torus group, 5, 9 totally disconnected, 5 Tychonoff product theorem, 2

Unitary matrices, 2 Zorn’s Lemma, 6