Manifolds, Vector Bundles, and Lie Groups

Michael E. Taylor

Contents0. Introduction1. Metric spaces and topological spaces2. Manifolds3. Vector bundles4. Sard’s theorem5. Lie groups6. The Campbell-Hausdorff formula7. Representations of Lie groups and Lie algebras8. Representations of compact Lie groups9. Representations of SU(2) and related groups

Introduction

This appendix provides background material on manifolds, vector bundles,and Lie groups, which are used throughout the book. We begin with asection on metric spaces and topological spaces, defining some terms thatare necessary for the concept of a manifold, defined in §2, and for that ofa vector bundle, defined in §3. These sections contain mostly definitions;however, a few results about compactness are proved.

In §4 we establish the easy case of a theorem of Sard, a useful result inmanifold theory. It is used only once in the text, in the development ofdegree theory in Chapter 1, §19.

In §5 we introduce the concept of a Lie group G and its Lie algebrag and establish the correspondence between Lie subgroups of G and Liesubalgebras of g. We also define a Haar measure on a Lie group. In §6we establish an important relation between Lie groups and Lie algebras,known as the Campbell-Hausdorff formula.

In §7 we discuss representations of a Lie group and associated represen-tations of its Lie algebra. Some basic results on representations of compactLie groups are given in §8, and in §9 we specialize to the groups SU(2) andSO(3) and to some related groups, such as SO(4). Material in §9 is use-ful in Chapter 8, Spectral Theory, particularly in its study of the simplestquantum mechanical model of the hydrogen atom.

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1. Metric spaces and topological spaces

A metric space is a set X together with a distance function d : X ×X →[0,∞), having the properties that

(1.1)d(x, y) = 0 ⇐⇒ x = y,d(x, y) = d(y, x),

d(x, y) ≤ d(x, z) + d(y, z).The third of these properties is called the triangle inequality. An example ofa metric space is the set of rational numbers Q, with d(x, y) = |x− y|. An-other example is X = Rn, with d(x, y) =

√(x1 − y1)2 + · · ·+ (xn − yn)2.

If (xν) is a sequence in X, indexed by ν = 1, 2, 3, . . . (i.e., by ν ∈ Z+),one says xν → y if d(xν , y) → 0, as ν → ∞. One says (xν) is a Cauchysequence if d(xν , xµ) → 0 as µ, ν → ∞. One says X is a complete metricspace if every Cauchy sequence converges to a limit in X. Some metricspaces are not complete; for example, Q is not complete. One can takea sequence (xν) of rational numbers such that xν →

√2, which is not

rational. Then (xν) is Cauchy in Q, but it has no limit in Q.If a metric space X is not complete, one can construct its completion X̂

as follows. Let an element ξ of X̂ consist of an equivalence class of Cauchysequences in X, where we say (xν) ∼ (yν), provided d(xν , yν) → 0. Wewrite the equivalence class containing (xν) as [xν ]. If ξ = [xν ] and η = [yν ],we can set d(ξ, η) = limν→∞ d(xν , yν) and verify that this is well definedand makes X̂ a complete metric space.

If the completion of Q is constructed by this process, you get R, the setof real numbers.

A metric space X is said to be compact provided any sequence (xν) inX has a convergent subsequence. Clearly, every compact metric space iscomplete. There are two useful conditions, each equivalent to the charac-terization of compactness just stated, on a metric space. The reader canestablish the equivalence, as an exercise.

(i) If S ⊂ X is a set with infinitely many elements, then there is an accu-mulation point, that is, a point p ∈ X such that every neighborhood U ofp contains infinitely many points in S.

Here, a neighborhood of p ∈ X is a set containing the ball(1.2) Bε(p) = {x ∈ X : d(x, p) < ε},for some ε > 0.

1. Metric spaces and topological spaces 3

(ii) Every open cover {Uα} of X has a finite subcover.

Here, a set U ⊂ X is called open if it contains a neighborhood of each of itspoints. The complement of an open set is said to be closed. Equivalently,K ⊂ X is closed provided that(1.3) xν ∈ K, xν → p ∈ X =⇒ p ∈ K.It is clear that any closed subset of a compact metric space is also compact.

If Xj , 1 ≤ j ≤ m, is a finite collection of metric spaces, with metrics dj ,we can define a product metric space

(1.4) X =m∏

j=1

Xj , d(x, y) = d1(x1, y1) + · · ·+ dm(xm, ym).

Another choice of metric is δ(x, y) =√

d1(x1, y1)2 + · · ·+ dm(xm, ym)2.The metrics d and δ are equivalent; that is, there exist constants C0, C1 ∈(0,∞) such that(1.5) C0δ(x, y) ≤ d(x, y) ≤ C1δ(x, y), ∀ x, y ∈ X.

We describe some useful classes of compact spaces.

Proposition 1.1. If Xj are compact metric spaces, 1 ≤ j ≤ m, so is theproduct space X =

∏mj=1 Xj .

Proof. Suppose (xν) is an infinite sequence of points in X; let us writexν = (x1ν , . . . , xmν). Pick a convergent subsequence (x1ν) in X1, andconsider the corresponding subsequence of (xν), which we relabel (xν).Using this, pick a convergent subsequence (x2ν) in X2. Continue. Havinga subsequence such that xjν → yj in Xj for each j = 1, . . . , m, we thenhave a convergent subsequence in X.

The following result is called the Heine-Borel theorem:

Proposition 1.2. If K is a closed bounded subset of Rn, then K is com-pact.

Proof. The discussion above reduces the problem to showing that anyclosed interval I = [a, b] in R is compact. Suppose S is a subset of I withinfinitely many elements. Divide I into two equal subintervals, I1 = [a, b1],I2 = [b1, b], b1 = (a + b)/2. Then either I1 or I2 must contain infinitely

4

many elements of S. Say Ij does. Let x1 be any element of S lying in Ij .Now divide Ij in two equal pieces, Ij = Ij1 ∪ Ij2. One of these intervals(say Ijk) contains infinitely many points of S. Pick x2 ∈ Ijk to be one suchpoint (different from x1). Then subdivide Ijk into two equal subintervals,and continue. We get an infinite sequence of distinct points xν ∈ S, and|xν − xν+k| ≤ 2−ν(b − a), for k ≥ 1. Since R is complete, (xν) converges,say to y ∈ I. Any neighborhood of y contains infinitely many points in S,so we are done.

If X and Y are metric spaces, a function f : X → Y is said to becontinuous provided xν → x in X implies f(xν) → f(x) in Y .

Proposition 1.3. If X and Y are metric spaces, f : X → Y continuous,and K ⊂ X compact, then f(K) is a compact subset of Y .

Proof. If (yν) is an infinite sequence of points in f(K), pick xν ∈ K suchthat f(xν) = yν . If K is compact, we have a subsequence xνj → p in X,and then yνj → f(p) in Y .

If F : X → R is continuous, we say f ∈ C(X). A corollary of Proposition1.3 is the following:

Proposition 1.4. If X is a compact metric space and f ∈ C(X), then fassumes a maximum and a minimum value on X.

A function f ∈ C(X) is said to be uniformly continuous provided that,for any ε > 0, there exists δ > 0 such that

(1.6) x, y ∈ X, d(x, y) ≤ δ =⇒ |f(x)− f(y)| ≤ ε.An equivalent condition is that f have a modulus of continuity, in otherwords, a monotonic function ω : [0, 1) → [0,∞) such that δ ↘ 0 ⇒ ω(δ) ↘0 and such that

(1.7) x, y ∈ X, d(x, y) ≤ δ ≤ 1 =⇒ |f(x)− f(y)| ≤ ω(δ).Not all continuous functions are uniformly continuous. For example, ifX = (0, 1) ⊂ R, then f(x) = sin(1/x) is continuous, but not uniformlycontinuous, on X. There is a case where continuity implies uniform conti-nuity:

Proposition 1.5. If X is a compact metric space and f ∈ C(X), then fis uniformly continuous.

Proof. If not, there exist xν , yν ∈ X and ε > 0 such that d(xν , yν) ≤ 2−νbut

(1.8) |f(xν)− f(yν)| ≥ ε.

1. Metric spaces and topological spaces 5

Taking a convergent subsequence xνj → p, we also have yνj → p. Nowcontinuity of f at p implies f(xνj ) → f(p) and f(yνj ) → f(p), contradicting(1.8).

If X and Y are metric spaces, the space C(X,Y ) of continuous maps f :X → Y has a natural metric structure, under some additional hypotheses.We use

(1.9) D(f, g) = supx∈X

d(f(x), g(x)

).

This sup exists provided f(X) and g(X) are bounded subsets of Y , whereto say B ⊂ Y is bounded is to say d : B ×B → [0,∞) has bounded image.In particular, this supremum exists if X is compact. The following resultis useful in the proof of the fundamental local existence theorem for ODE,in Chapter 1.

Proposition 1.6. If X is a compact metric space and Y is a completemetric space, then C(X, Y ), with the metric (1.9), is complete.

We leave the proof as an exercise.The following extension of Proposition 1.1 is a special case of Tychonov’s

theorem.

Proposition 1.7. If {Xj : j ∈ Z+} are compact metric spaces, so is theproduct X =

∏∞j=1 Xj .

Here, we can make X a metric space by setting

(1.10) d(x, y) =∞∑

j=1

2−jdj(pj(x), pj(y))

1 + dj(pj(x), pj(y)),

where pj : X → Xj is the projection onto the jth factor. It is easy to verifythat if xν ∈ X, then xν → y in X, as ν → ∞, if and only if, for each j,pj(xν) → pj(y) in Xj .

Proof. Following the argument in Proposition 1.1, if (xν) is an infinitesequence of points in X, we obtain a nested family of subsequences

(1.11) (xν) ⊃ (x1ν) ⊃ (x2ν) ⊃ · · · ⊃ (xjν) ⊃ · · ·such that p`(xjν) converges in X`, for 1 ≤ ` ≤ j. The next step is a“diagonal construction.” We set

(1.12) ξν = xνν ∈ X.Then, for each j, after throwing away a finite number N(j) of elements, oneobtains from (ξν) a subsequence of the sequence (xjν) in (1.11), so p`(ξν)converges in X` for all `. Hence (ξν) is a convergent subsequence of (xν).

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We turn now to the notion of a topological space. This is a set X,together with a family O of subsets, called “open,” satisfying the followingconditions:

(1.13)

X, ∅ open,

Uj open, 1 ≤ j ≤ N ⇒N⋂

j=1

Uj open,

Uα open, α ∈ A ⇒⋃

α∈AUα open,

where A is any index set. It is obvious that the collection of open subsetsof a metric space, defined above, satisfies these conditions. As before, a setS ⊂ X is closed provided X \ S is open. Also, we say a subset N ⊂ Xcontaining p is a neighborhood of p provided N contains an open set Uthat in turn contains p.

If X is a topological space and S is a subset, S gets a topology as follows.For each U open in X, U ∩ S is declared to be open in S. This is calledthe induced topology.

A topological space X is said to be Hausdorff provided that any dis-tinct p, q ∈ X have disjoint neighborhoods. Clearly, any metric space isHausdorff. Most important topological spaces are Hausdorff.

A Hausdorff topological space is said to be compact provided the follow-ing condition holds. If {Uα : α ∈ A} is any family of open subsets of X,covering X (i.e., X =

⋃α∈A Uα), then there is a finite subcover, that is,

a finite subset {Uα1 , . . . , UαN : αj ∈ A} such that X = Uα1 ∪ · · · ∪ UαN .An equivalent formulation is the following, known as the finite intersectionproperty. Let {Sα : α ∈ A} be any collection of closed subsets of X. Ifeach finite collection of these closed sets has nonempty intersection, thenthe complete intersection

⋂α∈A Sα is nonempty. It is not hard to show

that any compact metric space satisfies this condition.Any closed subset of a compact space is compact. Furthermore, any

compact subset of a Hausdorff space is necessarily closed.Most of the propositions stated above for compact metric spaces have

extensions to compact Hausdorff spaces. We mention one nontrivial result,which is the general form of Tychonov’s theorem; for a proof, see [Dug].

Theorem 1.8. If S is any nonempty set (possibly uncountable) and if, forany α ∈ S, Xα is a compact Hausdorff space, then so is X =

∏α∈S Xα.

A Hausdorff space X is said to be locally compact provided every p ∈ Xhas a neighborhood N that is compact (with the induced topology).

A Hausdorff space is said to be paracompact provided every open cover{Uα : α ∈ A} has a locally finite refinement, that is, an open cover {Vβ :β ∈ B} such that each Vβ is contained in some Uα and each p ∈ X has

2. Manifolds 7

a neighborhood Np such that Np ∩ Vβ is nonempty for only finitely manyβ ∈ B. A typical example of a paracompact space is a locally compactHausdorff space X that is also σ-compact (i.e., X =

⋃∞n=1 Xn with Xn

compact). Paracompactness is a natural condition under which to constructpartitions of unity, as will be illustrated in the next two sections.

A map F : X → Y between two topological spaces is said to be continu-ous provided F−1(U) is open in X whenever U is open in Y . If F : X → Yis one-to-one and onto, and both F and F−1 are continuous, F is said to bea homeomorphism. For a bijective map F : X → Y , the continuity of F−1is equivalent to the statement that F (V ) is open in Y whenever V is openin X; another equivalent statement is that F (S) is closed in Y whenever Sis closed in X.

If X and Y are Hausdorff, and F : X → Y is continuous, then F (K)is compact in Y whenever K is compact in X. In view of the discussionabove, there arises the following useful sufficient condition for a continuousmap F : X → Y to be a homeomorphism. Namely, if X is compact, Y isHausdorff, and F is one-to-one and onto, then F is a homeomorphism.

2. Manifolds

A manifold is a Hausdorff topological space with an “atlas,” that is, acovering by open sets Uj together with homeomorphisms ϕj : Uj → Vj ,Vj open in Rn. The number n is called the dimension of M . We say thatM is a smooth manifold provided the atlas has the following property. IfUjk = Uj ∩ Uk 6= ∅, then the map(2.1) ψjk : ϕj(Ujk) → ϕk(Ujk),given by ϕk ◦ ϕ−1j , is a smooth diffeomorphism from the open set ϕj(Ujk)to the open set ϕk(Ujk) in Rn. By this, we mean that ψjk is C∞, with aC∞-inverse. If the ψjk are all C`-smooth, M is said to be C`-smooth. Thepairs (Uj , ϕj) are called local coordinate charts.

A continuous map from M to another smooth manifold N is said tobe smooth if it is smooth in local coordinates. Two different atlases onM , giving a priori two structures of M as a smooth manifold, are said tobe equivalent if the identity map on M is smooth from each one of thesetwo manifolds to the other. Actually, a smooth manifold is considered tobe defined by equivalence classes of such atlases, under this equivalencerelation.

One way manifolds arise is the following. Let f1, . . . , fk be smooth func-tions on an open set U ⊂ Rn. Let M = {x ∈ U : fj(x) = cj}, for a given(c1, . . . , ck) ∈ Rk. Suppose that M 6= ∅ and, for each x ∈ M , the gradi-ents ∇fj are linearly independent at x. It follows easily from the implicitfunction theorem that M has a natural structure of a smooth manifoldof dimension n − k. We say M is a submanifold of U . More generally,

8

let F : X → Y be a smooth map between smooth manifolds, c ∈ Y ,M = F−1(c), and assume that M 6= ∅ and that, at each point x ∈ M ,there is a coordinate neighborhood U of x and V of c such that the de-rivative DF at x has rank k. More pedantically, (U,ϕ) and (V, ψ) are thecoordinate charts, and we assume the derivative of ψ ◦ F ◦ ϕ−1 has rankk at ϕ(x); there is a natural notion of DF (x) : TxX → TcY , which willbe defined in the next section. In such a case, again the implicit functiontheorem gives M the structure of a smooth manifold.

We mention a couple of other methods for producing manifolds. For one,given any connected smooth manifold M , its universal covering space M̃has the natural structure of a smooth manifold. M̃ can be described asfollows. Pick a base point p ∈ M . For x ∈ M , consider smooth paths from pto x, γ : [0, 1] → M . We say two such paths γ0 and γ1 are equivalent if theyare homotopic, that is, if there is a smooth map σ : I × I → M(I = [0, 1])such that σ(0, t) = γ0(t), σ(1, t) = γ1(t), σ(s, 0) = p, and σ(s, 1) = x.Points in M̃ lying over any given x ∈ M consist of such equivalence classes.

Another construction produces quotient manifolds. In this situation, wehave a smooth manifold M and a discrete group Γ of diffeomorphisms onM . The quotient space Γ \M consists of equivalence classes of points ofM , where we set x ∼ γ(x) for each x ∈ M , γ ∈ Γ. If we assume that eachx ∈ M has a neighborhood U containing no γ(x), for γ 6= e, the identityelement of Γ, then Γ \M has a natural smooth manifold structure.

We next discuss partitions of unity. Supose M is paracompact. In thiscase, using a locally finite covering of M by coordinate neighborhoods, wecan construct ψj ∈ C∞0 (M) such that, for any compact K ⊂ M , onlyfinitely many ψj are nonzero on K (we say the sequence ψj is locally finite)and such that, for any p ∈ M , some ψj(p) 6= 0. Then

(2.2) ϕj(x) =(∑

k

ψk(x)2)−1

ψj(x)2

is a locally finite sequence of functions in C∞0 (M), satisfying∑

j ϕj(x) = 1.Such a sequence is called a partition of unity. It has many uses.

Using local coordinates plus such cut-offs as appear in (2.2), one can eas-ily prove that any smooth, compact manifold M can be smoothly imbeddedin some Euclidean space RN , though one does not obtain so easily Whit-ney’s optimal value of N (N = 2dim M + 1, valid for paracompact M , notjust compact M), proved in [Wh].

A more general notion than manifold is that of a smooth manifold withboundary. In this case, M is again a Hausdorff topological space, andthere are two types of coordinate charts (Uj , ϕj). Either ϕj takes Uj toan open subset Vj of Rn as before, or ϕj maps Uj homeomorphically ontoan open subset of Rn+ = {(x1, . . . , xn) ∈ Rn : xn ≥ 0}. Again appropriatetransition maps are required to be smooth. In case M is paracompact,there is again the construction of partitions of unity. For one simple but

3. Vector bundles 9

effective application of this construction, see the proof of the Stokes formulain §13 of Chapter I.

3. Vector bundles

We begin with an intrinsic definition of a tangent vector to a smooth man-ifold M , at a point p ∈ M . It is an equivalence class of smooth curvesthrough p, that is, of smooth maps γ : I → M , I an interval containing 0,such that γ(0) = p. The equivalence relation is γ ∼ γ1 provided that, forsome coordinate chart (U,ϕ) about p, ϕ : U → V ⊂ Rn, we have

(3.1)d

dt(ϕ ◦ γ)(0) = d

dt(ϕ ◦ γ1)(0).

This equivalence is independent of the choice of coordinate chart about p.If V ⊂ Rn is open, we have a natural identification of the set of tangent

vectors to V at p ∈ V with Rn. In general, the set of tangent vectors to Mat p is denoted TpM . A coordinate cover of M induces a coordinate coverof TM , the disjoint union of TpM as p runs over M , making TM a smoothmanifold. TM is called the tangent bundle of M . Note that each TpM hasthe natural structure of a vector space of dimension n, if n is the dimensionof M . If F : X → M is a smooth map between manifolds, x ∈ X, there isa natural linear map DF (x) : TxX → TpM , p = F (x), which agrees withthe derivative as defined in §1 of Chapter 1, in local coordinates. DF (x)takes the equivalence class of a smooth curve γ through x to that of thecurve F ◦ γ through p.

The tangent bundle TM of a smooth manifold M is a special case ofa vector bundle. Generally, a smooth vector bundle E → M is a smoothmanifold E, together with a smooth map π : E → M with the followingproperties. For each p ∈ M , the “fiber” Ep = π−1(p) has the structure of avector space, of dimension k, independent of p. Furthermore, there exists acover of M by open sets Uj , and diffeomorphisms Φj : π−1(Uj) → Uj ×Rkwith the property that, for each p ∈ Uj , Φj : Ep → {p} × Rk → Rk is alinear isomorphism, and if Uj` = Uj ∩ U` 6= ∅, we have smooth “transitionfunctions”

(3.2) Φ` ◦ Φ−1j = Ψj` : Uj` × Rk → Uj` × Rk,which are the identity on the first factor and such that for each p ∈ Uj`,Ψj`(p) is a linear isomorphism on Rk. In the case of complex vector bundles,we systematically replace Rk by Ck in the discussion above.

The structure above arises for the tangent bundle as follows. Let (Uj , ϕj)be a coordinate cover of M , ϕj : Uj → Vj ⊂ Rn. Then Φj : TUj → Uj×Rntakes the equivalence class of smooth curves through p ∈ Uj containing anelement γ to the pair

(p, (ϕj ◦ γ)′(0)

) ∈ Uj × Rn.

10

A section of a vector bundle E → M is a smooth map β : M → E suchthat π(β(p)) = p for all p ∈ M . For example, a section of the tangentbundle TM → M is a vector field on M . If X is a vector field on M ,generating a flow F t, then X(p) ∈ TpM coincides with the equivalenceclass of γ(t) = F tp.

Any smooth vector bundle E → M has associated a vector bundle E∗ →M , the “dual bundle” with the property that there is a natural duality ofEp and E∗p for each p ∈ M . In case E is the tangent bundle TM , this dualbundle is called the cotangent bundle and is denoted T ∗M .

More generally, given a vector bundle E → M , other natural construc-tions involving vector spaces yield other vector bundles over M , such astensor bundles ⊗jE → M with fiber ⊗jEp, mixed tensor bundles with fiber(⊗jEp

) ⊗ (⊗kE∗p), exterior algebra bundles with fiber ΛEp, and so forth.

Note that a k-form, as defined in Chapter 1, is a section of ΛkT ∗M . Asection of

(⊗jT )⊗ (⊗kT ∗)M is called a tensor field of type (j, k).A Riemannian metric tensor on a smooth manifold M is a smooth, sym-

metric section g of ⊗2T ∗M that is positive-definite at each point p ∈ M ;that is, gp(X, X) > 0 for each nonzero X ∈ TpM . For any fixed p ∈ M ,using a local coordinate patch (U,ϕ) containing p, one can construct apositive, symmetric section of ⊗2T ∗U . Using a partition of unity, we canhence construct a Riemannian metric tensor on any smooth, paracompactmanifold M . If we define the length of a path γ : [0, 1] → M to be

L(γ) =∫ 1

0

g(γ′(t), γ′(t)

)1/2dt,

then

(3.3) d(p, q) = inf{L(γ) : γ(0) = p, γ(1) = q}

is a distance function making M a metric space, provided M is connected.The notion of vector bundle often aids in making intrinsic definitions of

important mathematical concepts. As an illustration, we note the followingintrinsic characterization of the contact form κ on T ∗M , which was specifiedin local coordinates in (14.17) of Chapter 1. Let z ∈ T ∗M ; if π : T ∗M → Mis the natural projection, let p = π(z), so z ∈ T ∗p M . To define κ at z, asκ(z) ∈ T ∗z (T ∗M), we specify how it acts on a tangent vector v ∈ Tz(T ∗M).The specification is

(3.4) 〈v, κ(z)〉 = 〈(Dπ)v, z〉,

where Dπ : Tz(T ∗M) → TpM is the derivative of π, and the right side of(3.4) is defined by the usual dual pairing of TpM and T ∗p M . It is routine tocheck that this agrees with (14.17) of Chapter 1 in any coordinate systemon M . This establishes again the result of §14 of Chapter 1, that thesymplectic form σ = dκ is well defined on a cotangent bundle T ∗M .

4. Sard’s theorem 11

4. Sard’s theorem

Let F : Ω → Rn be a C1-map, with Ω open in Rn. If p ∈ Ω and DF (p) :Rn → Rn is not surjective, then p is said to be a critical point and F (p) acritical value. The set C of critical points can be a large subset of Ω, evenall of it, but the set of critical values F (C) must be small in Rn. This ispart of Sard’s theorem.

Theorem 4.1. If F : Ω → Rn is a C1-map, then the set of critical valuesof F has measure 0 in Rn.

Proof. If K ⊂ Ω is compact, cover K ∩ C with m-dimensional cubes Qj ,with disjoint interiors, of side δj . Pick pj ∈ C ∩ Qj , so Lj = DF (pj) hasrank ≤ n− 1. Then, for x ∈ Qj ,

F (pj + x) = F (pj) + Ljx + Rj(x), ‖Rj(x)‖ ≤ ρj = ηjδj ,where ηj → 0 as δj → 0. Now Lj(Qj) is certainly contained in an (n− 1)-dimensional cube of side C0δj , where C0 is an upper bound for

√m‖DF‖

on K. Since all points of F (Qj) are a distance ≤ ρj from (a translate of)Lj(Qj), this implies

meas F (Qj) ≤ 2ρj(C0δj + 2ρj)n−1 ≤ C1ηjδnj ,provided δj is sufficiently small that ρj ≤ δj . Now

∑j δ

nj is the volume

of the cover of K ∩ C. For fixed K, this can be assumed to be bounded.Hence

meas F (C ∩K) ≤ CK η,where η = max {ηj}. Picking a cover by small cubes, we make η arbitrarilysmall, so meas F (C ∩K) = 0. Letting Kj ↗ Ω, we complete the proof.

Sard’s theorem also treats the more difficult case when Ω is open inRm,m > n. Then a more elaborate argument is needed, and one requiresmore differentiability, namely that F is class Ck, with k = m − n + 1. Aproof can be found in [Stb]. The theorem also clearly extends to smoothmappings between separable manifolds.

Theorem 4.1 is applied in Chapter 1, in the study of degree theory. Wegive another application of Theorem 4.1, to the existence of lots of Morsefunctions. This application gives the typical flavor of how one uses Sard’stheorem, and it is used in a Morse theory argument in Appendix C. Theproof here is adapted from one in [GP]. We begin with a special case:

Proposition 4.2. Let Ω ⊂ Rn be open, f ∈ C∞(Ω). For a ∈ Rn, setfa(x) = f(x)−a ·x. Then, for almost every a ∈ Rn, fa is a Morse function,that is, it has only nondegenerate critical points.

12

Proof. Consider F (x) = ∇f(x); F : Ω → Rn. A point x ∈ Ω is a criticalpoint of fa if and only if F (x) = a, and this critical point is degenerateonly if, in addition, a is a critical value of F . Hence the desired conclusionholds for all a ∈ Rn that are not critical values of F .

Now for the result on manifolds:

Proposition 4.3. Let M be an n-dimensional manifold, imbedded in RK .Let f ∈ C∞(M), and, for a ∈ RK , let fa(x) = f(x)−a·x, for x ∈ M ⊂ RK .Then, for almost all a ∈ RK , fa is a Morse function.

Proof. Each p ∈ M has a neighborhood Ωp such that some n of thecoordinates xν on RK produce coordinates on Ωp. Let’s say x1, . . . , xn doit. Let (an+1, . . . , aK) be fixed, but arbitrary. Then, by Proposition 4.2, foralmost every (a1, . . . , an) ∈ Rn, fa has only nondegenerate critical pointson Ωp. By Fubini’s theorem, we deduce that, for almost every a ∈ RK ,fa has only nondegenerate critical points on Ωp. (The set of bad a ∈ RKis readily seen to be a countable union of closed sets, hence measurable.)Covering M by a countable family of such sets Ωp, we finish the proof.

5. Lie groups

A Lie group G is a group that is also a smooth manifold, such that the groupoperations G×G → G and G → G given by (g, h) 7→ gh and g 7→ g−1 aresmooth maps. Let e denote the identity element of G. For each g ∈ G, wehave left and right translations, Lg and Rg, diffeomorphisms on G, definedby

(5.1) Lg(h) = gh, Rg(h) = hg.

The set of left-invariant vector fields X on G, that is, vector fields satis-fying

(5.2) (DLg)X(h) = X(gh),

is called the Lie algebra of G, and is denoted g. If X, Y ∈ g, then the Liebracket [X,Y ] belongs to g. Evaluation of X ∈ g at e provides a linearisomorphism of g with TeG.

A vector field X on G belongs to g if and only if the flow F tX it generatescommutes with Lg for all g ∈ G, that is, g(F tXh) = F tX(gh) for all g, h ∈ G.If we set

(5.3) γX(t) = F tXe,we obtain γX(t + s) = FsX(F tXe) · e = (F tXe)(FsXe), and hence(5.4) γX(s + t) = γX(s)γX(t),

5. Lie groups 13

for s, t ∈ R; we say γX is a smooth, one-parameter subgroup of G. Clearly,(5.5) γ′X(0) = X(e).

Conversely, if γ is any smooth, one-parameter group satisfying γ′(0) =X(e), then F tg = g ·γ(t) defines a flow generated by the vector field X ∈ gcoinciding with X(e) at e.

The exponential map

(5.6) Exp : g −→ Gis defined by

(5.7) Exp(X) = γX(1).

Note that γsX(t) = γX(st), so Exp(tX) = γX(t). In particular, under theidentification g → TeG,(5.8) D Exp(0) : TeG −→ TeG is the identity map.

The fact that each element X ∈ g generates a one-parameter group hasthe following generalization, to a fundamental result of S. Lie. Let h ⊂ g bea Lie subalgebra, that is, h is a linear subspace and Xj ∈ h ⇒ [X1, X2] ∈ h.By Frobenius’s theorem (established in §9 of Chapter 1), through eachpoint p of G there is a smooth manifold Mp of dimension k = dim h,which is an integral manifold for h (i.e., h spans the tangent space of Mpat each q ∈ Mp). We can take Mp to be the maximal such (connected)manifold, and then it is unique. Let H be the maximal integral manifoldof h containing the identity element e.

Proposition 5.1. H is a subgroup of G.

Proof. Take h0 ∈ H and consider H0 = h−10 H; clearly, e ∈ H0. By leftinvariance, H0 is also an integral manifold of h, so H0 = H. This showsthat h0, h1 ∈ H ⇒ h−10 h1 ∈ H, so H is a group.

In addition to left-invariant vector fields on G, one can consider all left-invariant differential operators on G. This is an algebra, isomorphic to the“universal enveloping algebra” U(g), which can be defined as

(5.9) U(g) =⊗

gC/J,

where gC is the complexification of g and J is the two-sided ideal in thetensor algebra

⊗gC generated by {XY − Y X − [X,Y ] : X, Y ∈ g}.

There are other classes of objects whose left-invariant elements are ofparticular interest, such as tensor fields (particularly metric tensors) anddifferential forms.

Given any α0 ∈ ΛkT ∗e G, there is a unique k-form α on G, invariant underLg, that is, satisfying L∗gα = α for all g ∈ G, equal to α0 at e. In case

14

k = n = dim G, if ω0 is a nonzero element of ΛnT ∗e G, the correspondingleft-invariant n-form ω on G defines also an orientation on G, and hence aleft-invariant volume form on G, called a (left) Haar measure. It is uniquelydefined up to a constant multiple. Similarly one has a right Haar measure.It is very important to be able to integrate over a Lie group using Haarmeasure.

In many but not all cases left Haar measure is also right Haar measure;then G is said to be unimodular. Note that if ω ∈ Λn(G) gives a left Haarmeasure, then, for each g ∈ G, R∗gω is also a left Haar measure, so we musthave

(5.10) R∗gω = µ(g)ω, µ : G → (0,∞).Furthermore, µ(gg′) = µ(g)µ(g′). If G is compact, this implies µ(g) = 1for all g, so all compact Lie groups are unimodular.

There are some particular Lie groups that we want to mention. Letn ∈ Z+ and F = R or C. Then Gl(n, F ) is the group of all invertible n×nmatrices with entries in F . We set

(5.11) Sl(n, F ) = {A ∈ Gl(n, F ) : det A = 1}.We also set

(5.12)O(n) = {A ∈ Gl(n,R) : At = A−1},

SO(n) = {A ∈ O(n) : det A = 1},and

(5.13)U(n) = {A ∈ Gl(n,C) : A∗ = A−1},

SU(n) = {A ∈ U(n) : det A = 1}.The Lie algebras of the groups listed above also have special names. We

have gl(n, F ) = M(n, F ), the set of n×n matrices with entries in F . Also,

(5.14)

sl(n, F ) = {A ∈ M(n, F ) : Tr A = 0},o(n) = so(n) = {A ∈ M(n,R) : At = −A},u(n) = {A ∈ M(n,C) : A∗ = −A},

su(n) = {A ∈ u(n) : Tr A = 0}.There are many other important matrix Lie groups and Lie algebras with

special names, but we will not list any more here. See [Helg], [T], or [Var1]for such lists.

6. The Campbell-Hausdorff formula

The Campbell-Hausdorff formula has the form

(6.1) Exp(X) Exp(Y ) = Exp(C(X,Y )),

6. The Campbell-Hausdorff formula 15

where G is any Lie group, with Lie algebra g, and Exp: g → G is theexponential map defined by (5.7); X and Y are elements of g in a sufficientlysmall neighborhood U of zero. The map C : U × U → g has a universalform, independent of g. We give a demonstration similar to one in [HS],which was also independently discovered by [Str].

We begin with the case G = Gl(n,C) and produce an explicit formulafor the matrix-valued analytic function X(s) of s in the identity

(6.2) eX(s) = eXesY ,

near s = 0. Note that this function satisfies the ODE

(6.3)d

dseX(s) = eX(s)Y.

We can produce an ODE for X(s) by using the following formula, derivedin Exercises 7–10 of §4, Chapter 1:

(6.4)d

dseX(s) = eX(s)

∫ 10

e−τX(s)X ′(s)eτX(s) dτ.

As shown there, we can rewrite this as

(6.5)d

dseX(s) = eX(s)Ξ

(ad X(s)

)X ′(s).

Here, ad is defined as a linear operator on the space of n× n matrices by(6.6) ad X(Y ) = XY − Y X;the function Ξ is

(6.7) Ξ(z) =∫ 1

0

e−τz dτ =1− e−z

z,

an entire holomorphic function of z; and a holomorphic function of an op-erator is defined either as in Exercise 10 of that set, or as in §5 of AppendixA. Comparing (6.3) and (6.5), we obtain

(6.8) Ξ(ad X(s)

)X ′(s) = Y, X(0) = X.

We can obtain a more convenient ODE for X(s) as follows. Note that

(6.9) ead X(s) = Ad eX(s) = Ad eX ·Ad esY = ead X es ad Y .Now let Ψ(ζ) be holomorphic near ζ = 1 and satisfy

(6.10) Ψ(ea) =1

Ξ(a)=

a

1− e−a ,

explicitly,

(6.11) Ψ(ζ) =ζ log ζζ − 1 .

It follows that

(6.12) Ψ(ead Xes ad Y

)Ξ

(ad X(s)

)= I,

16

so we can transform (6.8) to

(6.13) X ′(s) = Ψ(ead Xes ad Y

)Y, X(0) = X.

Integrating gives the Campbell-Hausdorff formula for X(s) in (6.2):

(6.14) X(s) = X +∫ s

0

Ψ(ead Xet ad Y

)Y dt.

This is valid for ‖sY ‖ small enough, if also X is close enough to 0.Taking the s = 1 case, we can rewrite this formula as

(6.15) eXeY = eC(X,Y ), C(X, Y ) = X +∫ 1

0

Ψ(ead Xet ad Y

)Y dt.

The formula (6.15) gives a power series in ad X and ad Y which is norm-summable provided

(6.16) ‖ad X‖ ≤ x, ‖ad Y ‖ ≤ y,with ex+y − 1 < 1, that is,(6.17) x + y < log 2.

We can extend the analysis above to the case where X and Y are vectorfields on a manifold M , asking for a vector field X(s) such that

(6.18) F1X(s) = F1XFsY ,where F tX is the flow generated by X, evaluated at time t. If there is sucha family X(s), depending smoothly on s, material in §6 of Chapter 1, inplace of material in §4 cited above, leads to a formula parallel to (6.4),and hence to (6.8), in this context. However, we cannot always solve (6.8),because ad X(s) tends not to act as a bounded operator on a Banach spaceof vector fields, and in fact one cannot always solve (6.18) for X(s) is thiscase. However, if there is a finite-dimensional Lie algebra g of vector fieldscontaining X and Y , then the analysis (6.9)–(6.17) extends. We have

(6.19) F tXF tY = F tC(t,X,Y ),with

(6.20) C(t,X, Y ) = X +∫ 1

0

Ψ(ead tXead stY

)Y ds,

provided ‖ad tX‖ + ‖ad tY ‖ < log 2, the operator norm ‖ad X‖ beingcomputed using any convenient norm on g. In particular, if M = G is aLie group with Lie algebra g, and X, Y ∈ g, this analysis applies to yieldthe Campbell-Hausdorff formula for general Lie groups.

7. Representations of Lie groups and Lie algebras 17

7. Representations of Lie groups and Lie algebras

We define a representation of a Lie group G on a finite-dimensional vectorspace V to be a smooth map

(7.1) π : G −→ End(V )such that

(7.2) π(e) = I, π(gg′) = π(g)π(g′), g, g′ ∈ G.If F ∈ C0(G), that is, if F is continuous with compact support, we candefine π(F ) ∈ End(V ) by

(7.3) π(F )v =∫

G

F (g)π(g)v dg.

We get different results depending on whether left or right Haar measure isused. Right now, let us use right Haar measure. Then, for g ∈ G, we have

(7.4) π(F )π(g)v =∫

G

F (x)π(xg)v dx =∫

G

F (xg−1)π(x)v dx.

We also define the derived representation

(7.5) dπ : g −→ End(V )by

(7.6) dπ = Dπ(e) : TeG −→ End(V ),using the identification g ≈ TeG. Thus, for X ∈ g,

(7.7) dπ(X)v = limt→0

1t

[π(Exp tX)v − v].

The following result states that dπ is a Lie algebra homomorphism.

Proposition 7.1. For X,Y ∈ g, we have(7.8)

[dπ(X), dπ(Y )

]= dπ

([X, Y ]

).

Proof. We will first produce a formula for π(F )dπ(X), given F ∈ C∞0 (G).In fact, making use of (7.4), we have

(7.9)

π(F )dπ(X)v = limt→0

1t

∫

G

[F (g)π(g)π(Exp tX)− F (g)π(g)]v dg

= limt→0

1t

∫

G

[F

(g · Exp(−tX))− F (g)]π(g)v dg

= −π(XF )v,

18

where XF denotes the left-invariant vector field X applied to F . It followsthat

(7.10)π(F )

[dπ(X)dπ(Y )− dπ(Y )dπ(X)]v

= π(Y XF −XY F )v = −π([X, Y ]F )v,which by (7.9) is equal to π(F )dπ

([X, Y ]

)v. Now, if F is supported near

e ∈ G and integrates to 1, is is easily seen that π(F ) is close to the identityI, so this implies (7.8).

There is a representation of G on g, called the adjoint representation,defined as follows. Consider

(7.11) Kg : G −→ G, Kg(h) = ghg−1.Then Kg(e) = e, and we set

(7.12) Ad(g) = DKg(e) : TeG −→ TeG,identifying TeG ≈ g. Note that Kg ◦Kg′ = Kgg′ , so the chain rule impliesAd(g)Ad(g′) = Ad(gg′).

Note that γ(t) = g Exp(tX)g−1 is a one-parameter subgroup of G satis-fying γ′(0) = Ad(g)X. Hence

(7.13) Exp(t Ad(g)X) = g Exp(tX) g−1.

In particular,

(7.14) Exp((Ad Exp sY )tX

)= Exp(sY ) Exp(tX) Exp(−sY ).

Now, the right side of (7.15) is equal to F−sY ◦ F tX ◦ FsY (e), so by resultson the Lie derivative of a vector field given in (8.1)–(8.3) of Chapter 1, wehave

(7.15) Ad(Exp sY )X = FsY #X.If we take the s-derivative at s = 0, we get a formula for the derivedrepresentation of Ad, which is denoted ad, rather than d Ad. Using (8.3)–(8.5) of Chapter 1, we have

(7.16) ad(Y )X = [Y, X].

In other words, the adjoint representation of g on g is given by the Liebracket. We mention that Jacobi’s identity for Lie algebras is equivalentto the statement that

(7.17) ad([X, Y ]

)=

[ad(X), ad(Y )

], ∀ X, Y ∈ g.

If V has a positive-definite inner product, we say that the representation(7.1) is unitary provided π(g) is a unitary operator on V , for each g ∈ G(i.e., π(g)−1 = π(g)∗).

7. Representations of Lie groups and Lie algebras 19

We say the representation (7.1) is irreducible if V has no proper linearsubspace invariant under π(g) for all g ∈ G. Irreducible unitary represen-tations are particularly important. The following version of Schur’s lemmais useful.

Proposition 7.2. A unitary representation π of G on V is irreducible ifand only if, for any A ∈ End(V ),(7.18) π(g)A = Aπ(g), ∀ g ∈ G =⇒ A = λI.

Proof. First, suppose π is irreducible and A commutes with π(g) for allg. Then so does A∗, hence A + A∗ and (1/i)(A − A∗), so we may as wellsuppose A = A∗. Now, any polynomial p(A) commutes with π(g) for allg, so it follows that each projection Pλ onto an eigenspace of A commuteswith all π(g). Hence the range of Pλ is invariant under π, so if Pλ 6= 0, itmust be I, and A = λI.

Conversely, suppose the implication (7.18) holds. Then if W ⊂ V isinvariant under π, the orthogonal projection P of V onto W must commutewith all π(g), so P is a scalar multiple of I, hence either 0 or I. Thiscompletes the proof.

Corollary 7.3. Assume G is connected. Then a unitary representation ofG on V is irreducible if and only if, for any A ∈ End(V ),(7.19) dπ(X)A = A dπ(X), ∀ X ∈ g =⇒ A = λI.

Proof. We mention that

(7.20) π(Exp tX) = et dπ(X)

and leave the details to the reader.

Given a representation π of G on V , there is also a representation of theuniversal enveloping algebra U(g), defined as follows. If

(7.21) P =∑

µ≤mci1···iµXi1 · · ·Xiµ , Xj ∈ g,

with ci1···iµ ∈ C, we have

(7.22) dπ(P ) =∑

µ≤mci1···iµdπ(Xi1) · · · dπ(Xiµ).

Proposition 7.4. Suppose G is connected. Let P ∈ U(g), and assume(7.23) PX = XP, ∀ X ∈ g.

20

If π is an irreducible unitary representation of G on V , then dπ(P ) is ascalar multiple of the identity, that is,

dπ(P ) = λI.

Proof. Immediate from Corollary 7.3.

So far in this section we have concentrated on finite-dimensional repre-sentations. It is also of interest to consider infinite-dimensional represen-tations. One example is the right-regular representation of G on L2(G):

(7.24) R(g)f(x) = f(xg).

If G has right-invariant Haar measure, then R(g) is a unitary operator onL2(G) for each g ∈ G, and one readily verifies that R(g)R(g′) = R(gg′).However, the smoothness hypothesis made on π in (7.1) does not holdhere. When working with an infinite-dimensional representation π of G ona Banach space V , one makes instead the hypothesis of strong continuity:For each v ∈ V , the map g 7→ π(g)v is continuous from G to V , with itsnorm topology. If the map is C∞, one says v is a smooth vector for therepresentation v. For example, each f ∈ C∞0 (G) is a smooth vector for therepresentation (7.24). Of course, C∞0 (G) is dense in L

2(G). More generally,the set of smooth vectors for any strongly continuous representation π ofG on a Banach space V is dense in V . In fact, for F ∈ C∞0 (G), π(F ) isstill well defined by (7.3), and the space

(7.25) Gπ = {π(F )v : F ∈ C∞0 (G), v ∈ V }is readily verified to be a dense subspace of V consisting of smooth vec-tors. If V is finite dimensional, this implies that Gπ = V , so any stronglycontinuous, finite-dimensional representation of a Lie group automaticallypossesses the smoothness property used above.

The occasional use made of Lie group representations in this book willnot require much development of the theory of infinite-dimensional repre-sentations, so we will not go further into it here. One can find treatmentsin many places, including [HT], [Kn], [T], [Var2], and [Wal1].

8. Representations of compact Lie groups

Throughout this section, G will be a compact Lie group. If π is a represen-tation of G on a finite-dimensional complex vector space V , we can alwaysput an inner product on V so that π is unitary. Indeed, let ((u, v)) be anyHermitian inner product on V , and set

(8.1) (u, v) =∫

G

((π(g)u, π(g)v)) dg.

8. Representations of compact Lie groups 21

Note that if V1 is a subspace of V invariant under π(g) for all g ∈ G, and ifπ is unitary, then the orthogonal complement of V1 is also invariant. Thus,if π is not irreducible on V , we can decompose it, and we can obviouslycontinue this process only a finite number of times if dim V is finite. Thusπ breaks up into a direct sum of irreducible unitary representations of G.

Let π and λ be two representations of G, on V and W , respectively. Wesay they are equivalent if there is A ∈ L(V, W ), invertible, such that

(8.2) π(g) = A−1λ(g)A, ∀ g ∈ G.

If these representations are unitary, we say they are unitarily equivalent ifA can be taken to be unitary.

Suppose that π and λ are irreducible and unitary, and (8.2) holds. Thenπ(g)∗ = A∗λ(g)∗(A−1)∗, for all g ∈ G, so π(g) = (A∗A)π(g)(A∗A)−1. BySchur’s lemma, A∗A must be a (positive) scalar, say b2. Replacing A byb−1A makes it unitary. Breaking up a general π into irreducible represen-tations, we deduce that whenever π and λ are finite-dimensional, unitaryrepresentations, if they are equivalent, then they are unitarily equivalent.

We now derive some results known as Weyl orthogonality relations, whichplay an important role in the study of representations of compact Liegroups. To begin, let π and λ be two irreducible representations of acompact group G, on finite-dimensional spaces V and W , respectively.Consider the representation ν = π ⊗ λ on V ⊗W ′ ≈ L(W,V ), defined by

(8.3) ν(g)(A) = π(g)Aλ(g)−1, g ∈ G, A ∈ L(W,V ).

Let Z be the linear subspace of L(V, W ) on which ν acts trivially. We wantto specify Z. Note that A0 ∈ Z if and only if

(8.4) π(g)A0 = A0π(g), ∀ g ∈ G.

Since this implies that the range of A0 is invariant under π and Ker A0is invariant under λ, we see that either A0 = 0 or A0 is an isomorphismfrom W to V . In the latter case, we have π(g) = A0λ(g)A−10 , so therepresentations π and λ would have to be equivalent. In this case, forarbitrary A ∈ Z, we would have

π(g)A = Aλ(g) = AA−10 π(g)A0,

or π(g)AA−10 = AA−10 π(g), so Schur’s lemma implies that AA

−10 is a scalar.

We have proved the following result:

Proposition 8.1. If π and λ are finite-dimensional, irreducible representa-tions of G and if ν = π⊗λ, then the trivial representation occurs not at allin ν if π and λ are not equivalent, and it occurs acting on a one-dimensionalsubspace of V ⊗W ′ if π and λ are equivalent.

22

The next ingredient for the orthogonality relation is the study of theoperator

(8.5) P =∫

G

π(g) dg.

Here π is a finite-dimensional representation of the compact group G, notnecessarily irreducible, and dg denotes Haar measure, with total mass 1.Note that

(8.6) π(y)P =∫

G

π(yg) dg = P = Pπ(y),

for all y ∈ G. Hence

(8.7) P 2 = P∫

G

π(g) dg =∫

G

Pπ(g) dg = P,

so P is a projection. Also, if π is unitary, we see that P = P ∗.Now, if π is unitary, it gives a representation both on the rangeR(P ) and

on the kernel Ker P . It is clear from (8.5) that, given v ∈ V , ‖Pv‖ < ‖v‖unless π(g)v = v, for all g ∈ G. Consequently, π operates like the identityon R(P ), but we do not have π(g)v = v for all g ∈ G, for any nonzerov ∈ Ker P . We have proved:

Proposition 8.2. If π is a unitary representation of G on V , then P , givenby (8.5), is the orthogonal projection onto the subspace of V on which πacts trivially.

The following is a special case:

Corollary 8.3. If π is a nontrivial, irreducible, unitary representation,and P is given by (8.5), then P = 0.

We apply Proposition 8.2 to

(8.8) Q =∫

G

π(g)⊗ λ(g) dg,

with π and λ irreducible. By Proposition 8.1, we see that

(8.9) Q = 0 if π and λ are not equivalent.

On the other hand, if λ = π, then Q has as its range the set of scalarmultiples of the identity operator on V (if π acts on V ). Note that π ⊗ πleaves invariant the space of elements A ∈ L(V, V ) of trace zero, whichis the orthogonal complement (with respect to the Hilbert-Schmidt inner

8. Representations of compact Lie groups 23

product) of the space of scalars, so Q must annihilate this space. Thus Qis given by

(8.10) Q(A) = (d−1 Tr A)I, π = λ, d = dim V.

The identities (8.9) and (8.10) are equivalent to the Weyl orthogonalityrelations. If we express π and λ as matrices, with respect to some or-thonormal bases, we get the following theorem:

Theorem 8.4. Let π and λ be inequivalent irreducible, unitary represen-tations of G, on V and W , with matrix entries πij and λk`, respectively.Then

(8.11)∫

G

πij(g)λk`(g) dg = 0.

Also,

(8.12)∫

G

πij(g)πk`(g) dg = 0, unless i = k and j = `.

Furthermore,

(8.13)∫

G

|πij(g)|2 dg = d−1,

where d = dim V = Tr π(e).

Hence, if {πk} is a complete set of inequivalent, irreducible, unitaryrepresentations of G on spaces Vk, of dimension dk, then

(8.14) d1/2k πkij(g)

forms an orthonormal set in L2(G). The following is the Peter-Weyl theo-rem:

Theorem 8.5. The orthonormal set (8.14) is complete.

In other words, the linear span of (8.14) is dense. If G is given as a groupof unitary N×N matrices, this result is elementary. In fact, the linear spanof (8.14) is an algebra (take tensor products of πk and π` and decomposeinto irreducibles), and is closed under complex conjugates (pass from π toπ), so if we know it separates points (which is clear if G ⊂ U(N)), theStone-Weierstrass theorem applies.

If we do not know a priori that G ⊂ U(N), we can prove the theorem byconsidering the right-regular representation of G on L2(G):

(8.15) R(g)f(x) = f(xg).

24

If we endow G with a bi-invariant Riemannian metric and consider theassociated Laplace operator ∆, which is then a bi-invariant differentialoperator, we see that the representation R leaves invariant each eigenspaceE` of ∆. Now, E` is finite-dimensional, and the restriction R` of R to E`splits into irreducibles:

(8.16) E` = E`1 ⊕ · · · ⊕ E`N , N = N(`),

say R`∣∣E`m

= R`m. Thus there is a unitary map A : E`m → Vk, for somek = k(`,m), such that R`m = AπkA−1. If {ei} is an orthonormal basis ofVk with respect to which the matrix of πk(g) is

(πkij(g)

), then ui = A−1ei

gives an orthonormal basis of E`m, and we have

(8.17) ui(xg) =∑

j

πkij(g)uj(x).

In particular, taking x = e,

(8.18) ui(g) =∑

j

cjπkij(g), cj = uj(e).

This shows that each space E`m consists of finite linear combinations ofthe functions in (8.14). Since

L2(G) =⊕

`

E` =⊕

`

⊕m

E`m,

this proves Theorem 8.5.The following corollary will be useful in the next section.

Corollary 8.6. If G1 and G2 are two compact Lie groups, then the irre-ducible, unitary representations of G = G1 ×G2 are, up to unitary equiv-alence, precisely those of the form

(8.19) π(g) = π1(g1)⊗ π2(g2),

where g = (g1, g2) ∈ G, and πj ∈ Ĝj is a general, irreducible, unitaryrepresentation of Gj .

Proof. Given irreducible, unitary representations πj of Gj , the irreducibil-ity and unitarity of (8.19) are clear. It remains to prove the completeness ofthe set of such representations. For this, it suffices to show that the matrixentries of such representations have dense linear span in L2(G1×G2). Thisfollows from the general elementary fact that tensor products of orthonor-mal bases of L2(G1) and L2(G2) form an orthonormal basis of L2(G1×G2).

9. Representations of SU(2) and related groups 25

9. Representations of SU(2) and related groups

The group SU(2) is the group of 2× 2, complex, unitary matrices of deter-minant 1, that is,

(9.1) SU(2) ={(

z1 z2−z2 z1

): |z1|2 + |z2|2 = 1, zj ∈ C

}.

As a set, SU(2) is naturally identified with the unit sphere S3 in C2. ItsLie algebra su(2) consists of 2× 2, complex, skew-adjoint matrices of tracezero. A basis of su(2) is formed by

(9.2) X1 =12

(i 00 −i

), X2 =

12

(0 1−1 0

), X3 =

12

(0 ii 0

).

Note the commutation relations

(9.3) [X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2.

The group SO(3) is the group of linear isometries of R3 with determinant1. Its Lie algebra so(3) is spanned by elements J`, ` = 1, 2, 3, which generaterotations about the x`-axis. One readily verifies that these satisfy the samecommutation relations as in (9.3). Thus SU(2) and SO(3) have isomorphicLie algebras. There is an explicit homomorphism

(9.4) p : SU(2) −→ SO(3),which exhibits SU(2) as a double cover of SO(3). One way to construct p isthe following. The linear span g of (9.2) over R is a three-dimensional, realvector space, with an inner product given by (X, Y ) = − Tr XY . It is clearthat the representation p of SU(2) by a group of linear transformations ong given by p(g) = gXg−1 preserves this inner product and gives (9.4). Notethat Ker p = {I,−I}.

If we regard Xj as left-invariant vector fields on SU(2), set

(9.5) ∆ = X21 + X22 + X

23 ,

a second-order, left-invariant differential operator. It follows easily from(9.3) that Xj and ∆ commute:

(9.6) ∆Xj = Xj∆, 1 ≤ j ≤ 3.Suppose π is an irreducible unitary representation of SU(2) on V . Then

π induces a skew-adjoint representation dπ of the Lie algebra su(2) andan algebraic representation of the universal enveloping algebra. By (9.6),dπ(∆) commutes with dπ(Xj), j = 1, . . . , 3. Thus, if π is irreducible,Proposition 7.4 implies

(9.7) dπ(∆) = −λ2I,

26

for some λ ∈ R. (Since dπ(∆) is a sum of squares of skew-adjoint operators,it must be negative.) Let

(9.8) Lj = dπ(Xj).

Now we will diagonalize L1 on V . Set

(9.9) Vµ = {v ∈ V : L1v = iµv}, V =⊕

iµ∈ spec L1Vµ.

The structure of π is defined by how L2 and L3 behave on Vµ. It is conve-nient to set

(9.10) L± = L2 ∓ iL3.We have the following key identity, as a direct consequence of (9.3):

(9.11) [L1, L±] = ±iL±.Using this, we can establish the following:

Lemma 9.1. We have

(9.12) L± : Vµ −→ Vµ±1.In particular, if iµ ∈ spec L1, then either L+ = 0 on Vµ or µ + 1 ∈ specL1, and also either L− = 0 on Vµ or µ− 1 ∈ spec L1.

Proof. Let v ∈ Vµ. By (9.11) we haveL1L±v = L±L1v ± iL±v = i(µ± 1)L±v,

which establishes the lemma. The operators L± are called ladder operators.

To continue, if π is irreducible on V , we claim that spec i−1L1 mustconsist of a sequence

(9.13) spec i−1L1 = {µ0, µ0 + 1, . . . , µ0 + k = µ1},with

(9.14) L+ : Vµ0+j → Vµ0+j+1 isomorphism, for 0 ≤ j ≤ k − 1,and

(9.15) L− : Vµ1−j → Vµ1−j−1 isomorphism, for 0 ≤ j ≤ k − 1.In fact, we can compute

(9.16) L−L+ = L22 + L23 + i[L3, L2] = −λ2 − L21 − iL1

on V , and

(9.17) L+L− = −λ2 − L21 + iL1

9. Representations of SU(2) and related groups 27

on V , so

(9.18)L−L+ = µ(µ + 1)− λ2 on Vµ,L+L− = µ(µ− 1)− λ2 on Vµ.

Note that since L2 and L3 are skew-adjoint, L+ = −L∗−, soL+L− = −L∗−L−, L−L+ = −L∗+L+.

Thus

Ker L+ = Ker L−L+, Ker L− = Ker L+L−.

These observations establish (9.13)–(9.15).Considering that dπ acts on the linear span of {v, L+v, . . . Lµ1−µ0+ v} for

any nonzero v ∈ Vµ0 , and that irreducibility implies this must be all of V ,we have

(9.19) dim Vµ = 1, µ0 ≤ µ ≤ µ1.From (9.18) we see that µ1(µ1 + 1) = λ2 = µ0(µ0 − 1). Hence,

(9.20) µ1 − µ0 = k =⇒ µ0 = −k2 , µ1 =k

2,

and we have

(9.21) dim V = k + 1, λ2 =14k(k + 2) =

14(dim V 2 − 1).

A nonzero element v ∈ V such that L+v = 0 is called a “highest-weightvector” for the representation π of SU(2) on V . It follows from the analysisabove that all highest-weight vectors for an irreducible representation onV belong to the one-dimensional space Vµ1 .

The calculations above establish that an irreducible, unitary representa-tion π of SU(2) on V is determined uniquely up to equivalence by dim V .We are ready to prove the following:

Proposition 9.2. There is precisely one equivalence class of irreducible,unitary representations of SU(2) on Ck+1, for each k = 0, 1, 2, . . . .

We will realize each such representation, which is denoted Dk/2, on thespace

(9.22) Pk = {p(z) : p homogeneous polynomial of degree k on C2},with SU(2) acting on Pk by(9.23) Dk/2(g)f(z) = f(g−1z), g ∈ SU(2), z ∈ C2.Note that, for X ∈ su(2),

(9.24) dDk/2(X)f(z) =d

dtf(e−tXz

)∣∣t=0

= −(∂1f, ∂2f) ·X(

z1z2

),

28

where ∂jf = ∂f/∂zj . A calculation gives

(9.25)

L1f(z) = − i2(z1∂1f − z2∂2f),

L2f(z) = −12(z2∂1f − z1∂2f),

L3f(z) = − i2(z2∂1f + z1∂2f).

In particular, for

(9.26) ϕkj(z) = zk−j1 z

j2 ∈ Pk, 0 ≤ j ≤ k,

we have

(9.27) L1ϕkj = i(−k

2+ j

)ϕkj ,

so

(9.28) V = Pk =⇒ span ϕkj = V−k/2+j , 0 ≤ j ≤ k.Note that

(9.29) L+f(z) = −z2∂1f(z), L−f(z) = z1∂2f(z),so

(9.30) L+ϕkj = −(k − j)ϕk,j+1, L−ϕkj = jϕk,j−1.We see that the structure of the representation Dk/2 of SU(2) on Pk

is as described in (9.12)–(9.21). The last detail is to show that Dk/2 isirreducible. If not, then Pk splits into a direct sum of several irreduciblesubspaces, each of which has a one-dimensional space of highest-weightvectors, annihilated by L+. But as seen above, within Pk, only multiplesof zk2 are annihilated by L+, so the representaiton Dk/2 of SU(2) on Pk isirreducible.

We can deduce the classification of irreducible, unitary representationsof SO(3) from the result above as follows. We have the covering homomor-phism (9.4), and Ker p = {±I}. Now each irreducible representation djof SO(3) defines an irreducible representation dj ◦ p of SU(2), which mustbe equivalent to one of the representations Dk/2 described above. On theother hand, Dk/2 factors through to yield a representation of SO(3) if andonly if Dk/2 is the identity on Ker p, that is, if and only if Dk/2(−I) = I.Clearly, this holds if and only if k is even. Thus all the irreducible, unitaryrepresentations of SO(3) are given by representations D̃j on P2j , uniquelydefined by

(9.31) D̃j(p(g)

)= Dj(g), g ∈ SU(2).

It is conventional to use Dj instead of D̃j to denote such a representationof SO(3). Note that Dj represents SO(3) on a space of dimension 2j + 1,

9. Representations of SU(2) and related groups 29

and

(9.32) dDj(∆) = −j(j + 1).Also, we can classify the irreducible representations of U(2), using the

results on SU(2). To do this, use the exact sequence

(9.33) 1 → K → S1 × SU(2) → U(2) → 1,where “1” denotes the trivial multiplicative group, and

(9.34) K = {(ω, g) ∈ S1 × SU(2) : g = ω−1I, ω2 = 1}.The irreducible representations of S1 × SU(2) are given by(9.35) πmk(ω, g) = ωmDk/2(g) on Pk,with m, k ∈ Z, k ≥ 0. Those giving a complete set of irreducible represen-tations of U(2) are those for which πmk(K) = I, that is, those for which(−1)mDk/2(−I) = I. Since Dk/2(−I) = (−1)kI, we see the condition isthat m + k be an even integer.

We now consider the representations of SO(4). First note that SO(4) iscovered by SU(2)×SU(2). To see this, equate the unit sphere S3 ⊂ R4,with its standard metric, to SU(2), with a bi-invariant metric. Then SO(4)is the connected component of the identity in the isometry group of S3.Meanwhile, SU(2)×SU(2) acts as a group of isometries, by(9.36) (g1, g2) · x = g1xg−12 , gj ∈ SU(2).Thus we have a map

(9.37) τ : SU(2)× SU(2) −→ SO(4).This is a group homomorphism. Note that (g1, g2) ∈ Ker τ implies g1 =g2 = ±I. Furthermore, a dimension count shows τ must be surjective, so(9.38) SO(4) ≈ SU(2)× SU(2)/{±(I, I)}.

As shown in §8, if G1 and G2 are compact Lie groups, and G = G1×G2,then the set of all irreducible, unitary representations of G, up to unitaryequivalence, is given by

(9.39) {π(g) = π1(g1)⊗ π2(g2) : πj ∈ Ĝj},where g = (g1, g2) ∈ G and Ĝj parameterizes the irreducible, unitaryrepresentations of Gj . In particular, the irreducible unitary representationsof SU(2)×SU(2), up to equivalence, are precisely the representations of theform

(9.40) γk`(g) = Dk/2(g1)⊗D`/2(g2), k, ` ∈ {0, 1, 2, . . . },acting on Pk ⊗ P` ≈ Ck+1 ⊗ C`+1. By (9.38), the irreducible, unitaryrepresentations of SO(4) are given by all γk` such that k + ` is even, since,for p0 = (−I,−I) ∈ SU(2)×SU(2), γk`(p0) = (−1)k+`I.

30

We next consider the problem of decomposing the tensor-product repre-sentations Dk/2 ⊗ D`/2 of SU(2) (i.e., the composition of (9.40) with thediagonal map SU(2)↪→SU(2)×SU(2)) into irreducible representations. Wemay as well assume that ` ≤ k. Note that πk` = Dk/2 ⊗D`/2 acts on

(9.41)Pk` = {f(z, w) : polynomial on C2 × C2,

homogeneous of degree k in z, ` in w},as

(9.42) πk`(g)f(z, w) = f(g−1z, g−1w).

Parallel to (9.25) and (9.29), we have, on Pk`,

(9.43)L1f =− i2(z1∂z1f − z2∂z2f + w1∂w1f − w2∂w2f),

L+f =− z2∂z1f − w2∂w1f, L−f = z1∂z2f + w1∂w2f.To decompose Pk` into irreducible subspaces, we specify Ker L+. In fact,a holomorphic function f(z, w) annihilated by L+ is of the form

(9.44) f(z, w) = g(z2, w2, w2z1 − z2w1),and the kernel of L+ in Pk` is the linear span of(9.45) ψk`µ(z, w) = z

k−µ2 w

`−µ2 (w2z1 − z2w1)µ, 0 ≤ µ ≤ `.

A calculation gives

(9.46) L1ψk`µ =i

2(k + `− 2µ)ψk`µ.

It follows that, for fixed k, `, 0 ≤ ` ≤ k, and for each µ = 0, . . . , `, ψk`µ isthe highest-weight vector of a representation equivalent to D(k+`−2µ)/2, sowe have(9.47)

Dk/2 ⊗D`/2 ≈⊕̀µ=0

D(k+`−2µ)/2 = D(k−`)/2 ⊕D(k−`)/2+1 ⊕ · · · ⊕D(k+`)/2.

This is called the Clebsch-Gordon series.Extensions of the results presented here to more general compact Lie

groups, due mainly to E. Cartan and H. Weyl, can be found in a numberof places, including [T], [Var1], and [Wal1].

References

[Dug] J. Dugundji, Topology, Allyn and Bacon, New York, 1966.[GP] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood-

Cliffs, New Jersey, 1974.

References 31

[HS] M. Hausner and J. Schwartz, Lie Groups; Lie Algebras, Gordon andBreach, London, 1968.

[Helg] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces,Academic Press, New York, 1978.

[HT] R. Howe and E. Tan, Non-Abelian Harmonic Analysis, Springer-Verlag,New York, 1992.

[Hus] D. Husemuller, Fibre Bundles, McGraw-Hill, New York, 1966.[Kn] A. Knapp, Representation Theory of Semisimple Groups, Princeton Univ.

Press, Princeton, N. J., 1986.[Stb] S. Sternberg, Lectures on Differential Geometry, Prentice Hall, Engle-

wood Cliffs, N. J., 1964.[Str] R. Strichartz, The Campbell-Hausdorff formula and DuHamel’s principle,

Preprint.[T] M. Taylor, Noncommutative Harmonic Analysis, AMS, Providence, R. I.,

1986.[Var1] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations,

Springer-Verlag, New York, 1984.[Var2] V. S. Varadarajan, An Introduction to Harmonic Analysis on Semisimple

Lie Groups, Cambridge Univ. Press, Cambridge, 1986.[Wal1] N. Wallach, Harmonic Analysis on Homogeneous Spaces, Marcel Dekker,

New York, 1973.[Wal2] N. Wallach, Real Reductive Groups, I, Academic Press, New York, 1988.[Wh] H. Whitney, Sphere spaces, Proc. NAS, USA 21(1939), 462–468.