39. Bundles with Regular Structure Groups .... . . . . . . . . . . 422

40. Rudiments of Lie Theory for Regular Lie Groups . . . . . . . . . . 426

The theory of in¬nite dimensional Lie groups can be pushed surprisingly far: Ex-

ponential mappings are unique if they exist. In general, they are neither locally

surjective nor locally injective. A stronger requirement (leading to regular Lie

groups) is to assume that smooth curves in the Lie algebra integrate to smooth

curves in the group in a smooth way (an ˜evolution operator™ exists). This is due

to [Milnor, 1984] who weakened the concept of [Omori et al., 1982]. It turns out

that regular Lie groups have strong permanence properties. In fact, up to now

all known Lie groups are regular. Connections on smooth principal bundles with

a regular Lie group as structure group have parallel transport (39.1), and for ¬‚at

connections the horizontal distribution is integrable (39.2). So some (equivariant)

partial di¬erential equations in in¬nite dimensions are very well behaved, although

in general there are counter-examples in every possible direction (some can be found

in (32.12)).

The actual development is quite involved. We start with general in¬nite dimensional

Lie groups in section (36), but for a detailed study of the evolution operator of

regular Lie groups (38.4) we need in (38.10) the Maurer-Cartan equation for right

(or left) logarithmic derivatives of mappings with values in the Lie group (38.1),

and this we can only get by looking at principal connections. Thus, in the second

section (37) bundles, connections, principal bundles, curvature, associated bundles,

and all results of principal bundle geometry which do not involve parallel transport

are developed. Finally, we then prove the strong existence results mentioned above

and treat regular Lie groups in section (38), and principal bundles with regular

structure groups in section (39). The material in this chapter is an extended version

of [Kriegl, Michor, 1997].

36. Lie Groups

36.1. De¬nition. A Lie group G is a smooth manifold and a group such that the

multiplication µ : G — G ’ G and the inversion ν : G ’ G are smooth. If not

36.1

370 Chapter VIII. In¬nite dimensional di¬erential geometry 36.3

stated otherwise, G may be in¬nite dimensional. If an implicit function theorem is

available, then smoothness of ν follows from smoothness of µ.

We shall use the following notation:

µ : G — G ’ G, multiplication, µ(x, y) = x.y.

µa : G ’ G, left translation, µa (x) = a.x.

µa : G ’ G, right translation, µa (x) = x.a.

ν : G ’ G, inversion, ν(x) = x’1 .

e ∈ G, the unit element.

36.2. Lemma. The kinematic tangent mapping T(a,b) µ : Ta G — Tb G ’ Tab G is

given by

T(a,b) µ.(Xa , Yb ) = Ta (µb ).Xa + Tb (µa ).Yb ,

and Ta ν : Ta G ’ Ta’1 G is given by

’1 ’1

Ta ν = ’Te (µa ).Ta (µa’1 ) = ’Te (µa’1 ).Ta (µa ).

Proof. Let insa : G ’ G — G, insa (x) = (a, x) be the right insertion, and let

insb : G ’ G — G, insb (x) = (x, b) be the left insertion. Then we have

T(a,b) µ.(Xa , Yb ) = T(a,b) µ.(Ta (insb ).Xa + Tb (insa ).Yb ) =

= Ta (µ —¦ insb ).Xa + Tb (µ —¦ insa ).Yb = Ta (µb ).Xa + Tb (µa ).Yb .

Now we di¬erentiate the equation µ(a, ν(a)) = e; this gives in turn

’1

0e = T(a,a’1 ) µ.(Xa , Ta ν.Xa ) = Ta (µa ).Xa + Ta’1 (µa ).Ta ν.Xa ,

’1 ’1

Ta ν.Xa = ’Te (µa )’1 .Ta (µa ).Xa = ’Te (µa’1 ).Ta (µa ).Xa .

36.3. Invariant vector ¬elds and Lie algebras. Let G be a (real) Lie group. A

(kinematic) vector ¬eld ξ on G is called left invariant, if µ— ξ = ξ for all a ∈ G, where

a

µa ξ = T (µa’1 )—¦ξ—¦µa as in (32.9). Since by (32.11) we have µ— [ξ, ·] = [µ— ξ, µ— ·], the

—

a a a

space XL (G) of all left invariant vector ¬elds on G is closed under the Lie bracket,

so it is a sub Lie algebra of X(G). Any left invariant vector ¬eld ξ is uniquely

determined by ξ(e) ∈ Te G, since ξ(a) = Te (µa ).ξ(e). Thus, the Lie algebra XL (G)

of left invariant vector ¬elds is linearly isomorphic to Te G, and the Lie bracket on

XL (G) induces a Lie algebra structure on Te G, whose bracket is again denoted by

[ , ]. This Lie algebra will be denoted as usual by g, sometimes by Lie(G).

We will also give a name to the isomorphism with the space of left invariant vector

¬elds: L : g ’ XL (G), X ’ LX , where LX (a) = Te µa .X. Thus, [X, Y ] =

[LX , LY ](e).

A vector ¬eld · on G is called right invariant, if (µa )— · = · for all a ∈ G. If ξ

is left invariant, then ν — ξ is right invariant, since ν —¦ µa = µa’1 —¦ ν implies that

(µa )— ν — ξ = (ν —¦µa )— ξ = (µa’1 —¦ν)— ξ = ν — (µa’1 )— ξ = ν — ξ. The right invariant vector

¬elds form a sub Lie algebra XR (G) of X(G), which also is linearly isomorphic to

Te G and induces a Lie algebra structure on Te G. Since ν — : XL (G) ’ XR (G) is an

isomorphism of Lie algebras by (32.11), Te ν = ’ Id : Te G ’ Te G is an isomorphism

between the two Lie algebra structures. We will denote by R : g = Te G ’ XR (G)

the isomorphism discussed, which is given by RX (a) = Te (µa ).X.

36.3

36.7 36. Lie groups 371

36.4. Remark. It would be tempting to apply also other kinds of tangent bundle

functors like D and D[1,∞) , where one gets Lie algebras of smooth sections, see

(32.8). Some results will stay true like (36.3), (36.5). In general, one gets strictly

larger Lie algebras for Lie groups, see (28.4). But the functors D and D[1,∞) do not

respect products in general, see (28.16), so e.g. (36.2) is wrong for these functors.

36.5. Lemma. If LX is a left invariant vector ¬eld and RY is a right invariant

one, then [LX , RY ] = 0. So if the ¬‚ows of LX and RY exist, they commute.

Proof. We consider the vector ¬eld 0 — LX ∈ X(G — G), given by (0 — LX )(a, b) =

(0a , LX (b)). Then T(a,b) µ.(0a , LX (b)) = Ta µb .0a + Tb µa .LX (b) = LX (ab), so 0 — LX

is µ-related to LX . Likewise, RY — 0 is µ-related to RY . But then 0 = [0 —

LX , RY — 0] is µ-related to [LX , RY ] by (32.10). Since µ is surjective, [LX , RY ] = 0

follows.

36.6. Lemma. Let • : G ’ H be a smooth homomorphism of Lie groups. Then

• := Te • : g = Te G ’ h = Te H is a Lie algebra homomorphism.

Proof. For X ∈ g and x ∈ G we have

Tx •.LX (x) = Tx •.Te µx .X = Te (• —¦ µx ).X

= Te (µ•(x) —¦ •).X = Te (µ•(x) ).Te •.X = L• (X) (•(x)).

So LX is •-related to L• (X) . By (32.10), the ¬eld [LX , LY ] = L[X,Y ] is •-related

to [L• (X) , L• (Y ) ] = L[• (X),• (Y )] . So we have T • —¦ L[X,Y ] = L[• (X),• (Y )] —¦ •. If

we evaluate this at e the result follows.

36.7. One parameter subgroups. Let G be a Lie group with Lie algebra g. A

one parameter subgroup of G is a Lie group homomorphism ± : (R, +) ’ G, i.e. a

smooth curve ± in G with ±(s + t) = ±(s).±(t), and hence ±(0) = e.

Note that a smooth mapping β : (’µ, µ) ’ G satisfying β(t)β(s) = β(t + s) for

|t|, |s|, |t + s| < µ is the restriction of a one parameter subgroup. Namely, choose

0 < t0 < µ/2. Any t ∈ R can be uniquely written as t = N.t0 + t for 0 ¤ t < t0

and N ∈ Z. Put ±(t) = β(t0 )N β(t ). The required properties are easy to check.

Lemma. Let ± : R ’ G be a smooth curve with ±(0) = e. Let X ∈ g. Then the

following assertions are equivalent.

‚

(1) ± is a one parameter subgroup with X = ‚t 0 ±(t).

(2) ±(t) is an integral curve of the left invariant vector ¬eld LX and also an

integral curve of the right invariant vector ¬eld RX .

(3) FlLX (t, x) := x.±(t) (or FlLX = µ±(t) ) is the (unique by (32.16)) global ¬‚ow

t

of LX in the sense of (32.13).

(4) FlRX (t, x) := ±(t).x (or FlRX = µ±(t) ) is the (unique) global ¬‚ow of RX .

t

Moreover, each of these properties determines ± uniquely.

Proof. (1) ’ (3) We have

d d d

|0 x.±(t + s) = ds |0 x.±(t).±(s)

dt x.±(t) = ds

d d

|0 µx.±(t) ±(s) = Te (µx.±(t) ). ds |0 ±(s)

= = LX (x.±(t)).

ds

36.7

372 Chapter VIII. In¬nite dimensional di¬erential geometry 36.8

Since it is obviously a ¬‚ow, we have (3).

ν—ξ

= ν ’1 —¦ Flξ —¦ν by (32.16). Therefore, we have by (36.3)

(3) ” (4) We have Flt t

—

(FlRX (x’1 ))’1 = (ν —¦ FlRX —¦ν)(x) = Flν RX

(x)

t t t

= Fl’LX (x) = FlLX (x) = x.±(’t).

’t

t

So FlRX (x’1 ) = ±(t).x’1 , and FlRX (y) = ±(t).y.

t t

(3) and (4) together clearly imply (2).

(2) ’ (1) This is a consequence of the following result.

Claim. Consider two smooth curves ±, β : R ’ G with ±(0) = e = β(0) which

satisfy the two di¬erential equations

d

dt ±(t) = LX (±(t))

d

dt β(t) = RX (β(t)).

Then ± = β, and it is a 1-parameter subgroup.

We have ± = β since

= T µβ(’t) .LX (±(t)) ’ T µ±(t) .RX (β(’t))

d

dt (±(t)β(’t))

= T µβ(’t) .T µ±(t) .X ’ T µ±(t) .T µβ(’t) .X = 0.

Next we calculate for ¬xed s

’ s)β(s)) = T µβ(s) .RX (β(t ’ s)) = RX (β(t ’ s)β(s)).

d

dt (β(t

Hence, by the ¬rst part of the proof β(t ’ s)β(s) = ±(t) = β(t).

The statement about uniqueness follows from (32.16), or from the claim.

36.8. De¬nition. Let G be a Lie group with Lie algebra g. We say that G admits

an exponential mapping if there exists a smooth mapping exp : g ’ G such that

t ’ exp(tX) is the (unique by (36.7)) 1-parameter subgroup with tangent vector

X at 0. Then we have by (36.7)

(1) FlLX (t, x) = x. exp(tX).

(2) FlRX (t, x) = exp(tX).x.

(3) exp(0) = e and T0 exp = Id : T0 g = g ’ Te G = g since T0 exp .X =

|0 exp(0 + t.X) = dt |0 FlLX (t, e) = X.

d d

dt

(4) Let • : G ’ H be a smooth homomorphism between Lie groups admitting

exponential mappings. Then the diagram

wh

•

g

u u

expG expH

wH

•

G

commutes, since t ’ •(expG (tX)) is a one parameter subgroup of H, and

d G G H

dt |0 •(exp tX) = • (X), so •(exp tX) = exp (t• (X)).

36.8

36.10 36. Lie groups 373

36.9. Remarks. [Omori et al., 1982, 1983, etc.] gave conditions under which a

smooth Lie group modeled on Fr´chet spaces admits an exponential mapping. We

e

shall elaborate on this notion in (38.4) below. They called such groups ˜regular

Fr´chet Lie groups™. We do not know any smooth Fr´chet Lie group which does not

e e

admit an exponential mapping.

If G admits an exponential mapping, it follows from (36.8.3) that exp is a di¬eo-

morphism from a neighborhood of 0 in g onto a neighborhood of e in G, if a suitable

inverse function theorem is applicable. This is true, for example, for smooth Banach

Lie groups, also for gauge groups, see (42.21) but it is wrong for di¬eomorphism

groups, see (43.3).

If E is a Banach space, then in the Banach Lie group GL(E) of all bounded linear

automorphisms of E the exponential mapping is given by the series exp(X) =

∞1 i

i=0 i! X .

If G is connected with exponential mapping and U ‚ g is open with 0 ∈ U , then

one may ask whether the group generated by exp(U ) equals G. Note that this is a

normal subgroup. So if G is simple, the answer is yes. This is true for connected

components of di¬eomorphism groups and many of their important subgroups, see

[Epstein, 1970], [Thurston, 1974], [Mather, 1974, 1975, 1984, 1985], [Banyaga, 1978].

Results on weakened versions of the Baker-Campbell-Hausdor¬ formula can be

found in [Wojty´ski, 1994].

n

36.10. The adjoint representation. Let G be a Lie group with Lie algebra

g. For a ∈ G we de¬ne conja : G ’ G by conja (x) = axa’1 . It is called the

conjugation or the inner automorphism by a ∈ G. This de¬nes a smooth action of

G on itself by automorphisms.

The adjoint representation Ad : G ’ GL(g) ‚ L(g, g) is given by Ad(a) =

(conja ) = Te (conja ) : g ’ g for a ∈ G. By (36.6), Ad(a) is a Lie algebra ho-

momorphism, moreover

’1 ’1

Ad(a) = Te (conja ) = Ta (µa ).Te (µa ) = Ta’1 (µa ).Te (µa ).

Finally, we de¬ne the (lower case) adjoint representation of the Lie algebra g, ad :

g ’ gl(g) := L(g, g) by ad := Ad = Te Ad.

We shall also use the right Maurer-Cartan form κr ∈ „¦1 (G, g), given by κr = g

’1

Tg (µg ) : Tg G ’ g; similarly the left Maurer-Cartan form κl ∈ „¦1 (G, g) is given

by κl = Tg (µg’1 ) : Tg G ’ g.

g

Lemma.

(1) LX (a) = RAd(a)X (a) for X ∈ g and a ∈ G.

(2) ad(X)Y = [X, Y ] for X, Y ∈ g.

(3) dAd = (ad —¦ κr ).Ad = Ad.(ad —¦ κl ) : T G ’ L(g, g).

’1

Proof. (1) LX (a) = Te (µa ).X = Te (µa ).Te (µa —¦ µa ).X = RAd(a)X (a).

36.10

374 Chapter VIII. In¬nite dimensional di¬erential geometry 36.10

Proof of (2). We need some preparation. Let V be a convenient vector space. For

f ∈ C ∞ (G, V ) we de¬ne the left trivialized derivative Dl f ∈ C ∞ (G, L(g, V )) by

(4) Dl f (x).X := df (x).Te µx .X = (LX f )(x).

For f ∈ C ∞ (G, R) and g ∈ C ∞ (G, V ) we have

(5) Dl (f.g)(x).X = d(f.g)(Te µx .X)

= df (Te µx .X).g(x) + f (x).dg(Te µx .X)

= (f.Dl g + Dl f — g)(x).X.

From the formula

Dl Dl f (x)(X)(Y ) = Dl (Dl f ( ).Y )(x).X

= Dl (LY f )(x).X = LX LY f (x)

follows

Dl Dl f (x)(X)(Y ) ’ Dl Dl f (x)(Y )(X) = L[X,Y ] f (x) = Dl f (x).[X, Y ].

(6)

We consider now the linear isomorphism L : C ∞ (G, g) ’ X(G) given by Lf (x) =

Te µx .f (x) = Lf (x) (x) for f ∈ C ∞ (G, g). If h ∈ C ∞ (G, V ) we get (Lf h)(x) =

Dl h(x).f (x). For f, g ∈ C ∞ (G, g) and h ∈ C ∞ (G, R) we get in turn, using (6) and

(5), generalized to the bilinear pairing L(g, R) — g ’ R,

(Lf Lg h)(x) = Dl (Dl h( ).g( ))(x).f (x)

= Dl Dl h(x)(f (x))(g(x)) + Dl h(x).Dl g(x).f (x)

2

([Lf , Lg ]h)(x) = Dl h(x).(f (x), g(x)) + Dl h(x).Dl g(x).f (x)’

2

’ Dl h(x).(g(x), f (x)) ’ Dl h(x).Dl f (x).g(x)

= Dl h(x). [f (x), g(x)]g + Dl g(x).f (x) ’ Dl f (x).g(x)

[Lf , Lg ] = L [f, g]g + Dl g.f ’ Dl f.g .

(7)

Now we are able to prove the second assertion of the lemma. For X, Y ∈ g we will

apply (7) to f (x) = X and g(x) = Ad(x’1 ).Y . We have Lg = RY by (1), and

[Lf , Lg ] = [LX , RY ] = 0 by (36.5). So

0 = [LX , RY ](x) = [Lf , Lg ](x)

= L([X, (Ad —¦ ν)Y ]g + Dl ((Ad —¦ ν)( ).X).Y ’ 0)(x)

[X, Y ] = [X, Ad(e)Y ] = ’Dl ((Ad —¦ ν)( ).X)(e).Y

= d(Ad( ).X)(e).Y = ad(X)Y.

Proof of (3). Let X, Y ∈ g and g ∈ G, and let c : R ’ G be a smooth curve with

c(0) = e and c (0) = X. Then we have

‚ ‚

‚t |0 Ad(c(t).g).Y ‚t |0 Ad(c(t)).Ad(g).Y

(dAd(RX (g))).Y = =

(ad —¦ κr )(RX (g)).Ad(g).Y,

= ad(X)Ad(g)Y =

and similarly for the second formula.

36.10

37.1 37. Bundles and connections 375

36.11. Let : G — M ’ M be a smooth left action of a Lie group G, so ∨ : G ’

Di¬(M ) is a group homomorphism. Then we have partial mappings a : M ’ M

and x : G ’ M , given by a (x) = x (a) = (a, x) = a.x.

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by ζX (x) =

Te ( x ).X = T(e,x) .(X, 0x ).

Lemma. In this situation, the following assertions hold:

ζ : g ’ X(M ) is a linear mapping.

(1)

(2) Tx ( a ).ζX (x) = ζAd(a)X (a.x).

RX — 0M ∈ X(G — M ) is -related to ζX ∈ X(M ).

(3)

[ζX , ζY ] = ’ζ[X,Y ] .

(4)

Proof. (1) is clear.

(b) = abx = aba’1 ax =

x ax

(2) We have conja (b), so

a

Tx ( a ).ζX (x) = Tx ( a ).Te ( x ).X = Te ( x

—¦ ).X

a

ax

= Te ( ).Ad(a).X = ζAd(a)X (ax).

(3) We have —¦ (Id — a ) = —¦ (µa — Id) : G — M ’ M , so

ζX ( (a, x)) = T(e,ax) .(X, 0ax ) = T .(Id —T ( a )).(X, 0x )

= T .(T (µa ) — Id).(X, 0x ) = T .(RX — 0M )(a, x).

(4) [RX — 0M , RY — 0M ] = [RX , RY ] — 0M = ’R[X,Y ] — 0M is -related to [ζX , ζY ]

by (3) and by (32.10). On the other hand, ’R[X,Y ] — 0M is -related to ’ζ[X,Y ] by

(3) again. Since is surjective we get [ζX , ζY ] = ’ζ[X,Y ] .

36.12. Let r : M — G ’ M be a right action, so r∨ : G ’ Di¬(M ) is a group anti

homomorphism. We will use the following notation: ra : M ’ M and rx : G ’ M ,

given by rx (a) = ra (x) = r(x, a) = x.a.

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by ζX (x) =

Te (rx ).X = T(x,e) r.(0x , X).

Lemma. In this situation the following assertions hold:

ζ : g ’ X(M ) is a linear mapping.

(1)

Tx (ra ).ζX (x) = ζAd(a’1 )X (x.a).

(2)

0M — LX ∈ X(M — G) is r-related to ζX ∈ X(M ).

(3)

(4) [ζX , ζY ] = ζ[X,Y ] .

37. Bundles and Connections

37.1. De¬nition. A (¬ber) bundle (p : E ’ M, S) = (E, p, M, S) consists of

smooth manifolds E, M , S, and a smooth mapping p : E ’ M . Furthermore, each

37.1

376 Chapter VIII. In¬nite dimensional di¬erential geometry 37.2

x ∈ M has an open neighborhood U such that E | U := p’1 (U ) is di¬eomorphic to

U — S via a ¬ber respecting di¬eomorphism:

w

‘ ψ

E|U U —S

‘p“ &

‘ &pr

)

& 1

U.

E is called total space, M is called base space or basis, p is a ¬nal surjective smooth

mapping, called projection, and S is called standard ¬ber. (U, ψ) as above is called

a ¬ber chart.

A collection of ¬ber charts (U± , ψ± ) such that (U± ) is an open cover of M , is called

a ¬ber bundle atlas. If we ¬x such an atlas, then ψ± —¦ ψβ ’1 (x, s) = (x, ψ±β (x, s)),

where ψ±β : (U± ©Uβ )—S ’ S is smooth, and where ψ±β (x, ) is a di¬eomorphism of

S for each x ∈ U±β := U± © Uβ . These mappings ψ±β are called transition functions

of the bundle. They satisfy the cocycle condition: ψ±β (x) —¦ ψβγ (x) = ψ±γ (x) for

x ∈ U±βγ and ψ±± (x) = IdS for x ∈ U± . Therefore, the collection (ψ±β ) is called a

cocycle of transition functions.

Given an open cover (U± ) of a manifold M and a cocycle of transition functions

(ψ±β ) we may construct a ¬ber bundle (p : E ’ M, S), as in ¬nite dimensions.