38.3. Let G be a Lie group with Lie algebra g. For a closed interval I ‚ R and for

X ∈ C ∞ (I, g) we consider the ordinary di¬erential equation

g(t0 ) = e

(1) ‚ ‚

= Te (µg(t) )X(t) = RX(t) (g(t)), or κr ( ‚t g(t)) = X(t),

‚t g(t)

for local smooth curves g in G, where t0 ∈ I.

38.3

38.3 38. Regular Lie groups 409

Lemma.

(2) Local solutions g of the di¬erential equation (1) are uniquely determined.

(3) If for ¬xed X the di¬erential equation (1) has a local solution near each

t0 ∈ I, then it has also a global solution g ∈ C ∞ (I, G).

(4) If for all X ∈ C ∞ (I, g) the di¬erential equation (1) has a local solution near

one ¬xed t0 ∈ I, then it has also a global solution g ∈ C ∞ (I, G) for each

X. Moreover, if the local solutions near t0 depend smoothly on the vector

¬elds X (see the proof for the exact formulation), then so does the global

solution.

(5) The curve t ’ g(t)’1 is the unique local smooth curve h in G which satis¬es

h(t0 ) = e

‚ ‚

or κl ( ‚t h(t)) = ’X(t).

‚t h(t) = Te (µh(t) )(’X(t)) = L’X(t) (h(t)),

Proof. (2) Suppose that g(t) and g1 (t) both satisfy (1). Then on the intersection

of their intervals of de¬nition we have

’1

’1

= ’T (µg1 (t) ).T (µg(t)’1 ).T (µg(t) ).T (µg(t) ).X(t)

‚

‚t (g(t) g1 (t))

+ T (µg(t)’1 ).T (µg1 (t) ).X(t) = 0,

so that g = g1 .

Proof of (3) It su¬ces to prove the claim for every compact subinterval of I, so let

I be compact. If g is a local solution of (1) then t ’ g(t).x is a local solution of

the same di¬erential equation with initial value x. By assumption, for each s ∈ I

there is a unique solution gs of the di¬erential equation with gs (s) = e; so there

exists δs > 0 such that gs (s + t) is de¬ned for |t| < δs . Since I is compact there

exist s0 < s1 < · · · < sk such that I = [s0 , sk ] and si+1 ’ si < δsi . Then we put

for s0 ¤ t ¤ s1

gs0 (t)

±

for s1 ¤ t ¤ s2

gs1 (t).gs0 (s1 )

g(t) := . . .

for si ¤ t ¤ si+1

gs (t).gs (si ) . . . gs (s1 )

i i’1 0

...

which is smooth by the ¬rst case and solves the problem.

Proof of (4) Given X : I ’ g we ¬rst extend X to a smooth curve R ’ g, using

(24.10). For t1 ∈ I, by assumption, there exists a local solution g near t0 of the

translated vector ¬eld t ’ X(t1 ’ t0 + t), thus t ’ g(t0 ’ t1 + t) is a solution near

t1 of X. So by (3) the di¬erential equation has a global solution for X on I.

Now we assume that the local solutions near t0 depend smoothly on the vector

¬eld. So for any smooth curve X : R ’ C ∞ (I, g) we have:

For every compact interval K ‚ R there is a neighborhood UX,K of t0

in I and a smooth mapping g : K — UX,K ’ G with

g(k, t0 ) = e

‚

= Te (µg(k,t) ).X(k)(t) for all k ∈ K, t ∈ UX,K .

‚t g(k, t)

38.3

410 Chapter VIII. In¬nite dimensional di¬erential geometry 38.4

Given a smooth curve X : R ’ C ∞ (I, g) we extend (or lift) it smoothly to X : R ’

C ∞ (R, g) by (24.10). Then the smooth parameter k from the compact interval K

passes smoothly through the proofs given above to give a smooth global solution

g : K — I ’ G. So the ˜solving operation™ respects smooth curves and thus is

˜smooth™.

Proof of (5) One can show in a similar way that h is the unique solution of (5) by

di¬erentiating h1 (t).h(t)’1 . Moreover, the curve t ’ g(t)’1 = h(t) satis¬es (5),

since

’1

’1

= ’T (µg(t)’1 ).T (µg(t) ).T (µg(t) ).X(t) = T (µg(t)’1 ).(’X(t)).

‚

‚t (g(t) )

38.4. De¬nition. Regular Lie groups. If for each X ∈ C ∞ (R, g) there exists

g ∈ C ∞ (R, G) satisfying

±

g(0) = e

‚ g(t)

‚t g(t) = Te (µ )X(t) = RX(t) (g(t)),

(1)

‚

or κr ( ‚t g(t)) = δ r g(‚t ) = X(t)

then we write

evolr (X) = evolG (X) := g(1),

G

Evolr (X)(t) := evolG (s ’ tX(ts)) = g(t),

G

and call them the right evolution of the curve X in G. By lemma (38.3), the solution

of the di¬erential equation (1) is unique, and for global existence it is su¬cient that

it has a local solution. Then

Evolr : C ∞ (R, g) ’ {g ∈ C ∞ (R, G) : g(0) = e}

G

is bijective with inverse δ r . The Lie group G is called a regular Lie group if evolr :

C ∞ (R, g) ’ G exists and is smooth. We also write

evoll (X) = evolG (X) := h(1),

G

Evoll (X)(t) := evoll (s ’ tX(ts)) = h(t),

G G

if h is the (unique) solution of

±

h(0) = e

‚

‚t h(t) = Te (µh(t) )(X(t)) = LX(t) (h(t)),

(2)

‚

or κl ( ‚t h(t)) = δ l h(‚t ) = X(t).

Clearly, evoll : C ∞ (R, g) ’ G exists and is also smooth if evolr does, since we have

evoll (X) = evolr (’X)’1 by lemma (38.3).

38.4

38.5 38. Regular Lie groups 411

If f ∈

Let us collect some easily seen properties of the evolution mappings.

C ∞ (R, R) then we have

Evolr (X)(f (t)) = Evolr (f .(X —¦ f ))(t).Evolr (X)(f (0)),

Evoll (X)(f (t)) = Evoll (X)(f (0)).Evoll (f .(X —¦ f ))(t).

If • : G ’ H is a smooth homomorphism between regular Lie groups then the

diagrams

wC wC

•— •—

C ∞ (R, g) ∞

C ∞ (R, g) ∞

(R, h) (R, h)

u u u u

(3) evolG evolH EvolG EvolH

wH wC

• •—

C ∞ (R, G) ∞

G (R, H)

‚

commutes, since ‚t •(g(t)) = T •.T (µg(t) ).X(t) = T (µ•(g(t)) ).• .X(t). Note that

each regular Lie group admits an exponential mapping, namely the restriction of

evolr to the constant curves R ’ g. A Lie group is regular if and only if its universal

covering group is regular.

This notion of regularity is a weakening of the same notion of [Omori et al., 1982,

1983, etc.], who considered a sort of product integration property on a smooth Lie

group modeled on Fr´chet spaces. Our notion here is due to [Milnor, 1984]. Up to

e

now the following statement holds:

All known Lie groups are regular.

Any Banach Lie group is regular since we may consider the time dependent right

invariant vector ¬eld RX(t) on G and its integral curve g(t) starting at e, which

exists and depends smoothly on (a further parameter in) X. In particular, ¬nite

dimensional Lie groups are regular. For di¬eomorphism groups the evolution oper-

ator is just integration of time dependent vector ¬elds with compact support, see

section (43) below.

38.5. Some abelian regular Lie groups. For (E, +), where E is a convenient

1

vector space, we have evol(X) = 0 X(t)dt, so convenient vector spaces are regular

abelian Lie groups. We shall need ˜discrete™ subgroups, which is not an obvious

notion since (E, +) is not a topological group: the addition is continuous only as a

mapping c∞ (E —E) ’ c∞ E and not for the cartesian product of the c∞ -topologies.

Next let Z be a ˜discrete™ subgroup of a convenient vector space E in the sense that

there exists a c∞ -open neighborhood U of zero in E such that U © (z + U ) = … for

all 0 = z ∈ Z (equivalently (U ’ U ) © (Z \ 0) = …). For that it su¬ces e.g., that Z

is discrete in the bornological topology on E. Then E/Z is an abelian but possibly

non Hausdor¬ Lie group. It does not su¬ce to take Z discrete in the c∞ -topology:

Take as Z the subgroup generated by A in RN—c0 in the proof of (4.26).(iv).

Let us assume that Z ful¬lls the stronger condition: there exists a symmetric c∞ -

open neighborhood W of 0 such that (W + W ) © (z + W + W ) = … for all 0 = z ∈ Z

(equivalently (W + W + W + W ) © (Z \ 0) = …). Then E/Z is Hausdor¬ and thus an

38.5

412 Chapter VIII. In¬nite dimensional di¬erential geometry 38.6

abelian regular Lie group, since its universal cover E is regular. Namely, for x ∈ Z,

/

we have to ¬nd neighborhoods U and V of 0 such that (Z + U ) © (x + Z + V ) = ….

There are two cases. If x ∈ Z + W + W then there is a unique z ∈ Z with

x ∈ z + W + W , and we may choose U, V ‚ W such that (z + U ) © (x + V ) = …;

then (Z + U ) © (x + Z + V ) = …. In the other case, if x ∈ Z + W + W , then we

/

have (Z + W ) © (x + Z + W ) = ….

Notice that the two conditions above and their consequences also hold for gen-

eral (non-abelian) (regular) Lie groups instead of E and their ˜discrete™ normal

subgroups (which turn out to be central if G is connected).

It would be nice if any regular abelian Lie group would be of the form E/Z described

above. A ¬rst result in this direction is that for an abelian Lie group G with Lie

algebra g which admits a smooth exponential mapping exp : g ’ G one can easily

‚

check by using (38.2) that ‚t (exp(’tX). exp(tX + Y )) = 0, so that exp is a smooth

homomorphism of Lie groups.

Let us consider some examples. More examples can be found in [Banaszczyk, 1984,

1986, 1991]. For the ¬rst one we consider a discrete subgroup Z ‚ RN . There

exists a neighborhood of 0, without loss of generality of the form U — RN\n for

U ‚ Rn , with U © (Z \ 0) = …. Then we consider the following diagram of Lie group

homomorphisms

wR wR

N\n N\n

0

u u u

wR w R /Z (S 1 )k — RN\(n’k)

N N

Z

∼ π

u u u u

=

wR w R /π(Z)

n n

(S 1 )k — Rn’k

π(Z)

which has exact lines and columns. For the right hand column we use a diagram

chase to see this. Choose a global linear section of π inverting π|Z. This factors to

a global homomorphism of the right hand side column.

As next example we consider Z(N) ‚ R(N) . Then, obviously, R(N) /Z(N) = (S 1 )(N) ,

which is a real analytic manifold modeled on R(N) , similar to the ones which are

treated in section (47). The reader may convince himself that any Lie group covered

by R(N) is isomorphic to (S 1 )(A) — R(N\A) for A ⊆ N.

As another example, one may check easily that ∞ /(ZN © ∞ ) = (S 1 )N , equipped

with the ˜uniform box topology™; compare with the remark at the end of (27.3).

38.6. Extensions of Lie groups. Let H and K be Lie groups. A Lie group G

is called a smooth extension of H with kernel K if we have a short exact sequence

of groups

p

i

{e} ’ K ’ G ’ H ’ {e},

’’

(1)

38.6

38.6 38. Regular Lie groups 413

such that i and p are smooth, and one of the following two equivalent conditions is

satis¬ed:

(2) p admits a local smooth section s near e (equivalently near any point), and

i is initial (27.11).

(3) i admits a local smooth retraction r near e (equivalently near any point),

and p is ¬nal (27.15).

Of course, by s(p(x))i(r(x)) = x the two conditions are equivalent, and then G is

locally di¬eomorphic to K — H via (r, p) with local inverse (i —¦ pr1 ).(s —¦ pr2 ).

Not every smooth exact sequence of Lie groups admits local sections as required in

(2). Let, for example, K be a closed linear subspace in a convenient vector space

G which is not a direct summand, and let H be G/K. Then the tangent mapping

at 0 of a local smooth splitting would make K a direct summand.

p

i

Theorem. Let {e} ’ K ’ G ’ H ’ {e} be a smooth extension of Lie groups.

’ ’

Then G is regular if and only if both K and H are regular.

Proof. Clearly, the induced sequence of Lie algebras also is exact,

p

i

0 ’ k ’ g ’ h ’ 0,

’’

with a bounded linear section Te s of p . Thus, g is isomorphic to k — h as convenient

vector space.

Let us suppose that K and H are regular. Given X ∈ C ∞ (R, g), we consider

‚

Y (t) := p (X(t)) ∈ h with evolution curve h satisfying ‚t h(t) = T (µh(t) ).Y (t) and

h(0) = e. By lemma (38.3) it su¬ces to ¬nd smooth local solutions g near 0 of

‚ g(t)

‚t g(t) = T (µ ).X(t) with g(0) = e, depending smoothly on X. We look for

solutions of the form g(t) = s(h(t)).i(k(t)), where k is a local evolution curve in K

‚

of a suitable curve t ’ Z(t) in k, i.e., ‚t k(t) = T (µk(t) ).Z(t), and k(0) = e. For

this ansatz we have

s(h(t)).i(k(t)) = T (µs(h(t)) ).T i. ‚t k(t) + T (µi(k(t)) ).T s. ‚t h(t)

‚ ‚ ‚ ‚

‚t g(t) = ‚t

= T (µs(h(t)) ).T i.T (µk(t) ).Z(t) + T (µi(k(t)) ).T s.T (µh(t) ).Y (t),

and we want this to be

T (µg(t) ).X(t) = T (µs(h(t)).i(k(t)) ).X(t) = T (µi(k(t)) ).T (µs(h(t)) ).X(t).

Using i —¦ µk = µi(k) —¦ i, one quickly sees that

’1

i .Z(t) := Ad s(h(t))’1 . X(t) ’ T (µs(h(t)) ).T s.T (µh(t) ).Y (t) ∈ ker p

solves the problem, so G is regular.

Let now G be regular. If Y ∈ C ∞ (R, h), then p —¦ Evolr (s —¦ Y ) = EvolH (Y ), by

G

∞

the diagram in (38.4.3). If U ∈ C (R, k) then p —¦ EvolG (i —¦ U ) = EvolH (0) = e, so

that EvolG (i —¦ U )(t) ∈ i(K) for all t and thus equals i(EvolK (U )(t)).

38.6

414 Chapter VIII. In¬nite dimensional di¬erential geometry 38.8

38.7. Subgroups of regular Lie groups. Let G and K be Lie groups, let G

be regular, and let i : K ’ G be a smooth homomorphism which is initial (27.11)

with Te i = i : k ’ g injective. We suspect that K is then regular, but we are only

able to prove this under the following assumption.

There are an open neighborhood U ‚ G of e and a smooth mapping

p : U ’ E into a convenient vector space E such that p’1 (0) = K © U

and p is constant on left cosets Kg © U .

Proof. For Z ∈ C ∞ (R, k) we consider g(t) = EvolG (i —¦ Z)(t) ∈ G. Then we have

‚ g(t)

‚t (p(g(t))) = T p.T (µ ).i (Z(t)) = 0 by the assumption, so p(g(t)) is constant

p(e) = 0, thus g(t) = i(h(t)) for a smooth curve h in H, since i is initial. Then

h = EvolH (Y ) since Te i is injective, and h depends smoothly on Z since i is

initial.

38.8. Abelian and central extensions. From theorem (38.6), it is clear that

any smooth extension G of a regular Lie group H with an abelian regular Lie group

(K, +) is regular. We shall describe EvolG in terms of EvolG , EvolK , and in terms

of the action of H on K and the cocycle c : H — H ’ K if the latter exists.

Let us ¬rst recall these notions. If we have a smooth extension with abelian normal

subgroup K,

p

i

{e} ’ K ’ G ’ H ’ {e},

’’

then a unique smooth action ± : H — K ’ K by automorphisms is given by

i(±h (k)) = s(h)i(k)s(h)’1 , where s is any smooth local section of p de¬ned near h.

If moreover p admits a global smooth section s : H ’ G, which we assume without

loss of generality to satisfy s(e) = e, then we consider the smooth mapping c :

H — H ’ K given by ic(h1 , h2 ) := s(h1 ).s(h2 ).s(h1 .h2 )’1 . Via the di¬eomorphism

K — H ’ G given by (k, h) ’ i(k).s(h) the identity corresponds to (0, e), the

multiplication and the inverse in G look as follows:

(1) (k1 , h1 ).(k2 , h2 ) = (k1 + ±h1 k2 + c(h1 , h2 ), h1 h2 ),

(k, h)’1 = (’±h’1 (k) ’ c(h’1 , h), h’1 ).

Associativity and (0, e)2 = (0, e) correspond to the fact that c satis¬es the following

cocycle condition and normalization

±h1 (c(h2 , h3 )) ’ c(h1 h2 , h3 ) + c(h1 , h2 h3 ) ’ c(h1 , h2 ) = 0,

(2)

c(e, e) = 0.

These imply that c(e, h) = 0 = c(h, e) and ±h (c(h’1 , h)) = c(h, h’1 ). For a central

extension the action is trivial, ±h = IdK for all h ∈ H.

If conversely H acts smoothly by automorphisms on an abelian Lie group K and

if c : H — H ’ K satis¬es (2), then (1) describes a smooth Lie group structure on

K — H, which is a smooth extension of H over K with a global smooth section.

38.8

38.10 38. Regular Lie groups 415

For later purposes, let us compute

(0, h1 ).(0, h2 )’1 = (’±h1 (c(h’1 , h2 )) + c(h1 , h’1 ), h1 h’1 ),

2 2 2

’1 ’1

’1

).Yh1 + T (c( , h’1 )).Yh1 , T (µh2 ).Yh1 ).

T(0,h1 ) (µ(0,h2 ) ).(0, Yh1 ) = (’T (±c(h2 ,h2 )

2

Let us now assume that K and H are moreover regular Lie groups. We consider

a curve t ’ X(t) = (U (t), Y (t)) in the Lie algebra g which as convenient vector

space equals k — h. From the proof of (38.6) we get that

g(t) := EvolG (U, Y )(t) = (0, h(t)).(k(t), e) = (±h(t) (k(t)), h(t)), where

h(t) := EvolH (Y )(t) ∈ H,

’1