Differential forms with values in groups

BULL. AUSTRAL. MATH. SOC. VOL. 25 ( 1 9 8 2 ) ,

I8FI5,

58AI5

357-386.

DIFFERENTIAL FORMS WITH VALUES IN GROUPS ANDERS KOCK

In the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold.

Natural geometric examples of such forms and the role

of their exterior differentiation is given.

The main result is a

comparison with the classical theory of Lie algebra valued forms.

In synthetic differential geometry, one encounters formal manifolds, and in these it makes sense to talk about two points being neighbours [JO]. In terms of this neighbour notion, it makes sense to talk about differential forms with values in a group

G . This is equivalent to a

classically considered notion of Lie algebra valued differential form (namely with values in the Lie algebra of

G ) , and the comparison between

these two notions is the main result presented here. definition of coboundary of

0-

and

However, the

1-forms with values in

G

is, both

from the analytic and geometric viewpoint, more natural than the classical Lie algebra valued notions. group

G

Thus, the Maurer-Cartan form

appears as the coboundary of the identity map of

zero-form).

In particular, it is closed,

Q G

for a Lie (which is a

dQ. = 0 , and this can be

reinterpreted as the Maurer-Cartan formula. Also, the two well known lemmas (cf. , for example, [6] or [4]) Received 12 November 1981. This research was partially supported by the Australian Research Grants Committee. A preliminary report on some of the results is printed in Cahiers Topologie Geom. Diffeventielle 22 (1981), 11*1-11(8. 357

358

Anders Kock

concerning maps from connected and simply connected manifolds into a Lie group-become just the statements that where H

H

H (M, G) = G

denotes "deRham cohomology with values in

(which can be expressed:

"closed

(z, x) • u>{y, z) • m(x, y) ,

(1.2) where

• denotes the multiplication of the value group.

(We choose the

ordering in (1.2) rather than the more forward-looking w(x, y) • u(y, z) • u(z, x) , because we want the value groups to be transformation groups which act from the left.)

The element (1.2) is denoted

du>(x, y, s) .

In fact,

du> is a

2-form with values in the group in question, and its vanishing (that is, having

e

as its only value) means precisely that integrating

any triangle (l.l) gives

w

around

e .

If we can perform the passage from infinitesimal to finite (nullhomotopic) closed curves, alluded to above, we have

(1.3)

w depends only on the end points of k ^k

(in simply connected domains

M ) and by the standard procedure, this in

turn leads to the construction of a function

f(y) • f(x)~

(l.M namely letting

f(z) =

/ : M -*• G

such that

= w(x, y) for x ~ y

w , where

a

is a point chosen once and for

>k(a,z) all, and

k{a, s)

is any curve starting in

(l.l») can be expressed

a

dw = 0 =» a) = df . As an example of how one may arrive at a

group

F

z . How

df = w , so that (1.3) says

(*)

on

and ending in

M

Oiff(F) x

F

of all bijective maps

1-form with values in the

F •+ F , consider a distribution

transverse to the fibers of the projection to

are formal manifolds;

this means that around each

M , where

M

(x, u) € M x F ,

V and

Differential

there is given a subset maps bijectively to is a unique

u'

forms

V(x, u) c M (x, u)

36 1

which by

M (x) . Thus, if u € F , and x ~ y

with

1-form on M

with values in

Diff(F)

M

in M , there

(y, u') (. M(x, u) . Thus, the pair

an automorphism of F , u *—*• u' , which we denote a

proj : M * F

x, y

defines

w(x, y) . Thus

w

is

(which is a very big group - not

a formal manifold). Heuristically, the "line" connecting u ) to the "line" connecting V(x, u) , and thus

V

curve integration of u dui(x, y, z) = e , where

(x, u)

to

x

to y

(y, u') , and this "line" lies in

defines an infinitesimal path-lifting. amounts to lifting of finite paths. (x, y, z)

M, means that (for any initial value

that of

V

To say

u € F ) , the triangle lifts to a To say u> = df

for some

(locally) can by a little combinatorics be seen to imply

arises from a foliation, the leaf through

y >—>• f(y)f(x)~

(u) ; "din = 0 =* w = df"

about integrability of distributions. condition

The finite

is an infinitesimal triangle (l.l) in

closed triangle ("no infinitesimal holonomy"). f : M -*• Diff(F)

lifts (for the given

dbs = 0

(x, u)

being the graph

in this case expresses a theorem

In §8 we will demonstrate that the

"is" the usual analytic condition in Frobenius' Theorem

about distributions. Now,

Diff(F)

does not "admit integration over finite intervals" in

the sense to be explained in §6, whence we cannot really derive (*), unless we can assert that the form in question takes values in a subgroup of Diff(F)

that does admit integration, or alternatively, if M

enough.

We discuss this in §8.

is small

2. Some infinitesimal arithmetic We assume satisfy Axiom

R W 1

to be of line type in the strong sense of [9], or of ['?].

As usual, we define

D(n) = { ( d l 5 . . . , dn) € FT I di • d. = 0 V i , o) c if1 . Mote that

D(2n) cz D(n) x D(n) .

An intermediate object i s

362

Anders Kock

..., dn), (6X, .... 6 J | [di

D(2n) =

&j + d..

6^

0)

A [di • ^. = 0) A (6i • 8j = 0) Vi, j} . We write Note also

d for (d , ..., d)

. Note

M (v) = u + D(n)

for u €fl".

0 ~ d ~ 6 ~ 0 if and only if (d, 6) € 0(2n) . Since

invertible in i? , we have furthermore (d, 6) t D(2n)

2 is

d. • 6. = 0 for any

and i = 1, ..., n . We have

PROPOSITION

2.1. Any map

D(n) •* if

with

0 «—> 0 is the

restriction of a unique linear map R -*• R . (For

n = 1 , this is the "basic ("Kock-Lawvere") axiom.) 2.2. Any map

PROPOSITION

(d, 0) i—> 0 j for all map

D{n) x D{n) •* if with

(0, d) i—• 0 and

d € D(n) , is the restriction of a unique bilinear

if1 x if1 ^ if . PROPOSITION

2.3. Any map

(d, 0) i—>• 0 j for all skew symmetric map

n

f : D{2n) •+ if

(0, d) •—* 0 and

x if -*- if .

We give the proof for the last one only. case

with

d € D(n) , is the restriction of a unique bilinear

m = 1 . The Weil algebra defining

It suffices to consider the

D(2n) is

*[*!• .-., * n , y i> •••• ^ / J where

k

is the ground field (assumed to be of characteristic not equal to

2 ) , and J is the ideal generated by all

X. • X., Y. ' Y. , and •&•

'•

3

0

X. ' Y . + X. • Y. . A k-linear basis for this Weil algebra i s given by i0 3 ithe classes (mod J) of the polynomials 1 , X . , , . . . , X , Y , . . . , Y , v[X. - Y. - X. ' 1 n' 1' ' n' i j j

' Y.) . . ^-'^ w e

C , and associates

e

of the simplex are equal.

Such forms have, in another context, been considered by Bkouche and Joyal who use them (with (unpublished).

G = R ) to define deRham cohomology

We shall be interested mainly in non-commutative groups

G , and therefore only in low dimensions A

0-form on

coboundary,

M

with values in

df , is the

G

k = 0, 1, 2 . is just a map

1-form given by

df(x, y) = fly) - f(x)-1 If

a) is a

f : M -*• G . Its

for x ~ y .

1-form, we define dirt(x, y, z) = ui(z, x) ' w(z/, 2 ) • w(x, y)

for

x ~ y ~ z ~ x ; under suitable assumptions on

G , du

will be a

364

Anders Kock

2-form on the

M , of. Proposition U.I. We denote (for

?c-form whose only value is

g : M -*• M'

e . Clearly

k = 0, 1, 2 ) by

0

d(df) = 0 . Since any map

between formal manifolds preserves the property of being

neighbours:

x ~ y =* g(x) ~ g(y) , a

k-form

Clearly,

u) on

M'

immediately gives rise to a fe-form g*(w)

on

A/ .

g*(du) = d{g*u) .

If the value group identity map

G

i : G "*• G

Maurer-Cartan form

^

itself is a formal manifold, we have the

as a

on

0-form on

G . Its coboundary

di- is the

G :

tt(x, y) = di{x, y) = y • x~X

for

x ~ y .

Clearly

(3.D

dn = o ;

we shall see in §5 that this is really the Maurer-Cartan formula. Also, for any

f :M + G ,

(3.2)

df = d(/*i) = f*(di) = f*Sl . 4.

If

w is a

Certain infinitesimal curve integrals

C-valued

1-form on a formal manifold

M , we want t o

consider (U.I)

w(j/, a;) • w(x, j/)

for

x ~ j/

for

x ~ y ~ s ~ x .

and (U.2)

i»)(3, x) • u(!/, z) If

M = K

• u{x, y)

(or any etale subobject thereof), these can also be

written (U.3)

w(x+d, x) • o)(x, x+d)

for

d € 0(w)

and (U.U)

w(x+6, x) • w(x+d, x+6) • u(x, x+d)

These are to be thought of as infinitesimal !:

forth

and infinitesimal

l:

for

(d, 6) € 5(2n) .

"curve integral back and

curve integral around a triangle11, respectively.

Differential

forms

365

We also want to consider the infinitesimal curve integral around a parallelogram: (U.5) w(x+6, a:) • w(x+d+6, x+6) • w(x+d, x+d+6) • m(x, x+d) for PROPOSITION

4.1.

bijeotive maps

Let

G

be a subgroup of the group Diff(F)

F •* F , where F

1-form with values in such

is a formal manifold. If

e , for all

(it.5) gives e Proof. u

of all

is a

x ~ y , (expressing

is alternating); and

(ii) (k.h) gives e , for all

fixed

w

G , we have:

(i) the value of (U.1) is that u

(d, 6) € D(n) x D(n) .

for all

(d, 6) € D(n) , if and only if (d, 6) € D{n) * DM

.

For all of the five expressions, we consider their effect on a

£ F . We choose a frame around

local, we may as well assume we also choose a frame around respectively; For fixed

FcB

u

and since the question is

(etale subset);

x , reducing them to C+.3) and (i*.it)

in particular, we assume

M c R

(etale subset).

y €M , u £F , w(j/, -)(u) is a map

M A y ) •+ M A u ) ,

thus, by Proposition 2.1, given by a unique affine part we denote

for (U.l) and (U.2)

R

•*• FT

whose linear

A(y, u, -) , in other words co(x, x+d)(u) = u + A(x, u, d) .

Let us note that u(a;+d, x){u) = u + A(x+d, u, -d) = u + A(x, u, -d) + DxA{x, u, -d)(d) , where

DA

variable.

denotes partial Jacobian of Since

D A(x, u, s)(t)

A

with respect to its first

is bilinear in

S

and

t , and

d € D(n) , the last term vanishes, so that we have tu(x+d, x)(u) = u + A{x, u, -d) = u - A{x, u, d) . How let us calculate the effect of (1».3) on

u :

366

Anders

w(x+d, x) • w(x,

Kock

x+d)(u) = w(x+d,

x)[u+A(x,

u, d ) )

= u + 4 ( x , w, d ) - A(x,

U+A{X,

U, d ) , d)

= u + 4 ( x , w, d ) - 4 ( x , w, d) - DJi{x,

(where

u, d)A(x,

u, d)

D.A is partial Jacobian with respect to the second variable). The

last term here equals bilinear

0 because it again is from S(d, d) for some

B . So the effect of (It.3) on u is u . This proves (i).

To prove (ii) , we calculate {k.k) and (k.5) by the same technique; we get, after cancelling all S(d, d) and B(6, &) for B bilinear, that (k.h) - u = DxA{x, u, 6)(d) + D2(x, u, &){A(x, U, d)) , (d, 6) e D(2n) . Similarly, we get (U.6)

(U.5) - u = ^ ( x , u, 6)(d) - D±A(x, u, d)(6) + D2A(x, u, 6)[A(x, U, d)) - D2A(x, u, d)[A(x, u, • N

into

[t >—*• cp o t] . This is well-known; PROPOSITION arbitrary

5.2. Let M

x €M

3

and

F

be formal manifolds.

linear maps

taking

T M •+ Vect(F)

x

to

[that is,

id j T. (Diff(f)) ) , '

i*'

where

Vect(F)

id

is the vector space o/ aZZ- vector fields on

passage from (l) to (2) is given by sending *

There is, for

a natural bijective correspondenae between

M A x ) -*• Diff(F)

(1) maps (2)

see, for example, [JO], Remark 6.1.

cp : M

F . T?ze

-»• Diff(F)

to

: T M •*• Vect(F) , where

*(t)(w, d) = constructed is

u € F . i?-linear.

It is

easily seen to be homogeneous (that is commute with multiplication by scalars from is an

R ). But because

F

is infinitesimally linear,

V = Vect(F)

/?-module for which Proposition 1.10.2 of [/)] can be applied, to give

the linearity.

We next produce the passage from (2) to (l). We choose a

frame around

x 6 M , identifying

TM

. Let there be given a linear

with

K

M (x) with

D{n)

(n = dim M) , and

T M = if1 -£+ Vect(F) . cc Let *.

$. = $(e.) where

e.

is a vector field on

is the

ith

F . We define

canonical basis vector of cp : D(n) •*• Diff(F)

FT ;

by putting

If

(for

u i F , and

d = [d±, .. . , d)

£ D(n) )

cp(d, u) = * 1 ( d 1 , * 2 (d 2 , .-., \{dn, where

#.

is identified with a map

u)) ...) ,

D x F -*• F .

To see that the processes are mutually inverse, first note that the process

cp •—*• $

described in the proposition in coordinates can be

described as follows;

it suffices to describe

*. : 7s

368

Anders Kock *.(

N

which take (x, z) and (y, x) to u , and (2) bilinear skew symmetric maps T M x T M —-* T N U

XX

(the codomain is Vect(F) , if u = e € Diff(F) ) . Proof. Here we have to do the whole work in coordinates, by choosing frames, and then prove independence of the choice. Let us do the case N = Diff(F) . So identify M A x ) with D(2n) . For fixed u € F we have, by choosing a frame around

u and working with coordinates there,

cp(d, 6)(u) = u + Ajd, 6) , where A : D(2n) •+ FT (k = dim F) takes Now A

extends uniquely to a skew bilinear

(d, 0) and (0, 6) to zero. R

x

n •*• FT , by Proposition

2.3. Thus the information contained in cp is: to each bilinear

H

X FT -»• R

u , a skew

, which is also the information in (2). We omit the

details about independence of choice of frames, and so on. From Proposition 5.3 follows that we have a bijective correspondence between

2-forms

9 on M with values in G , and ordinary "linear"

2-forms

B on M with values in LG for any G which is either a formal

manifold, or Diff(F) with

F a formal manifold.

If we let at denote exterior differentiation of ordinary linear differential forms, we may ask what is the relationship between

So and

3J3 ? The answer involves the Lie bracket structure on LG (calculated via right invariant vector fields on G ). THEOREM

5.4. Let G be a group which is a formal manifold, or

Diff(F) for F a formal manifold.

Let M be a formal manifold, and w

370 a

Anders Kock

1-form on M with values

(5.1)

in

G.

(dii){wx, w2) = h&i{wv

Then, for

w , w €TM,

w2) + [5(uj , w(u2

Equivalently (5-2)

S

[u, Proof.

6](M, y) := [5>{tO,

Since the former case is easily reduced to the latter, we

shall only do the Diff(F) notation as in §U. For

case. We may do it in coordinates. We use

(d, 6) € 3(2n) , we calculated, in (U.7),

du(x, x+d, x+S)(u) = u + %C

(d, 6)

with

(5 3)

"

°x,u^l' W 2^ = D1A(X> "' W 2 )( W J " V ^ X ' "• W lH w 2 ) + D24(x, M , W2)(4(ar, u, w^^)) - ^ ( x , M , W ^ ^ a : , u, V*2))

for all

(w , W ) €fl"x Rn . Thus the skew bilinear

the coordinate information of dw at x two terms in (5-3) involving

DA

I?1 x F?1 ->• R

and u

is %C ( - , - ) . The x ,u will be shown to be 3w (w W )(M) ,

i

and the two terms involving [w (w ) , w (w )J(M)

x

j.

if

C> 4 will be shown to be

. Now the first is rather clear, since the vector

space structure on Vect(F) separately.

giving

Keeping

u

is calculated in T F

for each fixed

u € F

fixed and omitting it from notation, the

expression

is the standard expression for the exterior differential of the 1-form given by A . We now turn to calculating the tangent vector at u bracket.

Let us, for

[d , d)

€D x D

involving the Lie

arbitrary, calculate

Differential

^ K ) («> d±) = u + ^ t * .

M,