C y c l i c p e r m
(X , Y , Z )
T h e generalised versions of (2.4.7) and (2.4.8) still hold true: P r o p o s it io n 3 .2 .7
Let generalised S aletan con tractio n be given by (2.4.6), w ith [0,1] and (0,1] uniform ly replaced by [O,!]*7 an d (0,1]^ respectively. If S a le ta n ’s condition of (3.2.6) holds, th e n d q is a subalgebra of th e Lie algebra C q[0,1]°" w ith bracket tak en pointw ise, g 0 as given by (3.2.5) has th e S aletan Lie algebra stru c tu re of (3.2.6) , and dX is a hom om orphism .
P r o o f:
U niform ly su b stitu te [0, l]*7 for [0,1] in th e proof of (2.4.7. ■ P r o p o s it io n 3 .2 .8
Let generalised Saletan co n tractio n be given by (2.4.6) w ith [0,1] and (0,1] uniform ly replaced by [0, l]a an d (0, l]*7 respectively. T h en S a le ta n ’s condition of (2.8.6) holds if an d only if d q is a Lie algebra.
P r o o f:
See th e proof of (2.4.8) which still holds for a > 1. ■ O ur aim is to prove theorem s analogous to (2.4.17) a n d (2.4.19). To this end, we generalise d P and dXp. T he generalisation of d P as given by (2.4.10) is less obvious th a n for th e preceding results.
L e m m a 3 .2 .9
g reater th a n q. T h e n th e m a p ,
d P - . c m , \ ] ^ v i { g )
d P : X i->I E Eq
iff
r=0E’=,
i k‘=
n~
w here A'! =K 1\ K2\ ■ ■ ■ K at-
X - ( 0 )
d NX( t )
d t f ' d t “ 2 ■■■&“ ' t= 0(an d w here {X }, for X a polynom ial in £, denotes th e coset containing X by th e above m entioned ideal) is a Lie a lg e b r a h o m o m o r p h is m .
P r o o f:
Let X , Y be polynom ials on g, in t of order K\ , K2 respectively. Let X 1—> X
denote th e o p eratio n of dropping off term s in X of order above q in t ( th a t is, term s
a
t j 1 • • -t*” are dropped if kj > q). To prove th a t d P [ X , Y ] = [dPX, dPY], the i=l
problem is reduced to showing th a t,
[x,r] = [x,n
Now X = X + 0 ( t q+1) and Y = Y + 0 (^g+1) w here 0(£g+1) denotes term s t ^ t ^ 2 • • • t„ a , w ith K j > q. I Therefore [X, Y] = [X + 0(t^+1), Y + 0(£g+1)] = [X ,F ] + C1(^+1) = [X ,E ] establishing th e result. We generalise th e m ap d X p:
L e m m a 3 .2 .1 0
T h e m ap dXp : dP( dq) h-> g0 given by dXP : { d P ( X ) } •-> dXo (X ),
in th e n o ta tio n of (2.8.9), is well-defined an d satisfies dX = dXp o dP.
P r o o f:
T he proof of (2.4.11) rem ains valid w hen we replace th e first q + 1 term s of th e Taylor series of X and Y E C qg[0,1] by th e expressions for d P ( X ) and d P ( Y) given by (3.2.9) m aking t E [0, l]a. d ^ ( t ) has th e m ore general form ,
<7
ij = —q (taking ij < 0 w ith o u t loss of generality) an d th e p ro o f holds when
3= 1
we replace by a linear com bination of all derivatives, — r--- r— on account
Ol2 * Ol &
of C onvention (3.2.2(b)). H
We can now generalise T heorem (2.4.17):
T h e o r e m 3 .2 .1 1 (N ecessary and Sufficient C onditions for C ontractions)
Theorem (2.4.17) holds w ith the following uniform su b stitu tio n s: V q(g) for
V q(g), [0, e)* for [0, e) and ( 0 , l ] a for (0,1]; w here (0,1]°" and [0,1]°" have been su b stitu te d for (0,1] and [0,1] respectively in D efinition (2.4.13).
P r o o f:
We give th e necessary uniform su b stitu tio n s in th e proof of (2.4.17). All changes include [0,1 for [0,1], (0 ,1 ]^ for (0,1] and t m u ltiv ariate for t univariate.
P ro o f of (1) => (2): R eplace t lZ^l\ 0) by th e expression for d P ( Z ) in (3.2.9);
i = 0
P ro o f of (2) => (3): As p er th e changes specified above. P ro o f of (3) => (4): C hanges as given above.
P ro o f of (4) => (1): As given above.
P ro o f of (4) => (5): As given above; and [0, e) replaced by [0,6)^.
P ro o f of (5) => (1): As p er changes above. ■
T heorem 2.4.19 holds w ith very little m odification:
T h e o r e m 3 .2 .1 2 (D ifferentiable P ro p erties of C o n tractio n ) T heorem (2.4.19) is tru e for a > 1.
P r o o f:
M odify th e proof of (2.4.19) by replacing (0,1] an d [0,1] by (0, l]*7 an d [0, l]*7
respectively. M
C o m m e n ts 3 .2 .1 2 (a )
In (3.2.12), the dom ain of d Xp is contained in th e Lie algebra of m ultinom ials on g in a variables. In view of C om m ents (1.4.19) we expect th a t th e dom ain of th e globalisation X p of d X p will be contained in J q([0 ,1Y T he generalisation of T q(Q) of (2.5.1) is U J ?([0, l]a t G)o,g, denoted J g([0,1]CT, Q)0tg (as given by
geG
(1.4.18)), which is a subset of the q-}et bundle J g([ 0 ,1]°",Q). We will now show th a t J g([0,1]°\ Q )o,g is a Lie group, T h en suitably generalise th e m ap P of (2.5.21),
th en prove a condensed version of (2.5.30), our principal aim being to show th e role of je t bundles for th e case a > 1.
P r o p o s itio n 3 .2 .1 3
Let J ?([0, l]*7, £/)o,e? = U J g([0, l]*7, Q)o,g in th e n o ta tio n of (1.4.18), w ith
geG
elem ents { / i }, { /2}, f i G Cg°[0, l]'7, and p ro d u ct { / i } • { / 2} = { /1/2} where / i / 2
is th e m ap f i f 2 : [0, l]a —> Q, t ( { f i } is th e equivalence class of f i by th e equivalence relation (1.4.17))
P r o o f:
By form al analogy to th e proof (2.5.3) th a t T 2(Q) is a group it follows th a t J g([0, l]*7, Q)o,g is a group w ith respect to th e above p ro d u c t, w ith iden tity {e} 6 </g([0 ,1]°",£7)o,e? where e is th e m ap e : [0,1]* —►Q which is constantly th e identity. To o b tain th e m anifold stru c tu re of J g([0, l]*7, G)o,g, let { /} be an a rb itra ry elem ent of J g([0, l] CT,^ ) o )5, and (Xj-AO a ch art at g in Q. Let / x = X 0 / an d f l be its coordinate functions. For convenience, identify V qn (see (1.4.18)) as a coordinate space by suitable choice of basis polynom ials. Define th e m ap
tt: J q([0,l]a ,G)oto - + G
77 : {/} £
iG)o,g►
g,th en th e coordinate m aps are given by
X* : k
- ' W
- Rn xV ln
X*
•• { / } €J q([
0,1]",g)o,g
-> (x(ff), - P ,/ i ... ^ g /J ) ( ^ e (1.4.18)).T h e are surjective m aps by (1.4.18), which induce a topology on J g([0, l]*7, Q)o,g- O verlapping ch arts ( x i , tt-1 ( M ) , xi , y\ , . . . , y?) an d ( x 2 > ^ “ H-A/i), x2, y2, ■ • ■, 2/?)
at { /i} an d { /2} respectively, correspond to overlapping ch arts ( x i , A/*i) and (X27 «A/2) in £/. W riting x 2 0 ( x D -1 in coordinate form,
(x2,y\,
• • • , 2 / ? ) = ( ^ 2 ( ^ 1 ) , 3/2 (2/1 )> • • • >2/2 ( 2 / 1 ) ) -x 2 is a C°° function of x\ and th e y l2 are linear functions of yi, induced by a change of variables in th e Taylor series of . Hence overlapping charts are C°°- related . To see th a t th e p ro d u ct is C °°, let ( x j , 7r- 1 (-A/3), £3, y \ . . . , y j ) be a chart
a t { /1H /2}. T hen,
X3 ((xSr)_1(x l>!/l X x J ) _1(*2> !fe))