About interactions between Banach-Lie groups and finite dimensional ...

J. Math. Kyoto Univ. (JMKYAZ) 12-3 (1972) 543-570

About interactions between Banach-Lie groups and finite dimensional manifolds By Hideki °mom and Pierre de la HARPE (Communicated by Professor Nagata, August 1971)

1.

Introduction Consider a smooth connected compact manifold without boundary,

and let g) be its group of diffeomorphisms. As shown in [14] , a Frechet-Lie group. Furthermore, g

is

is know n to be a IHL-Lie

group; this notion is a sort of generalization of Sobolev chains in function spaces. (D e fin itio n of IH L-Lie groups: see [6] or [16] further properties in [17] and [18] ; the present paper however uses neither the notion of a Frechet-Lie group, nor that of an IHL-Lie group). These two structures are rather difficult to study, because both the implicit function theorem and the Frobenius theorem are invalid for general Frechet or IHL manifolds. For this reason, the following question arises naturally: "Does there exist a sufficienly larg e subgroup o f ..g) which carries a real Banach-Lie group structure ?" The answer to this question is unfortunately negative, as this paper will show. Consequently, in order to study groups of diffeomorphisms, one must deal with Frechet-Lie groups or IHL-Lie groups, however difficult it might be. Before we can state the theorem which answers the above question, we need a definition, to make precise the content of the words "sufficiently large". Let G be a connected Banach-Lie group, with Lie algebra g, which acts smoothly on a finite dimensional manifold M.

544

Hideki Omori and Pierre de la Harpe

Let mo be an arbitrary fixed point of M , and let Go be the isotropy subgroup of G at m . The action of G is said to be irreducible at mo if th e linear representation p o f Go in GL(T„, o M ) is irreducible, 0

where p ( g ) is the derivative of the map ml-->g•m at mo and where T M is the tangent space to M at m . Suppose now that Go is a 0

sub Banach-Lie group and a submanifold o f G, and let go be its Lie algebra. (In fact go will be defined page 7 without reference to G 0 , so that no extra assumption on Go is needed.) The action is said to be prim itive a t mo i f go is maximal among the proper closed subalgebras o f g. I f G acts transitively on M , then the action is irreducible (respectively primitive) at any point of M as soon as it is in some point. If so, we simply say that the action is irreducible

(resp. prim itiv e). Clearly, an irreducible action is always primitive, and many examples show that the converse does not h old. Finally, and in order to avoid restrictions implied by second countability arguments, we define a kind o f "strong transitivity" as follows. Any vector in g defines a vector field on M , hence a tangent vector at the base point m o in M ; the action is ample if it is transitive and if moreover this map from g to T”,, M is on to. Obviously, an irreducible action is ample as soon as the dimension of M is 2 or more. By a "sufficiently large" subgroup G of 2 , we mean a group G which acts amply and primitively. H o w e v e r, th e following result shows that 2 does not contain any such subgroup but finite dimensional ones.

L et G be a connected Banach-Lie group acting smoothly, effectively an d am ply o n a finite dim ensional manifold w ithout boundary . If th e action i s prim itiv e, then G m u st be a finite dimensional Lie group. Theorem I:

As a direct application of the method o f proof, we have a second result. Theorem II:

L et g be the L ie algebra o f an infinite dimen-

About interactions between Banach-Lie groups

545

sional Banach-Lie g ro u p G . Suppose g has no proper close finite codimensional ideal. Then g has no proper closed finite codimensional subalgebra. In particular any smooth action of G on a finite dimensional manifold is trivial. We introduce a convenient set up for these two theorems in section 2, and show how they both follow from a property of primitiv graded Lie algebras, stated as theorem III (p a g e 1 4 ). Theorem III itself is proved in section 3, by an elementary method resting on the classification o f E. Cartan's real primitive infinite Lie algebras, that we know from the works of Morimoto and Tanaka [24]. (It has been pointed ou t to us that the classification is independently due to S. Schnider: J . o f D iff. Geometry 4 (1970) 81-89.) The main idea o f our proof appears in a simpler form in the two following examples. E xam ple. Let g be the Lie algebra generated in the space of smooth functions from [01] to

aa

x

ax

R

a ••., x"

ax '

b y the elements ••• ax '

g cannot be made a normed Lie algebra; indeed, the relation

L

a ,ax. xa 1—a(n —1)x x" x ax

a

is not compatible with the existence of a bound X ' such that 11 [u, v] I g

jR X JR morphism of the associative Banach algebra L ( H ) , which keeps the identity, C (H ), L ° C (H ) an d {X E L °C (H ) I trac e (X ) = 0 } invariant. Abusively, we denote still by ÇOR the involutions induced on invariant subsets of L ( H ) and on L ( H ) / C ( H ) . We consider now the orthogonal com plex Banach-Lie algebras C(H ,JR)= {XEgt(H )Igo ( X ) R

- - X } C (H , J R ; C)--- {X E A t(H ; C)1çoR(X )= — X ) an d th e L ie algebra o(H, J ; L°C ) {X E g(H ; L°C )Ç o ( X)= — Xl . R

R

Lemma 5. 4. B . The Lie algebra &(H, J R ; L°C ) is simple.

P ro o f : W ith th e same notations as in the proof o f Lemma 5. 4. B, note that im (X ,) + im (X 7 ) an d k e r(X ) f l k e r(X 7 ) a re invariant b y J R ; chose F invariant by J R and proceed as before. 111 Lemma 5. 5. B . Let a be a non-zero ideal of C (H , J R ) . Then &(H, J ; L ° C ) a. R

P ro o f : A s above; note that

(II,

JE )

does not contain the

identity. III

L em m a 5. 6. B . A ny operator in C (H , J R ) can be w ritten as a finite sum of commutators of operators each in C(H , J R ) .

P ro o f : In three steps. S tep o n e : Let B be a set o f generators in an associative alge.R] . W e fir s t p r o v e b y induction that bra then [B,_R]= [B",.gZ ] c [B, R1 f o r any n E N * ; it is trivial for n - 1 ; suppose it is true for i = 1, 2, • • n - 1 ; choose X , • • •, X„ E B an d YE R ; then: ,

i

[X X i

X.Y)— (X2—. X„Y) X X„(Y X ) — (Y Xi) X,. • X„ G [B, .R] + [B- , _R] c [B,

„ ,

Y]

i

1

i

N ow

[R,

[B", _R] c [B, n eN .

_R] and step one is established.

.

566

Hidek i Om ori and Pierre de la Harpe

S te p tw o : C (H , Theorem 15.

JR )

is a set o f generators in L ( H ) . See [9]

L e t m= {X Step three: [ C ( H , J R ) , C (H , J R ) ] —C(H, J ). 5. 6. A By Lemma E g i(H ) Ç O R (X ) = X } ; then gi(H ) =C(H, JR) e m . and the two first steps, one has: R

Af(H)= [ni(H), gf(H)] = [ C(H, JR ),C (H , JR )em ] C [C (H , J R ) , C (H , J ) ] + [C(H, J R ) , ml . R

A s [0 (H J R ) , C(H, J R ) ] cC (H , conclusion follows. IM

a n d [0 ( H, J R ) , m] c m the

JR )

Lemma 5. 7. B (H erstein). Let .R be a simple associative real

alg e b ra w ith c e n te r L . Suppose th at the dimension of grZ, over is larg e r th an 1 6 . Let ço be an involutive antiautom orphism of and consider the Lie algebra C = { X R lço(X )— — X } . Let a be a Lie ideal in C . T h e n one h as e ith e r ac L or [C , Cl a. -

P ro o f : See [9 ] Theorem 2 6 . In fact, Herstein proves a much more general result, where c a n b e a n y rin g ; we haven't tried to find a shorter proof for the case _T.=L ( H ).

Theorm I V B . Let a be a non-trivial Lie id eal o f C (H , J R ) . T hen &)(H, J R ; DC) c a c ( H, J R ; C ) . In particular, any non-trivial id e al o f F..)( H , J R ) is inf inite codim ensional, the o n ly non-trivial closed ideal o f C(H, J R ) is C (H , J R ; C ), and the Banach-Lie algebra C (H ,JR )/ C (H , J R ; C) is algebraically sim ple. C (H , J R ) is th e L ie algebra o f th e Banach-Lie group 0(H , J R ) so that Theorems I and II apply. 0(H , J R ) is the closed sub BanachLie group an d submanifold o f G L (H ) given a s follows: le t {1} be th e bilinear from defined on H b y {y1 z} < y IJ R z> ; then 0(H, JR )

{X E G L (H ) { X y X .z } { y z} fo r all y, 2E H}

.

567

About interactions between Banach-Lie groups

5. C . Symplectic L ie algebras Recall that an anticonjugation in H is a semi-linear operator

—id and < jex AY> = for all x, y e H. It is th e n e a s y to show that there exists a n orthonormal basis E (e„)n z . o f H such that J ( E x e _ „ + x „ e „ ) = E

A

in H such that R

I

e

=

(2

.E N *

. E

N

*

. E

N

*

. E

N

*

In particular, any two anticonjugations in H are conjugate to each other by unitary operators. From now on, we choose a fixed anticonjugation JQ in H.

—JQX*JQ o f L ( H ) . We consider the sy m p/ectic com plex Banach-Lie algebras 4 (H , J ) = {X ( H ) ç2 ( X ) — X } 4(H, A; C )= - { X epf(H ; C)1 9 (X )= - X } and the Lie algebra 4(H, j0 ; P C ) . Lemmas 5. 4. C to 5. 7. C are as in section 5. B, and one has the following theorem. Let

ÇOQ

be the antiautomorphism

Q

Ç Q

Theorem I V C . Let a be a non-trivial Lie ideal o f 0 (H , j e ). T h e n 4 ( H , 0 ; L °C )c a c 4 (H , j(? ; C). In particular, any nontriv ial ideal o f 4(H, J 0 ) is inf inite codimensional, the only nontriv ial closed ideal o f W H ,J ) i s 4 (H ,J ; C), and the BanachL ie algebra 4 (H , J 0 ) / ( H, j0 ; C ) is algebraically sim ple. Q

R

H, J ) is the Lie algebra of the Banach-Lie group sp(H , J ), so that Theorems I a n d II apply. S P(H, J ) i s th e closed sub Banach-Lie group and submanif old o f G L (H ) given as follows: let } b e the bilinear form defined on H b y {Y12} ‹Y IA* ; then SP(H, ./Q) = {X E G L (H )1 {X y l X z } {y1z} for all y, zE. HI . 0

Q

Q

N.B.: i ( H ) , C (H ,J ) and 4(H , J (2 ) can naturally be conR

sidered as re al algebras, and so can the corresponding Lie groups, by restriction of the scalars. 5. R . R eal forms Let g be one of the complex Lie algebra o f operator described above. A conjugation in g is a map r : g—>g such that the map

Hideki Omori and Pierre de la Harpe

568

X—>r(— X*) is an isom orphism . A real f orm of g is an R-sub-Lie algebra go of g such that g= go eigo . It is well known that one can associate a conjugation to any real form and vice-versa. To any non-exceptional class o f finite dimensional simple real

Lie algebras with simple complexifications, one can associate a conjugation of one of gt ( H ) , ( H , J ) an d 0(H, J ). This procedure has been described in detail elsewhere (see [7] and references there), an d w e w ill n ot repeat it h ere; w e on ly m ention the following R

Q

result: Theorem IV R . L e t a be an ideal o f g o , w here g o i s a real

f o rm as described above. Then a is infinite codimensional. Theorem s I and II ap p ly to the L ie group corresponding to go. We suggest th e following definitions. B y a classical complex

Banach-Lie group o f bounded operators in H , we mean one of the three groups G L (H ), 0 (H , J ) and S p(H, J ). By a classical real Banach-Lie group o f bounded operators in H , we mean either one R

Q

of these same three groups after restriction of the scalars, or one of the groups attached to a real form go as above; the list o f these can be given as follows, using the notations of [7] T ype A I GL(ER) AIT U * ( E ) U(E, r, 00) with r E N U {c.} A III U ( E ) corresponds to r =0 T yp e B D I 0(E, r , 00) with rE N U fool 0 ( E ) corresponds to r =0 BD III 0 * (E ) T ype C I sp (E , R ) C II SP(E, r, 00) with rEN U { 00} s p ( E ) correponds to r =0 .

R

Q

This makes precise the content of Theorem IV , as written in the introduction.

A bout interactions between Banach-Lie groups 5

6

9

B y a classical B anach-L ie group o f com pact operators in H we mean one obtained as follows. Let G (H ) be a classical BanachLie group o f bounded operators in H, and let g(H ) be its Lie algebra. L et g(H, L °C) b e th e Lie algebra g (H )n g (H ; L ° C); g(11; L°C) is simple. Let a be a uniform crossnorm in the sense o f [22] , and let g(H, a ) be the completion of g(H ; L°C) with respect to a . Then

g(H, a ) is the Lie algebra of a sub Banach-Lie group o f G (H ) which is denoted by G(H, a ) . G(H, a ) i s by definition a classical BanachLie group o f compact operators in H, and satisfies the conditions of Theorems I and II. Note that if a is the uniform crossnorm, G(H, a) = G (H , C ); i f a is the Hilbert-Schmidt cross-norm, G(H, a) = G(H,

L 2C ) is a L*-group [7] . References [1] [2]

N. Bourbaki: Groupes et algèbres de Lie, Chap. I . Hermann 1960. D. Burghelea and N. H . Kuiper: H ilbert manifolds. Ann M ath 9 0 (1969) 379-

[3]

J. W . Calkin: Two-sided ideals and congruences in the ring of bounded operators in H ilbert space. Ann of M ath 4 2 (1941) 838-873.

417.

[4]

J. Dieudonné: Les fondements de l'analyse moderne. Gauthier-Villars 1963. : Eléments d'analyse, tome 2 . Gauthier-Villars 1968. D. G. Ebin and J. Marsden: Groups o f diffeomorphisms and the motion of an incompressible flu id. A n n of M ath. 9 2 (1970) 102-163. [ 7 ] P. de la Harpe: Classification des L*-aIgébres semi-simples réelles séparables. CR Acad. Sci. Paris. Ser. A 272 (1971) 1559-1561. [ 8 ] L. N . Herstein: O n the Lie and Jordan R ings of a sim ple associative ring. Am . J. Math. 77 (1955) 279-285. : Lie and Jordan systems in simple rings w ith involution. Am. 9 J. Math. 78 (1956) 629-649. Topics in R in g T h eory. U niversity of C hicago, M ath. Lect. [10] : Notes, 1965. [11] S. Kobayashi and T. N agano: A report on filtered L ie algebras. Proc. of the US-Japan Sem. in Diff. Geom., K yoto 1965. [12] : On filtered L ie algebras an d geom etric strures I I I , I V . J. of Math. and Mech. 1 4 (1965) 679-706 and 1 5 (1966) 163-175. [13] S. Lang: Introduction aux variétés diffrentiables. Dunod 1967. [14] J. A . L e s lie : O n a differential structure f o r th e group o f diffeomorphisms. Topology 6 (1967) 263-271. [15] E . Nelson: Topics in dynamics I : Flows. Math. Notes, Princeton Press, 1969. [16] H. Omori: On the grou p o f diffeomorphisms on a com p act m an ifold . Proc. Symp. Pure Math. AMS 1 5 (1970) 167-183. [17] : Local structures of groups o f diffeomorphisms. Jour. Math. Soc.

[5] [6]

Hideki Om ori and Pierre de la H arpe

570

[18] [19]

[20]

Japan 24 (1972) pp. 60-80. On smooth extension theorem s. Jour. Math. Soc. Japan 24 (1972) pp. 405-432. and P. de la H arpe: Opération de grou pes de Lie banachiques sur des varitétés différentielles de dim ension finie. CR Acad. Sci. Paris. Ser. A (19 juillet 1971). R . Ouzilou: Groupes classiques de Banach. CR Acad. Sci. Paris, Ser. A 272

:

(1971) 1459-1461. C. R. Putnam: Commutation properties of Hilbert space operators and related topics. Springer 1967. [22] R. Schatten: Norm ideals o f completely continuous operators. Springer 1960. [23] I. M. Singer and S. Sternberg: The infinite groups of Lie and Cartan, part I (The transitive groups). Journ. d'an. Math. 1 5 (1965) 1-114. [2 4 ] T . Morimoto and N. Tanaka: The classification of the real prim itive infinite L ie algebras. J. Math. Kyoto Univ. 10 (1970) 207-243.

[21]

MATHEMATICS INSTITUTE UNIVERSITY OF WARWICK CONVENTRY ( G . B .)