C OMPOSITIO M ATHEMATICA
J ANUSZ G RABOWSKI
Derivations of the Lie algebras of analytic vector fields
Compositio Mathematica, tome 43, n
o2 (1981), p. 239-252
<http://www.numdam.org/item?id=CM_1981__43_2_239_0>
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239
DERIVATIONS OF THE LIE ALGEBRAS OF ANALYTIC VECTOR FIELDS
Janusz Grabowski
© 1981 Sijthoff & Noordhon’ International Publishers - Alphen aan den Rijn
Printed in the Netherlands
0. Introduction
A linear operator D of a Lie
algebra
L into L such thatfor all
X,
Y E L is termed a derivation of L.The set
Der(L)
of all derivations of L is a Liesubalgebra
of the Liealgebra End(L)
of all linearmappings
of L.For each X E
L,
we denoteby adx
themapping
Y -[X, Y]
of L into L. Themapping
X -adx
is ahomomorphism
of L intoDer(L).
The derivations of the form adx are called the inner derivations of L. The set
adL
of all inner derivations of L is a Lie ideal ofDer(L).
Dimensions of
adL
inDer(L) give
us an idea of how much non-commutative is the Lie
algebra
L. Forexample,
if L iscommutative,
then
adL
=101
andDer(L)
=End(L).
On the otherhand,
for finite- dimensionalsemi-simple
Liealgebras
we haveDer(L)
=adL.
The notion of
semi-simplicity
is well understood and useful in the finite-dimensional case. It seems to us that the class of the Liealgebras
of vector fields is aninteresting
class of infinite-dimensional Liealgebras
which havesemi-simple-like properties.
Derivations of such Lie
algebras
were examinedby Avez,
Lich-nerowicz,
Diaz-Miranda[1],
Morimoto[7]
and Takens[9].
For exam-ple,
it was shownby
F. Takens that the derivations of the Liealgebra
"M of all Coo vector fields on a smooth manifold are inner.
The same has been
proved by
T. Morimoto for thefollowing
classes of classical Lie
algebras
of vector fields on smooth manifolds 0010-437X/8105039-14$00.20/0with additional structures:
(i)
the Liealgebras
of vector fields of constantdivergence
(ii)
the Liealgebras
of vector fieldspreserving
a hamiltonian structure up to constant factors(iii)
the Liealgebras
of vector fieldspreserving
a contact structure.The aim of this paper is to consider an
analytic
version of suchtheorems,
i.e. to prove thefollowing
THEOREM: Derivations
of
the Liealgebras of
allholomorphic vector fields
on Stein spaces are inner.Note that a local version of the above result follows from J. Heinze
[6].
In view of the fact that
real-analytic
manifolds arereal-analytic
submanifolds of Stein spaces
(Grauert [4])
we derive from the above theorem the similar fact for the case of thereal-analytic
vector fields:THEOREM: Derivations
of
the Liealgebras of
allreal-analytic
vectorfields
onreal-analytic manifolds
are inner.Because the first
cohomology
groupH’(L, L)
of a Liealgebra
L withadjoint representation
isequal
toDer(L)/adL,
the firstcohomology
groups
H’(L, L)
= 0 for Lbeing
the Liealgebras
of vector fields fromthe above theorems.
By using
the Stein spaces in thereal-analytic
case we overcomeseveral technical difficulties which appear because of nonexistence of
partition
ofunity
to localize whole arguments.It is well known that derivations of the
algebra C°°(N)
of all smooth functions on a smooth manifold N areexactly
the smooth vectorfields.
We prove
THEOREM: Derivations
of
thealgebra of holomorphic functions
onStein spaces are
holomorphic
vectorfields.
Grauert’s theorem allows us to prove also the same in
R-analytic
case:
THEOREM: Derivations
of
thealgebra C03C9(N) of
all realanalytic
functions
on areal-analytic manifold
N are thereal-analytic
vectorfields
on N.The Lie
algebras
of vector fields have been examined in[3].
It isproved
there thatautomorphisms
of such Liealgebras
are"inner",
i.e.they
aregenerated by diffeomorphisms (bianalytic
or biholomor-phic mappings)
of theunderlying
manifold.The result of this paper demonstrate
that,
like in the finite-dimen- sional case, also for the Liealgebras
of vector fields there is a closecorrespondence
between their Liealgebras
of derivations and their groups ofautomorphisms.
It is
interesting
if thepresented
results about derivations can be obtained via anexplicit
form of the above mentionedcorrespondence.
In this paper
they
areproved independently.
1. Notation and
preliminary
remarksLet A be the associative
algebra
of allholomorphic
functions on aStein space M of
complex
dimension r and let 611 be thecomplex
Liealgebra
of allholomorphic
vector fields on M, i.e. the vector fields which in local coordinates zl,..., Zr have the form 03A3ri=1fi(~/~zi), where f;
areholomorphic
functions of variables zl, ..., zr.Recall that W can be
regarded
as a Liealgebra
of derivations of S!1 and that GtC is a module over thering A
so that forall f,
g ~ A andX,
Y e OU we haveWe will use the
following
notation:1. For p E M, let
Ip
be the ideal of A of all functionsvanishing
atP.
2.
Let I" p
denote the ideal of Agenerated by
Note that
Ig
=Ip
andI p
C1;
if m ~ n.3. For natural n and p E M, let
where
jnp(X)
denotes the n -thjet
of X at p, and letUkp
= U for kbeing negative integers.
Note that
OU’
CUmp
if n ~ m andfor
integer n
and m.Thus
p
is a filtration of OU.Observe also that
W) are
submodules of the module OU over A.4. For an ideal J of
A,
let JGh denote the submodule of OUgenerated by {fX : f
E J and X EU}.
5. Let
(i)
C be theanalytic
sheaf over M of germs ofholomorphic
functions on M,
(ii) F
be theanalytic
sheaf over M of germs ofholomorphic
vector fields on M,
(iii) 9, p nbe
theanalytic
sheaf over M of germs ofholomorphic
vector fields which n-th
jets
at p vanish.The
algebras
considered are the spaces of sections ofrespective
sheaves: W =
T(M, F), Uup
=r(M, np)
and d =r(M, 0).
Note also that all the sheaves are coherent.
6. Let
jp (OU)= U/Ukp
and denoteby jp
the canonical pro-jection
from ’M tojp(U).
jp(IU)
has the natural Liealgebra
structure and a filtration{jp(Unp)}n~Z
which used as a system of fundamentalneighborhoods
of
jp(U)
makes it atopological
Liealgebra
called the formalalgebra
of Uat p.
The
topology
isseparated
andcomplete.
Note 1.1. For each p E M the Lie
algebra jp(U)
isisomorphic
tothe
complex
Liealgebra Y
of all formal vector fields in r-variables.PROOF:
By
Theorem A of Cartan0393(M, F)
= OU generates each stalkFp
as aOp
module.Let zi,..., zr ~ A be local coordinates in a
neighborhood
of p withzi(p)
= 0(there
are local coordinates on Stein spaces which areglobal
defined
functions).
For a
given Y = 03A3ri=1(03A303B1 ai,03B1z03B1)~i (we
denote as usual a =(al, ..., ar), a;
naturalnumbers, |03B1| = 03A3rk=1 03B1k
and z"=z"l ...z03B1rr)
chooseX,, ...,
Xn E OU andf k
= 03A303B1bk,aza
ECp
such thatjmp(03A3nk=1 kXk) = jmp(Y). Putting Ym = 03A3nk=1 (03A3|03B1|~m bk,03B1z03B1)Xk
we haveYm E OU
andYm 1
Y in thetopology
injp(U).
DCOROLLARY 1.2:
Àl i
={X
Eôhn-1: [X, Y] ~ Un-1p for
all Y EU} for
each p E M and
integer
n.To prove this it suffices to observe that the elements of 6U span the tangent space
Tp (M).
Note 1.3. For each p E M and natural n we have
PROOF: Let zl, ..., zr ~ A be local coordinates in a
neighborhood
ofp. If
X ~ InpU
thenX(zi) -
the local coordinates ofX -belong
toInp.
Thus jnp(X)
= 0 and we getThere are gi,..., gm E.91 such that
(gi,..., gm) : M ~ Cm
is an imbed-ding
of M(R.
Narasimham[8])
andgi(p)
= 0.Put
Q
={(i1,..., im ) : ik
naturaland 03A3mk=1 ik
= n +1}.
For j
=(ti,..., im)
EQ,
denotegi11
...gimm by gj.
Consider a sheaf
homomorphism : F1 ~ Pnp (where 1 = card( Q) and Fl is
theproduct
of lsheaves F)
defined on thestalks Flq
in thef ollowing
way:where
X;,q
EFq
andgq
denote the germs ofgi
at q. Let us see that Tis an
epimorphism.
In
fact,
if q ~ p, then there exists k suchthat gk(q) ~
0. Henceg q
isinvertible
for j
=(i1,..., im ) with ik = n
+ 1and iu
=0 for
u ~ k. Thisimplies
that T is onto at q.For the
point
p, there are gi1,..., gir such thatthey
are localcoordinates in a
neighborhood
of p, since(g1,...,gm):M ~ Cm
isregular
at eachpoint.
If Yp
E9, p n@p then Yp
has the formin local coordinates gi1,..., gi,, and all
germs âk
vanish at p with all their derivatives ofdegree
Sn.Then, âk
have the formfor some germs
j,k
ofholomorphic
functions. HenceTo see that T is "onto" at p, it suffices to put
Let 3- be the sheaf
Ker(T).
SinceF1, np
are coherentanalytic sheaves, 3-
is a coherentanalytic
sheaf over M as the kernel of anepimorphism
between two coherentanalytic
sheaves over M. Fromthe exact sequence of sheaf
homomorphism
we obtain the exact sequence of the groups of
cohomology
withcoefficients in the sheaves:
By
the Theorem B of CartanH’(M, Y)
= 0 and since the0-groups
ofcohomology
are the spaces of allglobal
sections of thesheaves,
we get the exact sequence:Hence T :
0393(M, Fk)
~0393(M, np)
is "onto" and for every Y~ Unp
thereare
X1,..., XI
E U such that Y =LjEQ gjxj.
Sincegj
EInp,
the notefollows
by
this and(1.2).
0Let
Ip,n
be the ideal of A of all functions withvanishing
n -thjets
atp E M.
Note 1.4. For each p E M and each natural n we have
The
proof
is almostparallel
to theproof
of Note 1.3 and we omit it.Note
only
that it suffices to consider a sheafhomomorphism
betweensheaves of germs of functions instead of sheaves of germs of vector fields.
2. Global
approach
Observe that if D if an inner derivation of ôh then
by (1.1) D(X)
EU0p
for each X EU1p.
LEMMA 2.1:
If
D is a derivationof
OU thenD(U4p)
CU0p for
eachpoint
p E M.PROOF: Put
03A9p
={X
EOU’ : D(fX) EE OU 0
for allf
EAI.
From the
equality
which holds true for all
g, f
E s!1 and X EU,
we getIf X E
U1p,
thenby (1.1)
theright
side termbelongs
toU0p
and thusX(f)X ~ 03A9p
forall f ~ A.
Denote
by 03C9p(X)
the ideal of Aequal
tof g
~ A :gX
E03A9p}.
Then
Observe that
ilp
is a Lie ideal ofU0p.
To see
that,
consider theequality
If
X E Dp, f E.:4, Y ~ U0p,
thenD(Y(f)X) ~ U0p, [D(yx),Y]e [U0p, U0p] C oug
and[ f X, D(Y)]
E[U1p, U]
CU0p.
Hence
D(f [X, Y])
EU0p
for eachf
EA,
i.e.Now assume that X E
03A9p,
Y~U0p
andf
E A. We haveand since
[X, f Y], f [X, Y]
Eilp by (2.2),
we conclude thatX(f)
E03C9p(Y).
Because if h Ewp(gy)
thenhg
Ewp(Y),
we getwhere
Ip03A9p(A)
is an ideal of Agenerated by
It is easy to see from
(2.1)
thath2X(f)X
E03A9p
for all h EI1p, f
~ Aand X E 611. Since
2hg
=(h
+g)2 - h2 - g2,
alsohgX(f)X
EDp
for allh, g ~ I1p, f
E s1 and X E OU.Thus, by (2.3),
for all X E 6U
and f, g ~ A.
The
algebra A
has the property that iff1,...,fm ~ A
have nocommon zeros, then there exist gl, ... , gm ~ A such
that 03A3mi=1 f aga
= 1(see
e.g.[5]
p.244).
We will have shown in the next lemma that there exist Xi, ...,
Xm
E 6U andfi, ..., f m
E.:Ji such thatXi(fi)
have no commonzeros. Then also
(Xi(fi))2
have no common zeros and there exist g1, ..., gm e 4 such thatLet h E
I4p. Then, by (2.4),
which means that
I4p C n
YEOUwp(Y).
Because
by
Note 1.3I4pU
=U4p,
the lemma isproved.
DLEMMA 2.2: There exist
fi, ..., fm
~ A andXi,
..., Xm E au such that03A3mi=1 Xi(fi) = 1.
We used this fact in
[3]
but theproof
in that paper is false. We willprove this here
using
results of thetheory
of coherentanalytic
sheaves on Stein spaces as in the
proof
of Note 1.3.PROOF: There is a
family
of functionsf 1, - - ., f,,
E.:il such that the rank of themapping
is ~ 1 at each
point
p E M.(Such
amapping
could even beregular
ateach
point
as in theproof
of Note1.3.)
It means that for each
point
p E M there is i E{1,
...,m}
and agerm
Xp
of aholomorphic
vector field at p such thatfor
Îi,P being
the germof f
i at p.Consider a sheaf
homomorphism 03C1 : Fm ~ O
defined on the stalkover p by
Since
by (2.5)
there isi,p
such thati,p (i,p)
isinvertible,
p is anepimorphism.
Denote
Ker(p) by 1-T
which is ananalytic
coherent sheaf over M. We have the exact sequence of sheafhomomorphisms:
which generate the exact sequence
Because the
mapping 03C1 : 0393(M, Fm) ~ 0393(M, ) = A
is"onto",
thereare
X1,...,
Xm E 6U suchthat 03A3mi=1 Xi(fi)
= 1. 03. The main result
LEMMA 3.1:
If
D is a derivationof
6U thenfor
anypoint
p E M there is induced anunique
continuous derivationDp of
theformal algebra
jp(U)
such that thefollowing diagram
commutesPROOF: We need to show that for each
integer n
there is aninteger
k
such that D(Ukp) ~ Unp.
Let us put k = n + 4 and use induction.
If n = 0 then
D(U4p)
CU0p by
Lemma 2.1. AssumeD(Ui+4p)
CUip
fori n.
Then,
for each X EUn+4p
and all Y E OU we have[X, Y] ~ Un+3p
and
[X, D(Y)] ~ Un+3p.
HenceD([X, Y]) ~ Un-1p by
the inductiveassumption
and we getfor all Y E OU. Thus
D(X) ~ Unp by Corollary
1.2. 0THEOREM 3.2: Derivations
of
the Liealgebra of
allholomorphic vector fields
on a Stein space are inner.PROOF: For a
given
derivation D we getby
above lemma the induced derivationsDp
ofjp(U). By
T. Morimoto[7] (or by
J. Heinze[6]) Dp
are inner as derivations of the Liealgebras
of f ormal vectorfields
(see
the Note1.1).
Let
Dp
=ad Yp
whereYp E jp(ru).
Let z1,..., zr be local coordinates from A in a
neighborhood
U of achosen p E M and
zi (p )
= 0.If X E ru then the
coefficients Xi
ofjp(X ) = 03A3ri=1 xi~i
and thecoefficients of
jp(D(X))
are formal power series in variables z1, ..., zrwhich converge on U.
Because
jp(D(X)) = [Yp,jp(X)],
it is not hard toverify
that thecoefficients of
Yp
also converge on U.In
fact,
ifYp = 03A3ri=1 yi~i
thenby Dp(zkX) = ykjp(X) + zkDp(X)
theformal power series ykx; converge on U for i = 1,..., r.
Choosing
X such thatj0p(xi) ~
0 we can divide ykx;by
xi and conclude that yk converges on U.Hence
Yp
defines aholomorphic
vector fieldYU
on U such that[YU, Xlul
=D(X)lu. Obviously,
such Yu is on Uexactly
one. Cover-ing
M with such coordinateneighborhoods,
we get in this way aglobal
defined Y E 6lLsatisfying
4. The
real-analytic
caseNow, let "s4 be the
algebra
of allreal-analytic
functions on areal-analytic
paracompact manifold N of dimension r, and let £ù6U be the Liealgebra
of allreal-analytic
vector fields on N.We are
going
to sketch aproof
of thefollowing
THEOREM 4.1: Derivations
of
£ù6U are inner.PROOF: The
proof
is almostparallel
to that in thecomplex
case.It is known
(F.
Bruhat and H.Whitney [2])
that N can beregarded
as a
real-analytic
submanifold of acomplex
manifold M such that « has acovering
with local coordinatepatches (U03B1 ; 03B1Z1,
...,«z,)
satis-fying
If f
E "s4, then we can extendf
to aholomorphic
functioni
on aneighborhood
U of N in .lU with the same coefficients of its power series in thepoints
of N.By
theorem of Grauert[4]
there is aneighborhood
V of N con-tained in U such that V is a Stein space.
By CJJ Ip,n
denote the ideal of allreal-analytic
functions on N whichvanish at
p E N
with all their derivatives ofdegree
:5n. Iff
ECJJIp,n,
then f
EIp,n
andby
Note1.4 f ~ Inp,
i.e. there aregj1,..., g’
holomor-phic
on V andvanishing
at p such thatThe functions
Re(gjk)
andIm(gL)
are realanalytic
andHence
f ~ 03C9Inp,
i.e.By
Grauert[4]
N can be imbedded in R’ for some m :where
f1,..., f m
E 03C9A.Let ai
(i
=1, ..., m)
be the coordinate vector fields in R’.Having
the natural scalarproduct
in IR m(which
is areal-analytic operation)
we can obtain(as
in[3],
p.17)
vector fields Xie W6U(i
=1, ..., m)
such thatXi,,
is theimage
of~i,p
under theorthogonal projection
of Rm onto the tangent spaceTp(N)
at eachp E N.
Considering
Y E 03C9U as a vector field on a submanifold in Rm wehave
Therefore
If
Y ~ 03C9Unp
thenY(fk) ~ 03C9Ip,n
andby (4.1)
and(4.2) Y ~ 03C9In03C9pU,
thatproves the Note 1.3 for the
real-analytic
case.To prove the
analog
of Note 1.1 observe that the germs ofXi,..., Xm
as above generate the module of germs ofreal-analytic
vector fields over the
algebra
of germs ofreal-analytic
functions ateach
point p E N.
Corollary 1.2,
Lemma2.1,
and Lemma 3.1 follow as well without anychanges
in theirproofs,
and Lemma 2.2 isproved
for thereal-analytic
case inProposition
3.5 of[3].
Then,
theproof
of Theorem 4.1 is the samereasoning
as in theproof
of Theorem 3.2. 05. Dérivations of the
algebras
of functionsBecause
by
Note 1.4Ip,l
=I p,
for Xbeing
a derivation of A(i.e.
alinear operator of sl into A
satisfying
theequality X(fg) = X(f)g +
fX(g)
forall f,
g EA)
we haveLet zl, ..., zr ~ A be local coordinates of M in a
neighborhood
of p and letf
E A. Thenand hence
by (5.1)
Putting
we get a vector field
X
on M such that(f) = X(f)
forall f
E s4.Because
X(zi)
~ A it is obvious thatX
is aholomorphic
vectorfield.
The
real-analytic
case follows as well if we use(4.1)
instead ofNote 1.4. Thus we obtain
THEOREM 5.1.: Derivations
of
thealgebra of
allholomorphic (real- analytic) functions
on a Stein space(real-analytic manifold)
M areexactly
theholomorphic (real-analytic) vector fields
on M.Note that in
[3]
6U and w6U areregarded
as Liesubalgebras
ofDer(A)
andDer("A)
which are modules over A and W.st1 and it suffices to obtain the results aboutautomorphisms
and ideals.Thus we know that 6U and w6U are not proper Lie
subalgebras,
i.e.6U =
Der(s4)
and w6U =Der(’A).
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(Oblatum 28-IV-1980 & 9-IX-1980) Institute of Mathematics
University of Warsaw
00-901 Warsaw, Poland