Abstract LIE ALGEBRAS OF VECTOR FIELDS AND GENERALIZED FOLIATIONS

LIE ALGEBRAS OF VECTOR FIELDS AND GENERALIZED FOLIATIONS


. Introduction.
A whole series of papers followed the classical result of Shanks and Pursell [15] which states that the Lie algebra X(M) of all smooth vector fields on a smooth manifold M determines the smooth structure of M, Le. the Lie algebras X(M1) and X(M2 ) are isomorphic if and only if MI and M2 are diffeomorphic .Some of these papers concern special geometric situations (hamiltonian, contact, group invariant, etc. vector fields), as for example the results of Omori [14, Chapter X], Abe [1], or Atkin and Grabowski [3], and for which specific tools were developed in each case.There is however a case when the answer is more or less complete in the whole generality.These are the Lie algebras of vector fields which are modules over the corresponding rings of functions (we shall call them modular) .Let us recall the work of Amemiya [2], our paper [5] where developed algebraic approach made possible to consider analytic cases as well, and finally the brilliant purely algebraic result of Skriabin [16].This final result states that, in case when modular Lie algebras of vector fields contain finite families of vector fields with no common zeros (we shall say that they are strongly non-singular), isomorphisms between them are generated by isomorphisms of corresponding algebras of functions, Le. diffeomorphisms of underlying manifolds .The standard model of a modular Lie algebra of vector fields is the Lie algebra X(_F) of all vector fields tangent te a given (generalized) foliation P.However, if we consider not the Lie algebra of leaf preserving vector fields X(_P) but the Lie algebra of foliation preserving vector fields £(7) (Le .vector fields which generate local flows mapping leaves into leaves), it is no longer modular, since vector fields of £(.P) have transversal parts "constant" along leaves (see Lemma 1 in Chapter 4), so that the general "modular" methods fail.It makes the description of isomorphisms a little bit harder and only partial solutions for smooth classical (regular) foliations were found (cf.Fukui and Tomita [4] and Rybicki [19]).In this note we present a purely algebraic approach to this question and preve the Shanks-Pursell type result for the Lie algebras of vector fields preserving foliations not only for classical, but also for a large caass of generalized foliations.The result includes as well smooth as real-analytic and holomorphic cases .

Statement of the main result .
Since we shall be interested mainly in certain algebraic properties of the objects in question, we shall deal at the same time with finite dimensional manifolds M of different classes of smoothness C: C = C°°, C', l-L, where C°°denotes the classical smooth case, C'-analytic case, and 7-l denotes holomorphic case for Stein manifolds .For details we refer to [3] .For instante, C(M) is the algebra of caass C functions on the manifold M of caass C. Note that the algebras C°°(M) and Cw (M) are real and the algebra 7-L(M) of holomorphic functions en the Stein manifold M is complex .It is well-known that the corresponding Lie algebra X(M) of all caass C vector fields can be regarded as the Lie algebra of derivations of C(M) (in analytic cases we refer to [6]) .Definition.A generalized foliation .P = {te% EA en a manifold M of caass C is a partition of M into connected submanifolds M = UaEA .Fa which are exactly the orbits of compositions of flows generated by local vector fields of caass C tangent to the leaves of P.
In other words, leaves of a generalized foliation consist of maximal integral manifolds of an involutive generalized distribution P C TM of caass C which is invariant with respect to the flows of local vector fields with values in P (cf.[17]).Note also that in analytic cases the assumption about invariance is superfluous (cf.[13]) .Generalized foliations will be called further simply foliations, while the classical foliations will be called regular foliations, since the dimension of leaves is constant.Denote by X(_P) the Lie algebra of vector fields tangent to the leaves of .`F(leaf preserving vector fields) .This Lie algebra is modular, Le. it has the natural structure of an C(M)-module .Consider the Lie normalizer Remark 1 .A "standard" normalizer consists of foliation preserving vector fields, but in the smooth case it can be larger.Consider for instante the C°°-foliation F of R consisting of one 1-dimensional leaf (-oo, 0) and non-negative points being 0-dimensional leaves .It is clear that N(.F) = X(R), but not all smooth vector fields (e.g.generating translations) are foliation preserving.
Given p E M denote by the .Fp the leaf containing p and by .F(p) the tangent space Tp-Fp.For points of M the obvious equivalente relation " ti " means that q E Fp (p and q belong to the same leaf of Y) . Definition.
We call a foliation F finitely generated if the C(M)module X(Y) is generated by a finite family of vector fields which span F(p) at every p E M and non-singular if the leaves of F are at least one-dimensional .
It is not hard to prove the following (cf .[7] or [8]) .
Theorem 1 .Regular foliations of class C are finitely generated.
Note that all foliations generated by hamiltonian vector fields of local Lie algebras of Kirillov (cf.[11]) or, in other terminology, generated by Jacobi or Poisson structures (cf.[9]) are finitely generated .
Our main result is the following .The above result may be reduced to the following .
It suffices to apply Theorem 5.5 of [5] or Theorem 3 .2 of [16] to see that every isomorphism P : X(JF1) -> X( .F2) is implemented by a foliation preserving diffeomorphism .
Remark that non-regular foliations have not to be finitely generated as for example the foliation from Remark 1 or the C°°-foliation F = {R \ {0, 1, 2 , 3, .. .},{ 0 }, {1}, { 2 }, { 3 }, . ..} of R, but this assumption seems to be rather technical .However, in analytic cases we do not oven know whether the Lie algebra X(.T) is not trivial .We believe yes and due to the Theorem A of Cartan and analogous result of Tognoli [18] for real-analytic case it suffices to prove the following .
Conjecture.The analytic sheaf of germs of analytic (real or complex) vector fields tangent to a given analytic foliation is locally finitely generated .
3. Algebraic preparation .Throughout this section A denotes an associative commutative unital algebra over a field k of characteristic :~2 and X denotes a subalgebra of the Lie algebra Der(A) of derivations of A with the commutator bracket .Note that Der(A) is an A-module in the obvious way so that we have the identity for all X, Y E Der(A), f E A. Definition.We call X C Der(A) modular if X is an A-submodule of Der(A) and strongly nowhere vanishing if X(A) = A (where clearly X(A) = span{X (f) : X E Xand f E A}) .
The last property may be written in the homological way as Ho(X, A) = 0. Our standard model is of course A = C(M)-the algebra of caass C functions on the manifold M and X = X (.F)-the Lie algebra of caass C vector fields on M preserving leaves of the foliation F. The Lie algebra X(.F) is clearly modular and it is strongly non-singular if and only if it contains a finite number of vector fields with no common zeros (cf.[5]).This is the case of F being finitely generated and non-singular .
Remark that a modular Lie algebra of vector fields is (in a little bit more general setting) called sometimes also a differential Lie algebra (cf .
By M(A) denote the set of all finite codimensional maximal ideals of A and by M (X) the set of all finite codimensional maximal Lie subalgebras of X. Proof: Since X(A) = A, there are Xl, .. ., X, ,, E X and fI, ..., f7 E A such that r_'j Xj(fi

It is well-known that in case of
) as the kernel of the adjoint representation of L in X/L.Put W := {f E A : fXi E U and ffiXi E U,¡ = 1, . .., m}.
Since dim(A/W) <_ 2m(k + k2) = n(k), J := AW = span{gf : g E A, f E W} is an ideal of A of codimension < n(k) .For X E X and f E W the brackets [f Xi , fiX] and [X, f fiXi] belong to L, so calculating their sum with the help of (3.1) we get fXi(fi )X + fX (fi )Xi E L.
Since E ' 1 Xi(fi) = 1, the ideal generated by {(Xi(fi))2 : i = 1, ...,m} equals A and (3 .4)implies finally JX C L. The ideal I := {f E A fX C L} includes J and is therefore of codimension <_ n(k) .Clearly IX C L and L is a Lie algebra, so in view of (3.1) and hence L(I) C I .This in turn implies L(A) C I as shows Lemma 4.2 in [5] .
Corollary 1.Every L E M(X) is of the form Xj for a unique J E M(A) .
The proof is straightforward .Proof. .According to Corollary 1 and our remarks at the beginning of this section, V(La ) = {X(.F)pi : i = 1, ...,r} for some p1, .. . .pr E M and all a E A. Due to Corollary 2, we have the inclusion J' (k) X(, T) C naEA La for J = n2-1 J(pi) being the ideal of functions vanishing at p1, . .., pr .This ideal is Cearly finite codimensional and one can see that J,(k) is finite codimensional as well (cf .Note 1.4 in [6] or [10] for C°°case) .The C(M)-module X(.97) is finitely generated, so Jn(k)X(.F) and hence n l EA La is finite codimensional .Recall that N(.F) is the Lie normalizer of X(Y) in X(M) and that "~" is the equivalente relation given by F. Lemma 1 .If p -q, then N(.7)P = N(.F)e .
In other words, a vector field of L is tangent to the whole leaf if at one point.
Lemma 2 .If L is a Lie ideal of N( .F) with L(p) ¢ _ F(p) for certain p E M, then for any qp we have Y(q) C L(q) and V(Lq ) = V(Lp), where V(L, q) = {K E M(L) : Lq C K} .Moreover nge .~,pLq is infinite codimensional in L .
Proof.. Take Y E L with Y(p) 1 .7(p) .According to Lemma 1 we have Y(q) 0 .F(q) for any q E Fp .Civen a finite set {q1, . . . qs} of points of Fp we can find f E C(M) vanishing at ql, ..., qs and such that Y(f)(gi) = ai, i = 1, .. .,s,are arbitrarily chosen .For any X E X(F) the vector field [Y, fX] belongs to L. Since [Y, fX](qi) = ajX(gi), the intersection ngEFP Lq is infinite codimensional and .F(q) C L(q) for any q E .Fp.Take now K E V(Lq) .We claim that LQ C K. Indeed, LQ is a Lie subalgebra including Lq and Lq ~K means that .7(q)¢ K(q) .K(q) C .97(q) would imply that K C Lq, but by the assumption L9 É L, so by the maximality of K we would have K = Lq and Y(q) C K(q) .Assume therefore that there is Z E K, Z(q) q .F(q) .Since L is a Lie ideal of N (.F), [Y, fX] E Lq and hence [Z, [Y, fX]] E K for any X E X (.F) and any f E C(M) vanishing at q and such that Y(f)(q) = 0. Chosing such an f satisfying additionally Z(f)(q) = 0 and Z(Y(f))(q) = 1 we have [Z, [Y, fX]](q) = X(q) and hence .77(q)C K(q) .Therefore V(Lq ) =
A = C(M) we have M(A) -M, where the correspondence is given by ME)p-J(p)={f EC(M) :f(p)=0}EM(A) (cf.[5, Proposition 3.5]) .ForICAputXI :={XEX :X(A)CI},V(I) :={JEM(A) : I C J}, ¡ := nJEV(z) J, and In := span{f i . . .fn : fi, . .., fn E I} .For L C X put V(L) := {K E M(X) : L C K} .Due to [5, Theorem 5 .1]elements of M(X(.F)) (.F finitely generated and non-singular) are of the form X(_F)p := {X E X(.F) : X(p) = 0}for certain p E M. However, for our purposes we shall need a little bit stronger result .Its algebraic version is the following .Theorem 4. Let X be a modular strongly non-singular Lie subalgebra of Der(A) .Then for any k = 1, 2, 3, . . .there exists n(k) such that for any Lie subalgebra L of X of codimension < k we have IX C L C Xp for an ideal I of A of codimension < n(k) .
Proof.Take L as above.According to Theorem 4 there is n(k) and an ideal I of A of codimension <_ n(k) such that IX C L and L(A) C I. Since V(L) C {Xj : J E B} implies V(L(A)) C B, we have nB C I. Because of the codimension of I, the descending series A D (I+I) D (I+ (Í)2) D . . .stabilizes at at most n(k)-th step, so I+(I)n( k ) = I+(I)n(k)+1 and by the Nakayama's Lemma (I)n(k) C I, Le. (n B)n(k)X C IX C L .a Theorem 5 .Given finitely generated non-singular foliation ., on a manifold M of class C and a positive integer k the intersection naEA La of a family of at most k codimensional Lie subalgebras of X( .97) is finite codimensional if V(La ) =V(Lp) for all cx,~3 E A .

4 .
Proof of Theorem 3 .Suppose now that .F is a finitely generated non-singular foliation on a manifold M of class C. For any Lie algebra L of vector fields on M and any p E M denote Lp := {X E L : X (p) = 0}, Lp := {X E L : X(p) E _F(p)}, L(p) := {X (p) : X E L} .