The m-Derivations of Analytic Vector Fields Lie Algebras

We know several embedding theorems in differential geometry, some of them are of John F. Nash in Riemannian manifolds [1,2], of Whitney [3] in differentiable manifolds and of Grauert in analytic manifolds cf. [4]. They make easy certain study on a differentiable manifold. Here, we are interested to a real or complex analytic n-dimensional manifold M and let F(M) be the ring of all analytic functions on M. We know that these manifolds can be considered as smooth manifolds. But certain property on a smooth manifold cannot be true on M, for example the global representation of a smooth function germ theorem. Grabowski had this problem when he studied derivations of the real or complex analytic vector fields Lie algebra cf. [5] and he used Stein manifolds to avoid technical difficulties in them. Here, we examine not only the derivations but the (m ≥ 2)-derivations (generalization of derivation’s notion) of a Lie subalgebra of the real or complex analytic vector fields Lie algebra on M, using Lie algebra tools. In advance, we state that the considered Lie algebras have enough sections more than constant ones in the Lie algebra of all analytic vector fields. Then, we consider only Stein spaces unless expressly stated in a complex analytic case. In the real analytic one, we don’t need more hypothesis because of the imbedding theorem of Grauert and Cartan theorems [6]. More precisely, any real analytic manifold can be considered as a closed submanifold of a certain l (a ” real Stein manifold”). Now, an m-derivation of a Lie algebra A is a linear map from A to itself which is distributive on the brackets


Introduction and Preliminary
We know several embedding theorems in differential geometry, some of them are of John F. Nash in Riemannian manifolds [1,2], of Whitney [3] in differentiable manifolds and of Grauert in analytic manifolds cf. [4]. They make easy certain study on a differentiable manifold. Here, we are interested to a real or complex analytic n-dimensional manifold M and let F(M) be the ring of all analytic functions on M. We know that these manifolds can be considered as smooth manifolds. But certain property on a smooth manifold cannot be true on M, for example the global representation of a smooth function germ theorem. Grabowski had this problem when he studied derivations of the real or complex analytic vector fields Lie algebra cf. [5] and he used Stein manifolds to avoid technical difficulties in them. Here, we examine not only the derivations but the (m ≥ 2)-derivations (generalization of derivation's notion) of a Lie subalgebra of the real or complex analytic vector fields Lie algebra on M, using Lie algebra tools. In advance, we state that the considered Lie algebras have enough sections more than constant ones in the Lie algebra of all analytic vector fields. Then, we consider only Stein spaces unless expressly stated in a complex analytic case. In the real analytic one, we don't need more hypothesis because of the imbedding theorem of Grauert and Cartan theorems [6]. More precisely, any real analytic manifold can be considered as a closed submanifold of a certain  l (a " real Stein manifold"). Now, an m-derivation of a Lie algebra A is a linear map from A to itself which is distributive on the brackets On the one hand, we have studied the m-derivations of polynomial vector fields Lie algebras on  n in studies of 7. Randriambololondrantomalala [7], an important Lie subalgebra of analytic vector fields, we found that Lie algebras of derivations are different to those of (m > 2)-derivations. One can see the following example, on  3 , the Lie -algebra is spanned by 2 , , , , , ( ) x and let's define the −linear x which is zero otherwise. It's clear that D is not a derivation, but a 3-derivation. On the other hand, all m-derivations of a distribution over the full or truncated rings of smooth functions on a differentiable manifold in literature of Randriambololondrantomalala [8], are derivations. These facts lead us to ask if a distribution Lie algebra on an analytic manifold has results as the one or the other above results. So, we will divide our paper into three parts. First, we take a real or complex analytic involutive distribution Ω over M. That is to say, Ω is a F(M)-submodule of the analytic vector fields Lie algebra ( ) M χ on M. We can find some examples of these distributions and the interests for studying their derivations in literature of Grabowski and Cartan [5,6]. Here, we find the Ω's centralizer and the derivative ideal of Ω. We can say also that the normalizer of Ω is a Lie subalgebra of analytic vector fields. In addition, we find out that all m-derivations of Ω (resp. of the normalizer of Ω) are inner with respect to a normalizer's vector field (resp. are inner). Second, assuming that Ω is an involutive distribution on M over a subring F of F(M), namely an F-submodule of ( ) M χ stable by the vector fields bracket, where F≠ F(M). One can consider a system of commuting vector fields on M as in studies of Randriambololondrantomalala [8] and all distribution Lie subalgebras of the Lie algebra of analytic vector fields which commute with this system. The normalizer of Ω is an analytic vector fields Lie algebra and contains locally all constant vector fields and Euler's vector field. But in general, we can't use the reasoning by Randriambololondrantomalala [7] to characterize m-derivations of Ω. We make more explicit all m-derivations of Ω and of some of its normalizer. Whereas, in the end, we discuss the Lie algebras of holomorphic vector fields, especially when the holomorphic manifold is not a Stein one, and Lie algebras of locally polynomial vector fields on an analytic manifold M. Their m-derivations as well as their normalizers can be characterized by using some results of Randriambololondrantomalala [7]. Therefore, we have found the m-derivations of all distributions over a set of full or truncated analytic functions with respect to the local coordinates on M. When m = 2, we deduce from our results some [5]'s theorems. Third, we can apply our theorems on Lie algebras of real or complex analytic vector fields on M, of generalized foliation on M cf. [9], a Lie subalgebra of analytic vector fields on  2 and on  2 , Riemann surfaces, etc. Relations between the Lie algebra of compactly supported vector fields and the compactness of M are discussed. Moreover, we emphasize the extensions of our theorems when the studied distributions are singular, by using the complexification of a real analytic manifold, Hartogs and Riemann extension theorems. Of course, in these circumstances, we can use theory of coherent sheaves Following the above notations, let M be a real or complex analytic n-dimensional manifold. In complex case, we regard a Stein manifold unless special mention. We denote by ( ) doesn't vanish, is dense on U (non-trivial means different to {0}). We can use certain results of Randriambololondrantomalala [7,8] because in the proofs of theorem of these papers we consider only analytic functions (polynomials, exponentials). In the same way, we don't need partition of the unity to make global some local results cf. [10]. In all sections of this article, we set an integer m ≥ 2, recall that D is an m-derivation of a Lie algebra A if for This D is said inner on a Lie algebra  containing A, if D is a Lie derivative with respect to an element of . Recall us another basic definition cf. [11].
From these assertions, every local ring of holomorphic functions around x ∈ M is spanned by holomorphic functions on M cf. [12].
Some results of the Lie algebra of compactly supported vector fields C c relative to a Stein manifold are the following, and C c is not trivial. Conversely, suppose that M isn't a compact set and there is X ∈ C c such that K = Supp(X) ≠ ∅. We can consider K ≠ M because M is not compact. Then, we have the nullity of X in the open set K ≠ ∅  . By analyticity, X vanishes in whole M. Hence, we have a contradiction about K ≠ ∅ and we obtain M is a compact set. It's clear that a Stein space is never a compact set by definition, then its Lie algebra of compactly supported vector fields is trivial.

The m-derivations defined by distributions on F(M)
Let Ω be a non-trivial involutive analytic distribution over the analytic functions ring on M. Let N be the normalizer of Ω in ( ) M χ , that is to say that the set of all X Ω ⊂ Ω , and  [13]. Thus, every vector field defined over B admits a continuous extension on M, and if this last one is analytic, then it's necessarily an element of the normalizer of Ω. We use this last fact when we deal with extension theorems.
We know by literature of Nagano's [14] result that Ω is integrable, then it yields a generalized foliation F on M cf. [10]. So, Ω is the Lie algebra of tangent vector fields to the foliation and L F the one of all foliation preserving vector fields. It is known that the normalizer N in of Ω contains L F cf. [10]. Hence, the restriction of the foliation in B is non singular.
, for all f ∈ F(M). It's not possible in a Stein manifold or in a real analytic manifold if X doesn't vanish identically over M and if Ω ≠ {0}. Along with this result, we can adapt the proof of Proposition 2.28 of studies of Randriambololondrantomalala [15] and assert that [Ω, Ω] = Ω.
Let's recall an Hartogs's extension theorem and Riemann extension theorem. [16]) Let be t ≥ 2 and D be a bounded domain in  t . In addition, K be a compact subset of D such that D − K is a connected set. Then all holomorphic functions f over D − K can be extended holomorphically to D. Proof. We can prove this assertion over B by Theorem 2.1 of studies of Randriambololondrantomalala [8] using Proposition 2.1 and partially Theorem 3.7 of literature of Randriambololondrantomalala [10]. For the corresponding extension theorem over M, we adopt the following arguments. We know that B is dense over M, then the restriction of B in each domain of a chart U is dense over U (U is a bounded set). The complement of this B∩U in U can be considered as a compact set of the chart such that B∩U is connected. In holomorphic case, when n ≥ 2, we use Hartogs's theorem in a domain of the chart, so the extension theorem over M holds. If n = 1, we know by the isolated zeros principle that the domain of chart contains only a finite number of zeros in the corresponding restriction of B. By continuity at these zeros, which are removable singularities, the Riemann extension theorem can be used. Of course, if B = M in real analytic situation, the extension theorem is applicable.

Theorem 2.3. (Riemann extension theorem) Let U be an open set in
The next theorem is due by Grauert cf. [4,12].
Theorem 2.6. Every real analytic manifold has a Stein complexification and can be analytically properly embedded into an Euclidean space N  .
The following complexification of a Lie subalgebra  of the real analytic vector fields Lie algebra of M is in the following sense: if M can

The m-derivations associated to a distribution over a subring of F(M)
Let be an atlas of M such that Ω is locally spanned by x ≤ ≤ with respect to the atlas (where 1 ≤ k < n). We can consider Ω to be a Lie algebra which commutes with a system S of commuting vector fields by the usual bracket. That is to say, and S is locally of rank n − k (0 < q ≤ n). It is easy to check Ω Ω Ω because of Randriambololondrantomalala's [8] result. So with the same reason, every m-derivation of Ω is local. Moreover, the normalizer N of Ω is locally isomorphic to ( , or ) l n k Ω ⊕ − g   as a vector space. We consider the closed 1−differential forms α i and w i over a (n − k)-dimensional distinguished connected chart of the generalized foliation generated by S, where = 1, , , F A denotes a module spanned by A over a ring F) cf. [8]. We have omitted all singular charts of the foliation because the open set R of all regular points is dense over M cf. [10], we have no problem for the extension of our results from R towards M as in the previous section. By adapting Theorem 3.12 of literature of Randriambololondrantomalala [8], we obtain easily Theorem 3.1. All m-derivations of Ω (resp. of N) are a sum of a Lie derivative with respect to one element of N and a derivation D (α,w) as denoted before (resp. are similar to m-derivations of Ω).
Hence, adopting the arguments of Theorem 3.19 of studies of Ravelonirina [17], we hold the following As we know, we can split the above Ω into a semi-direct sum of Lie algebras 1 S Ω and 2 S Ω as in studies of Randriambololondrantomalala [8], where they are modules on the ring F 0 (M) of constant functions over the leaves relative to the above generalized foliation. We notice that 2 S Ω is spanned by S on F 0 (M). We can reason on a distinguished chart U with the coordinates So, it's immediate that x ≤ ≤ where 0 < l < k + 1 (resp. k + 1 < l < n + 1). When F = F(M) (resp. in the other one if S has a constant rank (⊕ is a module direct sum and B′ is the set of the corresponding foliation basic forms of M).
When we regard all the above normalizers on a distinguished chart, they contain locally all constant fields and Euler vector field. So, we ask one question: could we adapt Theorem 3.6 and Theorem 3.9 in [7] to these normalizers? The following remark shows us that this argument is false. One consequence of the maximum principle is the following, if the holomorphic manifold M is compact, every holomorphic function on M is constant in every connected component of M. We know that M is locally connected, then each function over M is locally constant. Therefore, it's clear that if M is a compact and connected holomorphic manifold, the ring of all holomorphic functions on M is the complex constant functions ring. It's confirm that results of the following theorem complete our study about an involutive analytic distribution when F(M) is reduced to .
By adapting Randriambololondrantomalala's [7] theorems and taking account that the vector field found in the proof of Theorem 3.6 of Princy [7] is analytic, it follows that So, taking into account: the vanishing of the centralizer of P cf. [19] p.91; both the proofs of Theorem 2.12 of Ravelonirina [19], Corollary 3.12 of Randriambololondrantomalala [7] and Theorem 3.7 in literature of Randriambololondrantomalala [10], we obtain Corollary 3.11. The first space of Chevalley-Eilenberg's cohomology of P, of  and of N is respectively isomorphic to the following respective Lie algebras  / P, N / , {0}, where N is the normalizer of .
We set the real analytic vector field