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The m-Derivations of Analytic Vector Fields Lie Algebras | OMICS International
ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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The m-Derivations of Analytic Vector Fields Lie Algebras

Randriambololondrantomalala P*

Department of Mathematics and Computer Science, Faculty of Science, University of Antananarivo, BP 906, 101- Ambohitsaina Antananarivo, Madagascar

Corresponding Author:
Randriambololondrantomalala P
Department of Mathematics and Computer Science
Faculty of Science, University of Antananarivo, BP 906
101- Ambohitsaina Antananarivo, Madagascar
Tel: +261331246249
E-mail: [email protected]

Received date: November 23, 2015; Accepted date: December 17, 2015; Published date: December 19, 2015

Citation: Randriambololondrantomalala P (2015) The m-Derivations of Analytic Vector Fields Lie Algebras. J Generalized Lie Theory Appl S2:002. doi:10.4172/generalized-theory-applications.S2-002

Copyright: © 2015 Randriambololondrantomalala P. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

We consider a (real or complex) analytic manifold M. Assuming that F is a ring of all analytic functions, full or truncated with respect to the local coordinates on M; we study the (m ≥ 2)-derivations of all involutive analytic distributions over F and their respective normalizers.

Keywords

m-derivations; Analytic vector fields Lie algebras; Distributions; Generalized foliations; Stein manifolds; Compact holomorphic manifolds; Chevalley-Eilenberg’s cohomology; Compactly supported vector fields

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 imagel (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 image , where all Xi are in A. On the one hand, we have studied the m-derivations of polynomial vector fields Lie algebras on imagen 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 image3, the Lie image-algebra is spanned by image and let’s define the image−linear map D by image 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 χ (M) 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 χ (M) 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 image and on image, 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 made by Cartan [6] in a Stein case or pass into Grabowski’s conjecture cf. [9]. We interpret our results in Chevalley-Eilenberg cohomology sense when m = 2.

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 χ (M ) the Lie algebra of analytic vector fields on M and F(M) the ring of analytic functions on M. Throughout this paper, we take an atlas in which every chart are connected. Then, the open subset of a chart U where a non-trivial subset of χ (M ) 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 image , we get image This D is said inner on a Lie algebra image containing A, if D is a Lie derivative with respect to an element of image. Recall us another basic definition cf. [11].

Definition 1.1. A complex manifold M is a Stein manifold, if we have simultaneously the three following conditions: For every x ≠ y, both in M; there is a holomorphic function f over M such that f (x) ≠ f (y). For all x ∈ M, it exists n holomorphic functions (fi) over M such that dfi are linearly independent over image on x. If K is a compact set of M, the following set is compact (holomorphic convexity of M)

image

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 Cc relative to a Stein manifold are the following,

Proposition 1.2. A complex analytic manifold M is compact iff Cc is non trivial, particularly if M is a Stein holomorphic manifold, Cc is trivial.

Proof. It’s obvious that if M is compact, then image and Cc is not trivial. Conversely, suppose that M isn’t a compact set and there is X ∈ Cc 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 setimage 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 χ (M) , that is to say that the set of all imageuch that image , and image . We can choose a connected domain Ui of a chart. If we suppose that it exists an open set Oi in Ui where Ω vanishes, then image by analyticity. Otherwise, every open set in Ui contains an element of B. So, B∩Ui is dense over Ui. Moreover, the collection of Ui forms an atlas of M, then B is dense over M. The set B is an open analytic submanifold of M. Particularly, B is a Stein cf. [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 LF the one of all foliation preserving vector fields. It is known that the normalizer N in χ (M) of Ω contains LF cf. [10]. Hence, the restriction of the foliation in B is non singular.

Proposition 2.1. The centralizer of Ω vanishes and the derivative ideal of Ω coincides with Ω itself.

Proof. We say that image is in the centralizer of Ω if image; and the derivative ideal of denoted by [Ω, Ω] is the Lie algebra spanned by all brackets of two elements of Ω. Suppose there is an non zero element X of the centralizer, we have image , for all , 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.

Theorem 2.2. (Hartogs [16]) Let be t ≥ 2 and D be a bounded domain in imaget. 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.

Theorem 2.3. (Riemann extension theorem) Let U be an open set in image and z0 ∈ U. If image is holomorphic function such that z0 is a removable singularity of f, then f can be extended into an unique holomorphic function image in U where image.

Theorem 2.4. In holomorphic case, all m-derivations of Ω, LF, and of N are Lie derivatives with respect to elements belonging to N. In real analytic one, we have the same results if B = M.

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.

Definition 2.5. The complexification of a real analytic manifold M is a holomorphic manifold image such that there is a real analytic embedding f: M → image where image has a holomorphic atlas image and image . We have a Stein complexification if image is Stein.

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 image.

The following complexification of a Lie subalgebra image of the real analytic vector fields Lie algebra of M is in the following sense: if M can be embedded in a holomorphic manifold image, the complexification image of image in image is such that image

Theorem 2.7. If the complexification of Ω in a Stein space T is still an involutive distribution, then the first result of the Theorem 2.4 holds in real analytic case.

Proof. We use the complexification of M on a Stein space T. Consequently, let be image the complexification over T of Ω. Recall that image is an involutive distribution over T where its normalizer on image is denoted by N0. So, all m-derivations of Ω, of LF, and of N have their complexified m-derivations over T on respectively image , the Lie algebra of all foliation preserving vector fields image on T and N0. By the results of Theorem 2.4, these last m-derivations are Lie derivatives with respect to elements belonging to N0. We can affirm that image and image by image . Thus, we have the same result as in the first part of Theorem 2.4.

By definition, the first space of Chevalley-Eilenberg’s cohomology of a Lie algebra image denoted by image is image with Der(image) the Lie algebra of all derivations of imageand image the set of inner derivations of image.

Throughout this paper, we suppose that all hypothesis of Theorem 2.7 are satisfied or image = M, in real analytic case.

Following ideas of Theorem 3.7, Corollary 2.14 and Remark 2.15 of literature of Randriambololondrantomalala [10], we state

Corollary 2.8. The first space of Chevalley-Eilenberg’s cohomology of Ω, image, and of image is respectively isomorphic to the following respective Lie algebras, image/Ω, image/image {0}.

Remark 2.9. By Theorem 2.4, we deduce Theorem 3.2 and 4.1 of studies of Grabowski [5].

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 image over the ring F0 of real or complex functions depending only on imagewith 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,image such that image and S is locally of rank n − k (0 < q ≤ n). It is easy to check thatimage 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 image as a vector space. We consider the closed 1−differential forms as a vector space. We consider the closed 1−differential forms αi and wi over a (n − k)-dimensional distingUished connected chart of the generalized foliation generated by S, where image and an m-derivation of image such that image and image (S in this chart is image, image 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 image) are a sum of a Lie derivative with respect to one element of image 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

Corollary 3.2. When the rank of S is a positive constant n − k, the first spaces of Chevalley-Eilenberg’s cohomology of Ω and of N are both isomorphic to image with HR (B) is the de Rham cohomology of foliation basic forms of M.

As we know, we can split the above Ω into a semi-direct sum of Lie algebras image and image as in studies of Randriambololondrantomalala [8], where they are modules on the ring F0(M) of constant functions over the leaves relative to the above generalized foliation. We notice that image is spanned by S on F0(M). We can reason on a distingUished chart U with the coordinates image . The image is the set of all analytic functions depending only on image , image is spanned by image and image by image on image.

Now, we discuss the m-derivations of image . The normalizer N1 of this Lie algebra can be written as a direct sum of Lie algebras image , where image is the centralizer of image in image and the center of image is zero (image is locally spanned by image on the ring of all analytic functions depending only on image. By a similar argument of Nakanishi [18], we deduce that all m-derivations of N1 are a direct sum of those of image and of image. By Theorem 2.4, it’s clear that

Theorem 3.3. Each m-derivation of image (resp. of image1) is a Lie derivative with respect to an element of N1.

The normalizer image is locally the sum of the F0(U) -module spanned by image and a vector space generated by image . That is to say, its normalizer is N. In addition, its centralizer is image itself. Because of imageor image is nilpotent of order 1, we obtain easily

Theorem 3.4. Every endomorphism of image is an m-derivation of image.

So, it’s immediate that

Corollary 3.5. The first space of Chevalley-Eilenberg’s cohomology of image, image and of N1 are respectively isomorphic to the following respective Lie algebras, image , image, , {0} .

Let’s consider image the Lie subalgebra of Ω, spanned by Xi over a ring image . That is to say, F is locally the set of all analytic functions depending only on image where 0 < l < k + 1 (resp. k + 1 < l < n + 1). When F = F(M) (resp. image ), it is a special case of Lie algebras defined in Theorem 2.4 (resp. in Theorem 3.1) when the submodule is generated by Xi. In the distingUished local coordinates, image is spanned by image over F. The normalizer image3 of image coincides with the sum of image and image where the element of this last one is locally the following analytic vector fields image . In the first case, we can adapt Theorem 2.4 because all analytic functions depending on image where image , are in the base ring of image . In the following case, it is easy to see that Theorem 3.1 can be adapted to image . Thus

Theorem 3.6. In the first case, every m-derivation D of image is a Lie derivative with respect to a image3's element; in the second, it is a sum of a Lie derivative of an element of image3 and a D(α,w) analogous to that of Theorem 3.1. In addition, the corresponding extension theorems hold.

Corollary 3.7. The first space of Chevalley-Eilenberg’s cohomology of image is respectively isomorphic to image in the first case;

image

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.

Remark 3.8. On image3, we consider the Lie image-algebra A spanned by image image

Lemma 2.3 of literature of Randriambololondrantomalala [10] is not true for A, so the arguments proposed in the proof of Theorem 3.6 of Princy [7] don’t hold in this situation.

Whereas, let P be a Lie subalgebra consisting locally of polynomial vector fields in χ (M ) , where the Euler vector field and all constant vector fields are locally in P. Especially, M is not supposed to be a Stein in the holomorphic case. Let’s recall a well known theorem,

Theorem 3.9. (The maximum principle) [12] Let be M a connected holomorphic manifold and f a holomorphic function on M such that image , where , where z0∈ M for all z ∈ M; then f is a constant function.

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 image.

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

Theorem 3.10. When m is even, all m-derivations of image (resp. of the normalizer image of P in χ (M ) ) are a Lie derivative with respect to one and only one vector field belonging to image(resp. to the normalizer of image in χ (M ) ). If m is an odd number, they are sum of a Lie derivative with respect to one element of image (resp. of the normalizer of image) and an m-derivation of local homogeneous degree -2 of image.

So, taking into account: the vanishing of the centralizer of image 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 image, of image and of N is respectively isomorphic to the following respective Lie algebras image / image, N / image, {0}, where N is the normalizer of image.

Illustrations

Some illustrations of our theorems are given in this section.

Example 4.1. It’s clear that Theorem 2.4 works well on the Lie algebra of all analytic vector fields χ (M) , that is to say, all m-derivations of χ (M) are Lie derivatives by elements of χ (M) . We can define the Lie algebra of compactly supported real analytic vector fields C and this theorem holds for this last one. In particular, image for a non-trivial image. More, image if and only if M is compact. Obviously, we can use the above cited theorem on the Lie algebras of vector fields relative to a generalized foliation over M. We can cite some well known Stein spaces, imagen, an open poly-disc in imagen , non-compact Riemann surfaces, ... and bUild our results in these.

Example 4.2. Let be image a holomorphic compact connected manifold. It’s not a Stein manifold nor a submanifold of imageu for any u, it’s a compact Riemann surface. We choose the modified stereographic coordinates over this manifold. The image has an atlas composed by two charts (U, z1) and (V, z2) with the overlap image in U ∩ V. We remark that the Lie algebra image of vector fields on M spanned over image by Y1, Y2 and Y3 is the one of all polynomial vector fields in image, where

image image
image . By Theorem 3.10, all m-derivations of image are Lie derivatives with respect to a vector field in image itself. That is to say, if D is an m-derivation of image defined by image, image, image we have image

When we look at image as a real analytic manifold, we set the charts image and image with the overlap map

image

We set the real analytic vector field image on image and the Lie algebra A of real analytic vector fields which commute with Y3. This image consists of real analytic vector fields Y such that

image

where in the U ∩ V,

image and

image with Fk are arbitrary convenient functions of one variable. So, we can apply all theorems in Section 3 to image. Particularly H1(image) = {0} and image

Example 4.3. Indeed, Theorem 3.10 can be applied to a polynomial vector fields Lie algebra on the real analytic manifold imagen or the Stein manifold imagen having the corresponding hypothesis.

Example 4.4. We set the Lie algebra image over the Stein manifold image3 spanned over imageby image . The normalizer of image is image0= image+ R, where R is the space over image generated by image. It is permit to use Theorem 3.10 and when m is even, every m-derivation of image is inner on image0. If m is odd, the m-derivation is a sum of an inner derivation on image0 with a image-linear map D defined by image which is zero otherwise (α ∈ image). Moreover, all m-derivations of image0 are inner for all m ≥ 2. So,

image and image.

Remark 4.5. In the following example, Theorem 3.10 cannot be applied. We take the 2-torus image, which is a holomorphic connected compact manifold cf. [20], it’s not a Stein. All overlap maps are translations, that is to say, they are holomorphic. We can define globally the Lie algebra of all constant vector fields image on M and find that image is the Lie algebra of all holomorphic vector fields over M. All endomorphisms of each Lie subalgebra of image, which is inevitably nilpotent of order 1, are m-derivations of this subalgebra. The normalizer of this subalgebra or its centralizer is the Lie algebra of all vector fields over image. But image and image since image in smooth cases.

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