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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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The m-Derivations of Distribution Lie Algebras

Princy Randriambololondrantomalala*

Département de Mathématiques et Informatique, Faculté des Sciences, Universitéd’Antananarivo, Antananarivo 101, Madagascar

*Corresponding Author:
Princy Randriambololondrantomalala
Département de Mathématiques et Informatique
Faculté des Sciences, Universitéd’Antananarivo
Antananarivo 101, BP 906, Madagascar
Tel: +261 20 22 326 39
E-mail: [email protected]

Received date: November 14, 2014; Accepted date: March 07, 2015; Published date: March 16, 2015

Citation: Randriambololondrantomalala P (2015) The m-Derivations of Distribution LieAlgebras. J Generalized Lie Theory Appl 9:217. doi:10.4172/1736-4337.1000217

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

Let M be an N-dimensional smooth differentiable manifold. Here, we are going to analyze (m>1)-derivations of Lie algebras relative to an involutive distribution on subrings of real smooth functions on M. First, we prove that any (m>1)-derivations of a distribution omega on the ring of real functions on M as well as those of the normalizer of omega are Lie derivatives with respect to one and only one element of this normalizer, if omega doesn’t vanish everywhere. Next,suppose that N= n + q such that n>0, and let S be a system of q mutually commuting vector fields. The Lie algebra of vector fields $\mathfrak{A}_S$ on M which commutes with S , is a distribution over the ring()0MFof constant real functions on the leaves generated by S. We find that m-derivations of $\mathfrak{A}_S$ are local if and only if its derivative ideal coincides with $\mathfrak{A}_S$ itself. Then, we characterize all non local m-derivations of $\mathfrak{A}_S$. We prove that all m-derivations of $\mathfrak{A}_S$ and of the normalizer of $\mathfrak{A}_S$ are derivations. We will make these derivations and those of the centralizer of $\mathfrak{A}_S$ more explicit.

Keywords

m-derivations; Vector fields lie algebras; Distributions; Commuting vector fields; Generalized foliations; Compactly supported vector fields; μ-projected vector fields; Nullity space of curvature

Introduction and Preliminary

Let m be a natural integer greater than or equal to 2. We recall that a m-derivation D of a Lie image -algebra imageis an endomorphism of image , such that for all image

image

image

image

image

This map is inner with respect to Lie algebra image if D equals to a Lie derivative with respect to image;if image , it is an inner m-derivation. A standard m-derivation D is a sum of derivations of imageand image -linear maps of imageinto the center of image such that image

Is it sufficient to study derivation of Lie algebras? What is the reason for studying the more general notion:” (m>2)-derivation”? In other words, can we find (m>2)-derivations of a vector fields Lie algebra which are not derivations? In [1], we found m-derivations all polynomial vector fields Lie algebras P on image where P contains Euler vector fields E and all constant vector fields. We remark that all these m-derivations are derivations when m is even. If m is an odd number, m-derivations are generally sum of derivations and m-derivations with homogeneous degree -2. Overimage , we can take a simple example where the Lie image -algebra is spanned by image and the image linear map D is defined byimage and vanishing otherwise. It is a 3-derivation, but not a derivation. In [2], some graded Lie algebra m-derivations are discussed. Here, we are interested in m-derivations of distribution Lie algebra on a N-smooth manifold M over an M-real functions ring. We know that all smooth vector fields can be locally approximated to polynomial vector fields, so we think that all results in [1] are naturally true in the case of distributions. But, the results which follow are different. The differential operator theory see [3] is the main tool throughout our proofs.

We denote by F(M) the ring of all real functions on M, image the vector fields Lie algebras over M(resp. over the tangent bundle TM).

At first, we consider an involutive distribution Ω over F(M). That is to say, Ω is a F(M)-sub-module of the module of all vector fields on M. Assuming that the open set image equals M, we are looking for characteristics of m-derivations of Lie algebras relative to Ω and applications of the obtained results on some remarkable distributions. We propose to prove that each m-derivation of Ω (resp. of the normalizer in image of Ω) is simply a Lie derivative with respect to one and only one normalizer’s vector fields (resp. is inner). These theorems can be extended where image is dense over M.

Secondly, let be N=n+q with n ≥ 1 and q>0, S a system of q nonvanishing vector fields which commute mutually. We know by results in [4] that S yields a generalized foliation on M. We assume that all leaves are regular and we notice that image, the ring of real smooth functions which are constant on the leaves over M. Let U be a p-dimensional adapted chart domain relative to the foliation and image,where image if p≥(resp. whereimage if p=0). Then, there are two modules over image spanned by imageand imagegenerated by image . These previous modules are Lie algebras such that image is equal to the semi-direct product of these two algebras: image for all distinguished U. We can say that image is a smooth distribution of M over image . Throughout this paper, we assume that this chart is ( p>0) -dimensional in the sense of the foliation, unless expressly stated. Our aims are to characterize all m-derivations of image , of the normalizer of image and of the centralizer of image in χ(m) . The corresponding work where S ={0}has been done in the previous section. Because of image ’s lower central series constancy, which coincides with module direct sum of image , our work on these m-derivations is non-trivial. The main results of this section are: all m-derivation is local iff the derivative ideal of image is image itself, which is equivalent to the fact that image has non-vanishing elements over the whole M. Moreover, all m-derivations of image or of the normalizer image of image are sums of a Lie derivative with respect to one image's element, of one local m-derivation which takes its value in image depending on two non-vanishing 1-differential forms over image , and of a non-local m-derivation of image. We give some recommendations for constructing all these non local m-derivations. In addition, all image -linear maps of image the centralizer of image into itself are m-derivations. We characterize all local image -endomorphisms of image in the case where all elements of S are densely supported or image is spanned by singleton, and those which are non local. It is well known that the open set of all foliation regular points is dense in M, then one can extend these results where the foliation is singular and if the above 1-forms prolongs smoothly on M.

Several applications of our results about Lie algebras relative to: all vector fields, all compactly supported vector fields, generalized foliations, µ -projected vector fields cf. [5], k-nullity space of connection curvature, and vector fields Lie algebras on TM commuting with Liouville vector fields cf. [6]; are given at the end of this paper.

Throughout this article, the Lie derivative with respect to image is denoted Lx. We adopt the Einstein index summation and suppose that all considered objects are smooth.

The m-derivations of Lie algebras attached to Ω

According the hypothesis about Ω, we can affirm that Ω is a Lie sub-algebra of image . A generalization of [7,8]’s theorems in the sense of derivation or triple derivation can be stated as follows:

Theorem 2.1. All m-derivations of Ω (resp. of the normalizer of in image) are Lie derivative with respect to one and only one vector field of the normalizer of Ω (resp. is inner).

Proof. Assume that image such that image . By Frobenius theorem, we find one chart image which contains x and local coordinate system image where image. Letting D be an m-derivation of Ω, we know that the Lie algebra spanned by brackets of all elements in Ω is the derivative ideal of Ω denoted by [Ω; Ω]. Local behavior of D can be proved by adapting one of Proposition 2.4 in [7] and using that the derivative ideal of Ω is Ω itself. Therefore image is an m-derivation of image . Let’s give image,as we know, image thenimage is uniquely determined, where each Di is differential operator over the trivial bundle image cf. [3]. Thus, if necessary we can write image ,where A, B are multi-indices corresponding to coordinates.

Let’s apply image to image, where xn= y. By definition of m-derivations and when f is replaced by monomials, we have:

- If image , image except for image

- If image, image except for image where image are free image are free xn monomials.

By reasoning as in the previous, we compute image. It is easy, using both the previous relation and the previous proof, to obtain the nullity of image. By coordinates translations, we can affirm that each image is a differential operator of order 0 and D0 is a sum of one of order 1 with one otherimage of order 2.

Computing in the same way as the previous calculus, image gives:

- for image.

- for image except image, image means 1 is in j-th rank).

By these results, image Consequently,

image is a derivation of the image -sub-module spanned by image. Applying D0 to image, we have image. By coordinate’s translations, we can write that image

We take Proposition 2.6 of [7] and we have image Follwing the arguments of the proof of Theorem 2.7 in [7], we end the demonstration of the first assertion of our theorem. Taking that the derivative ideal of Ω is Ω itself into account, we can adapt the proof of Theorem 2.12 in [7] to state the second assertion.

Remark 2.2. These theorems are correct if we consider OΩ to be dense over M and if the corresponding vector of the Lie derivative relative to the m-derivation cited by Theorem 2.1 can be smoothly extended towards M.

The m-derivations of Lie algebras defined by image

We know that nil potency of order image of image forces any endomorphism of image to be an m-derivation. To avoid this triviality, we prove that:

Proposition 3.1. The lower central series of image are constant and equal to the module image

Proof. The lower central series of image is determined by image image and for all p>0 ,

image cf.[1]. By Proposition 3.7 of [4], the derivative ideal of image is image. From the linearity of brackets, the Jacobi identity and the fact that image is an ideal of image, we deduce image. Then, we deduce the result,

We assume the following conventions about the index, image, image,and each index indexed by 0 is fixed.

Proposition 3.2. Let D be a m-derivation of image and U a domain of distinguished chart such that if image over U vanishes, thenimage over U on image is zero.

Proof. Let D be a such m-derivation and X an element of image satisfying the above hypothesis. We assume thatimage, , then it exists an open set image containing z, such that the a0-th component of image on image is everywhere non zero. Let’s consider image such that image whereSupp image,and image are elements of image wi t h imageimage

By definition, we obtain

image(3.1)

With image , a contradiction.

Proposition 3.3. The centralizer image of image c oincides with the vector image spanned by S.

Proof. Recall that

image

Choose imageand let be U a distinguished connected chart domain of the foliation. When p =0, we have X =0 . For image, we put image. By the fact image, image for all a and each image. Therefore, image and image where all image. Assume image with image,image.It’s known that image so image for all image Then all image are in image and consequently they are constant, and image is a subset of the image -vector space spanned by S. The converse inclusion obvious.

Proposition 3.4. All non-local m-derivations of image vanish image and take their values in image.. Conversely, all image-endomorphisms D of imagewhich have these properties, is a m-derivation of image. All theses maps are standard m-derivations.

Proof.To simplify, we pose a such m-derivation D. Then there is image and a distinguished chart domain U so that image everywhere non-vanishing. Recall that the center of image is the intersection of its centralizer with itself. We reason by contradiction, we suppose thatimage doesn’t belong to the center of image . By Proposition 3.2, we claim that on Vz, the i0-th component of image is everywhere non vanishing. So, this component is not a constant function. Consequently, we can assume that its partial derivative with respect to a xa0 is non-zero at z. Then, we consider image to be elements of image such that image and image. By the m-derivation definition,

image

Where image, we have a contradiction. Moreover, Proposition 3.1 and the previous result lead to nullity of D over image

It is easy to prove the last assertions of our proposition.

We can note immediately that,

Lemma 3. For all image, if D is a k-derivation of Lie algebra image then the center C of image satisfies the following equation image

Proposition 3.6. Local m-derivations of image

Proof. We set a local m-derivation D, image is still an m-derivation. Without trivial case p =0 , let a, b, i be some fixed indices, we write

image

And

image(3.2)

By using the (3.2), Lemma 3.5 and Proposition 3.3, we deduce that each image is constant.

Let f be an element of image ,We remark that

image(3.3)

By mapping DU to (3.3) in the case where f is a polynomial of degree greater or equal than two, the previous result and the fact that image is a differential operator over image, proves that image for all i . Furthermore, combining the previous results and the obtained relation by

image

We see that image. Then, image

In addition,

image

and the previous statement leads to image for all U.

Proposition 3.7. The Lie algebra image is stabilized by m-derivations of image

Proof. We deduce the result from Propositions 3.4, 3.6.

Theorem 3.8. We have equivalences between:

1. All m-derivation of image is local.

2. There is an image and image such that image

3. The derivative ideal of image,image coincides with image itself.

Proof. In image, we use the same reasoning as the one of the proof of Theorem 3.11 in [4]. As for image we suppose that there is an image. Since image, then it exists k such that Xk is nonzero on the open set Uk, and image vanishing on Uk with image .So, it is immediate that the image-linear map defined by

image

Is a non-local m-derivation when image. Thus image

We reason in the same way as in [4] for image

Remark 3.9. We assert that if the derivative ideal of image

Remark 3.9. We assert that if the derivative ideal of image doesn’t coincide with image, then it exists image, zero on the open set where one Xk is non-vanishing. To realize a non local m-derivation D, we exploit the non-vanishing on image of the following image -linear map:

For image where

image

These results are immediate by using Theorem 3.8, Proposition 3.4 and the definition of non local image m-derivation.

Proposition 3.10. The normalizer image of image in image is locally isomorphic to image as a vector space, where p is the corresponding leaf local dimension. So image locally image. Moreover, all local m-derivations of image stabilize image.

Proof. We define image by the set of all vector fields X such that image. So, we are in a distinguished chart U, all image elements are obtained with direct use of the definition of the normalizer of image. Indeed, image is the sum of imageand the vector image-space spanned by image. It’s clear that, this last space is isomorphic to image. The two results which follow are easily proved by the same argument as the previous. As for the last assertion, let’s take image, image and D a local m-derivation of image. In accordance with the m-derivation definition, we have imageBy local equation image, Proposition 3.1 and the previous result where each Xi runs through over the respective sets, we affirm that image is subset of image.

Theorem 3.11. Given that we have a local m-derivation D of image towardsimage. We find 1 differential closed forms image over U, where image with image denoted image uch that image and image . Besides,

image

image The converse of this result is also true. Furthermore, the condition that the maps image, with image and image, are inner is equivalent to, for all i, image and image are exact. Then we get image Where image with imageGenerally if imagethen image where image

Proof. Agreeing with the above hypothesis, we pose image, where the image with image belong to image. By the relations of m-derivations which come from

image

we state that image for all image. We write the subsequent equality

image

Then, we can have image, for all image. . So, with the help of coordinate’s translations, we get the previous equality at other arbitrary points in U. Thus, each image is closed. By exploiting all these assertions, we can adapt the demonstrations of Proposition 3.14, 3.15 et 3.16 of [4] and we achieve our proof.

Let image be the set of pair of forms image quoted before. We will denote by image, the complement set of those of image such that α is exact and image. We might assume that image

Theorem 3.12. The form of m-derivations of image is image where image , for all distinguished chart U, image if the leaf dimension over U is zero; image otherwise. And D1 is a non local m-derivation analogous to the one of Remark 3.9. Particularly, these m-derivations are derivations.

Proof. Taking D an m-derivation of image , it is split into a sum of local m-derivation D0 and of a non-local m-derivation D1 of image . So, D1 has the same form as the one of Remark 3.9. We can write D0 as imagewith image the image -linear component of D0 mapping image, whereimageBy Proposition 3.7, image. In accordance with the same proposition, we can divide imageand the quotient m derivation of D0 is denoted image. The map image becomes an m-derivation of image by the splitting of D0 we know that image is locally isomorphic to image. Then image coincides with image where image. Consequently, image with image and image, image is defined by Theorem 3.11. With the help of Proposition 3.4, respectively Theorem 3.11, D1 respectively D0 is a derivation.

Proposition 3.13. All m-derivation of the Lie image -algebra of all linear fields taking value in the constant fields Lie image-algebra of imageis Lie derivative with respect to one constant field.

Proof. Let D be such m-derivation and image one coordinates system of image. We note that image withimage. It’s easy to verify that D vanishes if and only if imageis zero too by using the following equation

image(3.4)

Where E is the Euler vector field. Then, we write image. For different and fixed u,v , we exploit the obtained relation from

image

Therefore, we have image where image. In accordance with (3.4) when u=v, we state that image . In addition, (3.4) gives us image. Thus, we proved that image.

Proposition 3.14. All m-derivation of imagetaking its value in is a sum of m-derivations of image towards image and m-derivation of image to image.

Proof. Let D be a m-derivation of image towards image. It is known that every local m-derivation of image stabilizes image, image is a Lie algebra and there is a direct sum of modules image Moreover, every non local m-derivation of image vanishes on image.Then image with image the linear component map of D from image. By the fact that D is linear andimagetakes value in image;image, imageare m-derivations.

Proposition 3.15. All m-derivations from image towards image are Lie derivative with respect to an element of image

Proof. For image, we consider the open set image. We know that all element of is imageof the form cf i Xj where image such that image in Ui equals 0 for all j ≠i , equals 1 for j =i . First, we show that such an m-derivation D is local. We fix i, j belonging to image with i ≠ j . Only in image we can find an distinguished open set U such that image or image. It is immediate that image by applying D to this last expression, we obtain image. Now, we let the map D acts on the following bracket image.We have image and image. In addition, when we are unable to choose i ≠j , the proof is trivial. Thus, D is local.

Second, suppose that D is local. Agreeing with the result of Proposition 3.13, we achieve our proof.

Theorem 3.16. All m-derivations of the normalizer image of image have a form like the one of Theorem 3.12. Moreover, the normalizer of image is image itself.

Proof. Given image an m-derivation of image ,where D0 is local and D2 non local. By Proposition 3.10, image is a m-derivation of image. Let be X1Εimageand image, we write the equation relative to m-derivation corresponding to image. By the definition of m-derivations, the previous result and Theorem 3.12, we prove that

image(3.5)

image. Let’s denote by image the m-derivation defined by image, image belongs to the intersection of the centralizer of image with image. With Proposition 3.3, image becomes an element of image. In addition, we know that image. By using image in the relation of m-derivation similar to (3.5), the Proposition 3.1 and Proposition 3.3, we have image. Then, Proposition 3.14 and Proposition 3.15 split image to a sum of a derivation of image and LX with image,D2 is zero. Moreover, we can affirm that the normalizer of image coincides with image itself.

Proposition 3.17. Every endomorphism D of the commutative Lie algebra image is a m-derivation of image . If D is local, it is a Lie derivative with respect to one element of image . In the case where D is non local, then it is determined by the existence of i ≠ j such that image with image , where each image and image

Proof. The first assertion is obvious. Moreover, it is clear that the normalizer of image is image, and its centralizer is image. Let D be a local endomorphism of image . On U, we put image. Then D=LX with image and image , for all U. Thus, X belongs to image .In addition, if image is a non-local endomorphism of image , let’s write image . It is easy to see that image is non-local iff our last assertion is true.

With the help of the previous proposition, we can confirm immediately

Corollary 3.18. If all elements of S are densely supported over M or if S is reduced to a singleton, then all endomorphisms of image is local.

Applications

The following is a list of some Lie algebras for which our theorems hold.

We denote by image the Lie algebra of all compactly supported vector fields on M which is an involutive distribution over M. We know that the normalizer of image in χ(M) is χ(M) and imagesee [7].

We suppose that M is a differential manifold equipped with a nonsingular generalized foliation imagesee [1]. We denote image the involutive distribution of tangent vector fields to the foliation (resp. of compactly supported vector fields inc image). The normalizer of image in (M) is denoted image The foliation preserving vector fields Lie algebra is named image.

Here, V is a smooth manifold and μ is surjective smooth map from M to V see [5]. It is well known that the set of μ-projected vector fields image is a Lie algebra, and μ-zero-projected vector fields set image h is an involutive distribution of M. The normalizer of image in image is denoted image and we assume that image.

Now, let G be a connection in the Grifone sense over M cf. [9]. We can cite the curvature horizontal nullity distribution space image (R is the curvature), the distribution of horizontal projected vector fields in the curvature nullity space image.Their respective normalizers in image are designated by image ,imagesee [10] and we suppose that image

We call Nk the k-nullity space distribution of vector fields in the Finsler space considered by Bidabad [11] such that the nullity index doesn’t vanish everywhere. Let’s note that image is its normalizer in the vector fields Lie algebra.

Thus, replacing respectively image by image by image we state that ”All m-derivation of image(resp. of image ) is inner with respect to image(resp. is inner)”.

In addition, let’s consider the system S composed by the Liouville vector field C on TM. We work on TM without zero section, we find all m-derivation of image by our theorem, as well as its normalizer which is locally isomorphic to image. By density of the foliation regular points set defined by S, we obtain analogous results on TM. All image -linear maps of imageinto itself are local.

The author benefits an Ingrid Daubechies initiative scholarship in collaboration with” Institute for the Conservation of Tropical Environments” (ICTE) Madagascar.

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