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**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.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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

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

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

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 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. Over , we can take a simple example where the Lie -algebra is spanned by and the linear map D is defined by 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, 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 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 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 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 , 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 ,where if p≥(resp. where if p=0). Then, there are two modules over spanned by and generated by . These previous modules are Lie algebras such that is equal to the semi-direct product of these two algebras: for all distinguished U. We can say that is a smooth distribution of M over . 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 , of the normalizer of and of the centralizer of in χ(m) . The corresponding work where S ={0}has been done in the previous section. Because of ’s lower central series constancy, which coincides with module direct sum of , 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 is itself, which is equivalent to the fact that has non-vanishing elements over the whole M. Moreover, all *m-*derivations of or of the normalizer of are sums of a Lie derivative with respect to one 's element, of one local *m-*derivation which takes its value in depending on two non-vanishing 1-differential forms over , and of a non-local *m-*derivation of . We give some recommendations for constructing all these non local *m-*derivations. In addition, all -linear maps of the centralizer of into itself are *m-*derivations. We characterize all local -endomorphisms of in the case where all elements of S are densely supported or 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 is denoted L_{x}. We adopt the Einstein index summation and suppose that all considered objects are smooth.

According the hypothesis about Ω, we can affirm that Ω is a Lie sub-algebra of . 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 ) are Lie derivative with respect to one and only one vector field of the normalizer of Ω (resp. is inner).*

*Proof.* Assume that such that . By Frobenius theorem, we find one chart which contains x and local coordinate system where . 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 is an *m-*derivation of . Let’s give ,as we know, then is uniquely determined, where each D_{i} is differential operator over the trivial bundle cf. [3]. Thus, if necessary we can write ,where A, B are multi-indices corresponding to coordinates.

Let’s apply to , where x^{n}= y. By definition of *m-*derivations and when f is replaced by monomials, we have:

- If , except for

- If , except for where are free are free x^{n} monomials.

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

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

- for .

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

By these results, Consequently,

is a derivation of the -sub-module spanned by . Applying D_{0} to , we have . By coordinate’s translations, we can write that

We take Proposition 2.6 of [7] and we have 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 O_{Ω }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.

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

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

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

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

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

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

*Proof. *Let D be a such *m-*derivation and X an element of satisfying the above hypothesis. We assume that, , then it exists an open set containing z, such that the a_{0}-th component of on is everywhere non zero. Let’s consider such that whereSupp ,and are elements of wi t h

By definition, we obtain

(3.1)

With , a contradiction.

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

Proof. Recall that

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

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

Proof.To simplify, we pose a such *m-*derivation D. Then there is and a distinguished chart domain U so that everywhere non-vanishing. Recall that the center of is the intersection of its centralizer with itself. We reason by contradiction, we suppose that doesn’t belong to the center of . By Proposition 3.2, we claim that on V_{z}, the i_{0}-th component of is everywhere non vanishing. So, this component is not a constant function. Consequently, we can assume that its partial derivative with respect to a x^{a0} is non-zero at *z*. Then, we consider to be elements of such that and . By the *m-*derivation definition,

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

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

We can note immediately that,

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

Proposition 3.6. Local *m-*derivations of

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

And

(3.2)

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

Let f be an element of ,We remark that

(3.3)

By mapping D_{U} 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 is a differential operator over , proves that for all *i* . Furthermore, combining the previous results and the obtained relation by

We see that . Then,

In addition,

and the previous statement leads to for all U.

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

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

**Theorem 3.8.** *We have equivalences between:*

*1. All m-derivation of is local.*

*2. There is an and such that *

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

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

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

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

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

*Remark 3.9.* We assert that if the derivative ideal of doesn’t coincide with , then it exists , 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 of the following -linear map:

For where

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

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

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

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

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

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

we state that for all . We write the subsequent equality

Then, we can have , for all . . So, with the help of coordinate’s translations, we get the previous equality at other arbitrary points in U. Thus, each 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 be the set of pair of forms quoted before. We will denote by , the complement set of those of such that α is exact and . We might assume that

**Theorem 3.12.** *The form of m-derivations of is where , for all distinguished chart U, if the leaf dimension over U is zero; otherwise. And D _{1} 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 , it is split into a sum of local *m-*derivation D_{0} and of a non-local *m-*derivation D_{1} of . So, D_{1} has the same form as the one of Remark 3.9. We can write D_{0} as with the -linear component of D_{0} mapping , whereBy Proposition 3.7, . In accordance with the same proposition, we can divide and the quotient m derivation of D_{0} is denoted . The map becomes an *m-*derivation of by the splitting of D_{0} we know that is locally isomorphic to . Then coincides with where . Consequently, with and , is defined by Theorem 3.11. With the help of Proposition 3.4, respectively Theorem 3.11, D_{1} respectively D_{0} is a derivation.

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

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

(3.4)

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

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

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

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

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

Proof. For , we consider the open set . We know that all element of is of the form* cf ^{ i }X*

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 of have a form like the one of Theorem 3.12. Moreover, the normalizer of is itself.*

Proof. Given an *m-*derivation of ,where D_{0} is local and D2 non local. By Proposition 3.10, is a *m-*derivation of . Let be X_{1}Εand , we write the equation relative to *m-*derivation corresponding to . By the definition of *m-*derivations, the previous result and Theorem 3.12, we prove that

(3.5)

. Let’s denote by the *m-*derivation defined by , belongs to the intersection of the centralizer of with . With Proposition 3.3, becomes an element of . In addition, we know that . By using in the relation of *m-*derivation similar to (3.5), the Proposition 3.1 and Proposition 3.3, we have . Then, Proposition 3.14 and Proposition 3.15 split to a sum of a derivation of and L_{X} with ,D_{2} is zero. Moreover, we can affirm that the normalizer of coincides with itself.

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

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

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

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

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

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 is a Lie algebra, and μ-zero-projected vector fields set h is an involutive distribution of M. The normalizer of in is denoted and we assume that .

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

We call N^{k} 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 is its normalizer in the vector fields Lie algebra.

Thus, replacing respectively by by we state that ”All *m-*derivation of (resp. of ) is inner with respect to (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 by our theorem, as well as its normalizer which is locally isomorphic to . By density of the foliation regular points set defined by S, we obtain analogous results on TM. All -linear maps of into 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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