alexa The m-Derivations of Distribution Lie Algebras
ISSN: 1736-4337

Journal of Generalized Lie Theory and Applications
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Research Article

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.

 

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.

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