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2015 The $m$-Derivations of Distribution Lie Algebras
Princy Randriambololondrantomalala
Author Affiliations +
J. Gen. Lie Theory Appl. 9(1): 1-6 (2015). DOI: 10.4172/1736-4337.1000217


Let $M$ be a 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 $Ω$ on the ring of real functions on $M$ as well as those of the normalizer of $Ω$ are Lie derivatives with respect to one and only one element of this normalizer, if $Ω$ 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 $F_0(M)$ of constant real functions on the leaves generated by $S$. We find that $m$-derivations of ${\mathfrak{A}_s}$ is local if and only if its derivative ideal coincides with ${\mathfrak{A}_s}$ itself. Then, we characterize all non local $m$-derivation of ${\mathfrak{A}_s}$. We prove that all $m$-derivations of ${\mathfrak{A}_s}$ and 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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Princy Randriambololondrantomalala. "The $m$-Derivations of Distribution Lie Algebras." J. Gen. Lie Theory Appl. 9 (1) 1 - 6, 2015.


Published: 2015
First available in Project Euclid: 30 September 2015

zbMATH: 06499576
MathSciNet: MR3624039
Digital Object Identifier: 10.4172/1736-4337.1000217

Primary: 17B40 , 17B66
Secondary: 47B47 , 53B15 , 53B40 , 53C12

Keywords: $m$-derivations , Commuting vector fields , Compactly supported vector fields , distributions , Generalized foliations , Nullity space of curvature , Vector fields lie algebras , μ-projected vector fields

Rights: Copyright © 2015 Ashdin Publishing (2009-2013) / OMICS International (2014-2016)

Vol.9 • No. 1 • 2015
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