Journal of Generalized Lie Theory and Applications

The $m$-Derivations of Distribution Lie Algebras

Princy Randriambololondrantomalala

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Gen. Lie Theory Appl., Volume 9, Number 1 (2015), 6 pages.

First available in Project Euclid: 30 September 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B66: Lie algebras of vector fields and related (super) algebras 17B40: Automorphisms, derivations, other operators
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 53B15: Other connections 47B47: Commutators, derivations, elementary operators, etc. 53B40: Finsler spaces and generalizations (areal metrics)

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


Randriambololondrantomalala, Princy. The $m$-Derivations of Distribution Lie Algebras. J. Gen. Lie Theory Appl. 9 (2015), no. 1, 6 pages. doi:10.4172/1736-4337.1000217.

Export citation