## Journal of Generalized Lie Theory and Applications

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

### The $m$-Derivations of Distribution Lie Algebras

Princy Randriambololondrantomalala

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 30 September 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.jglta/1443617957

**Digital Object Identifier**

doi:10.4172/1736-4337.1000217

**Mathematical Reviews number (MathSciNet)**

MR3624039

**Zentralblatt MATH identifier**

06499576

**Subjects**

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)

**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

#### Citation

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. https://projecteuclid.org/euclid.jglta/1443617957