The $m$-Derivations of Distribution Lie Algebras

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.