Source: Journal of Generalized Lie Theory and Applications, Volume 9, Number 1, 6 pages.
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.