జర్నల్ ఆఫ్ జెనరలైజ్డ్ లై థియరీ అండ్ అప్లికేషన్స్

The m-Derivations of Distribution LieAlgebras

Abstract

Princy Randriambololondrantomalala

Let M be an 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 omega on the ring of real functions on M as well as those of the normalizer of omega are Lie derivatives with respect to one and only one element of this normalizer, if omega 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()0MFof constant real functions on the leaves generated by S. We find that m-derivations of $\mathfrak{A}_S$ are local if and only if its derivative ideal coincides with $\mathfrak{A}_S$ itself. Then, we characterize all non local m-derivations of $\mathfrak{A}_S$. We prove that all m-derivations of $\mathfrak{A}_S$ and of the normalizer of $\mathfrak{A}_S$ are derivations. We will make these derivations and those of the centralizer of $\mathfrak{A}_S$ more explicit.