Les m-derivations des algebres de Lie de champs de vecteurs polynomiaux

Abstract

Let be \(m\ge 2\) a natural integer. We study the m-derivations of the Lie algebras \(\mathfrak {P}\) of polynomial vector fields on \(\mathbb {R}^{n}\) which contain all constant vector fields and the Euler’s vector field. They are Lie derivative with respect to a \(\mathbb {R}^{n}\) polynomial vector fields on the normalizer of \(\mathfrak {P}\), when m is even. If m is odd, all m-derivation of \(\mathfrak {P}\) is a sum of a Lie derivative with respect to a normalizer’s vector fields and, a m-derivation of \(\mathfrak {P}\) which acts on a quadratic vector fields to give a constant vector fields and which vanishes otherwise. We give a necessary and sufficient condition for a linear map of \(\mathfrak {P}\) into it self to be a previous last type of m-derivation. Generally, under some conditions on \(\mathfrak {P}\), all m-derivation of the normalizer of \(\mathfrak {P}\) is inner. Hence, the m-derivation of Lie algebra of all polynomial vector fields respectively of affine vector fields on \(\mathbb {R}^n\), is an inner m-derivation.