La propri\'et\'e de Dixmier pour les alg\`ebres de Lie de champs de vecteurs

Abstract

Given a linear representation $\rho : \mathfrak{g} \longrightarrow \mathfrak{g}\ell(V)$ of a Lie algebra $\mathfrak{g}$, one can define a linear representation $\rho_m : \mathfrak{g}_m \longrightarrow \mathfrak{g}\ell(V^m)$ of the generalized Takiff algebra $\mathfrak{g}_m$. It is proved here that the vector fields defined by $\rho_m$ on $V^m$ do have the Dixmier property if those defined by $\rho$ have the same property. Examples where the result applies are given and in particular, those of the adjoint or coadjoint representations of Takiff algebras.