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

TL;DR

This paper extends the Dixmier property from a Lie algebra's linear representation to its generalized Takiff algebra representations, with applications to adjoint and coadjoint representations.

Contribution

It proves that the Dixmier property is preserved under the construction of generalized Takiff algebras for certain representations.

Findings

Dixmier property holds for vector fields of generalized Takiff algebras if it holds for the original algebra.

Results apply to adjoint and coadjoint representations of Takiff algebras.

Provides explicit examples demonstrating the applicability of the main theorem.

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.