Groupes de Takiff généralisés associés a un groupe de Lie. Applications à $\mathbf{SL}_{2}(\mathbb{R})$

Abstract

Let $G$ be a locally compact Lie group, $\mathcal{G}$ its Lie algebra and $n$ an integer $\geq 1$. We build a locally compact Lie group, noted $G_{n}$, whose Lie algebra is the generalized Takiff algebra of order $n$ associated with $\mathcal{G}$. We investigate some properties of this group. As application, we show that $SL_{2}(\mathbb{R}^{n+1})$ is the generalized Takiff group of order $n$ associated with $SL_{2}(\mathbb{R})$, where $\mathbb{R}^{n+1}$ is equiped with an appropriate algebras structure.