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Groupes de Takiff généralisés associés à un groupe de Lie. Applications à SL2 (R)

Volume 10, Number 1 (2010), 7 - 17

Groupes de Takiff généralisés associés à un groupe de Lie. Applications à SL2 (R)

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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 $G$. 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.