Generalized Takiff groups associated with a Lie group. Applications to \(\text{SL}_{2}(\mathbb{R})\). (Groupes de Takiff généralisés associés à un groupe de Lie. Applications à \(\text{SL}_{2}(\mathbb{R})\).) (French. English summary) Zbl 1241.22006
Summary: 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, denoted by \(G_{n}\), whose Lie algebra is the generalized Takiff algebra of order \(n\) associated with \(\mathcal{G}\). We investigate some properties of this group. As an 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 algebra structure.
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