On semi-direct extensions of the Heisenberg group

• Autores: Giovanni Calvaruso
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 72, Fasc. 1, 2021, págs. 1-23
• Idioma: inglés
• DOI: 10.1007/s13348-019-00277-y
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• Resumen

  • Any {\mathcal {S}} \in \mathfrak {sp}(1,{\mathbb {R}}) induces canonically a derivation S of the Heisenberg Lie algebra {\mathfrak {h}} and so, a semi-direct extension G_{{\mathcal {S}}}=H \rtimes \exp ({\mathbb {R}}S) of the Heisenberg Lie group H (Müller and Ricci in Invent Math 101: 545–582, 1990). We shall explicitly describe the connected, simply connected Lie group G_{{\mathcal {S}}} and a family g_a of left-invariant (Lorentzian and Riemannian) metrics on G_{{\mathcal {S}}}, which generalize the case of the oscillator group. Both the Lie algebra and the analytic description will be used to investigate the geometry of (G_{{\mathcal {S}}},g_a), with particular regard to the study of nontrivial Ricci solitons.