On semi-direct extensions of the Heisenberg group


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.

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The author wishes to thank Professor Fulvio Ricci for bringing Ref. [19] to his attention.

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Correspondence to Giovanni Calvaruso.

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Giovanni Calvaruso partially supported by funds of the University of Salento and GNSAGA.

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Calvaruso, G. On semi-direct extensions of the Heisenberg group. Collect. Math. 72, 1–23 (2021). https://doi.org/10.1007/s13348-019-00277-y

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  • Heisenberg group
  • Semi-direct extensions
  • Oscillator group
  • Ricci solitons

Mathematics Subject Classification

  • 53C50
  • 53C21
  • 35A01