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.

任何\（{\ mathcal {S}} \在\ mathfrak {SP}（1，{\ mathbb {R}}）\）诱导规范地推导小号海森堡李代数的\（{\ mathfrak {H}} \ ），然后是Heisenberg Lie组H（Müller和Ricci in ）中的半直接扩展\（G _ {{\ mathcal {S}}} = H \ rtimes \ exp（{\ mathbb {R}} S）\） Invent Math 101：545–582，1990）。我们要清楚地描述了连接，只要连接李群\（G _ {{\ mathcal {S}}} \）和家庭\（G_A \）左不变的（洛伦兹和黎曼）的指标\（G _ {{\ mathcal {S}}} \），它概括了振荡器组的情况。李代数和解析描述都将用于研究\（（G _ {{\ mathcal {S}}}，g_a）\）的几何，特别是要研究非平凡的Ricci孤子。