On the Isometry Groups of Invariant Lorentzian Metrics on the Heisenberg Group

This work concerns the invariant Lorentzian metrics on the Heisenberg Lie group of dimension three ${{\rm H}_3(\mathbb{R})}$ and the bi-invariant metrics on the solvable Lie groups of dimension four. We start with the indecomposable Lie groups of dimension four admitting bi-invariant metrics and which act on ${{\rm H}_3(\mathbb{R})}$ by isometries and we study some geometrical features on these spaces. On ${{\rm H}_3(\mathbb{R})}$ , we prove that the property of the metric being proper naturally reductive is equivalent to the property of the center being non-degenerate. These metrics are Lorentzian algebraic Ricci solitons.