On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups

Abstract

We study $2$-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group $N \rtimes H^{\operatorname{aut}}$. Replacing the standard non-degenerate center assumption with the weaker condition that the commutator ideal be non-degenerate, we develop a framework that extends the construction to the Lorentzian context and covers both the non-degenerate and degenerate center cases. In the degenerate case, we show that the associated Lie algebra is a central extension of a semidirect product whose Riemannian factor is naturally reductive. Furthermore, we obtain invariant decompositions of the defining representation, including a distinguished Lorentzian factor, and provide an explicit description of the isotropy algebra and the identity component of the isometric automorphism group. These results complete the structural description of naturally reductive $2$-step Lorentzian nilpotent Lie groups under the assumption of non-degeneracy in the commutator ideal.