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 $N⋊H^{\mathrm{aut}}$. 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 representation-theoretic construction to the Lorentzian setting 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 in the non-degenerate case. In the degenerate case, we introduce a Riemannian quotient representation and show that the kernel of this representation is a central abelian ideal whose elements act by null rotations. Finally, we use these results to characterize the isotropy algebra of the group of isometric automorphisms.

This is a new version of a previously uploaded paper where we correct some minor mistakes and improve some of the results, 31 pages