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

  • Brian Luporini
  • Silvio Reggiani
  • Francisco Vittone
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Abstract

We study $2$-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group Nrtimes Haut. 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.

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Cite

BibTeXRIS
BibTeX
@misc{luporini2026structure,
  title = {{On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups}},
  author = {Brian Luporini and Silvio Reggiani and Francisco Vittone},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.18017},
  eprint = {2607.18017},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.18017},
}
RIS
TY  - UNPB
TI  - On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups
AU  - Brian Luporini
AU  - Silvio Reggiani
AU  - Francisco Vittone
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.18017
UR  - https://arxiv.org/abs/2607.18017
AB  - We study $2$-step nilpotent Lie groups with naturally reductive left-invariant Lorentzian metrics with respect to the presentation group Nrtimes H^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.
ER  -