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PreprintCC BY-NC-ND 4.0arXiv

Gray 𝒜-metrics on nilpotent Lie groups and the quasicentroid condition

  • Edison Alberto Fernández-Culma
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Abstract

We study left-invariant Gray 𝒜-metrics (cyclic-parallel Ricci tensor) on nilpotent Lie groups, focusing on the algebraic condition that the Ricci operator belongs to the quasicentroid QC(n) of the Lie algebra n. We review and develop the abstract structure theory of quasicentroids, and prove a Central Decomposition Theorem: any nilpotent Riemannian Lie algebra with Ric ∈ QC(n) decomposes as a central product of ideals ij = Eλj ⊕ z, where z is the center of n and Eλj are the negative Ricci eigenspaces. Endowed with the restricted metric, each factor ij is itself a Gray 𝒜-space. Finally, we investigate when the general 𝒜-metric condition reduces to Ric ∈ QC(n). We prove this equivalence unconditionally for two-step nilpotent algebras and whenever the metric admits a nice basis, and provide a general characterization via the vanishing of two trace identities.

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Cite

BibTeXRIS
BibTeX
@misc{fernandezculma2026gray,
  title = {{Gray $\mathcalA$-metrics on nilpotent Lie groups and the quasicentroid condition}},
  author = {Edison Alberto Fernández-Culma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39184},
  eprint = {2610.39184},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39184},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Gray 𝒜-metrics on nilpotent Lie groups and the quasicentroid condition
AU  - Edison Alberto Fernández-Culma
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39184
UR  - https://arxiv.org/abs/2610.39184
AB  - We study left-invariant Gray 𝒜-metrics (cyclic-parallel Ricci tensor) on nilpotent Lie groups, focusing on the algebraic condition that the Ricci operator belongs to the quasicentroid QC(n) of the Lie algebra n. We review and develop the abstract structure theory of quasicentroids, and prove a Central Decomposition Theorem: any nilpotent Riemannian Lie algebra with Ric ∈ QC(n) decomposes as a central product of ideals i_j = E_λ_j ⊕ z, where z is the center of n and E_λ_j are the negative Ricci eigenspaces. Endowed with the restricted metric, each factor i_j is itself a Gray 𝒜-space. Finally, we investigate when the general 𝒜-metric condition reduces to Ric ∈ QC(n). We prove this equivalence unconditionally for two-step nilpotent algebras and whenever the metric admits a nice basis, and provide a general characterization via the vanishing of two trace identities.
ER  -