PreprintCC BY 4.0arXiv
On flat metric nilpotent Lie algebras
Abstract
We study flat left-invariant pseudo-Riemannian metrics on nilpotent Lie groups. Associated with a flat metric Lie algebra (g,⟨·,·⟩) there is a canonical affine action of the corresponding group on the vector space g, whose linear part is the Levi-Civita representation L. Combining this action with Scheuneman's theorem on pure translations in nilpotent affine groups, we prove that a flat metric nonabelian nilpotent Lie algebra has a nontrivial kernel of L, that is, ker L≠ 0. As a consequence, every n-dimensional flat metric nonabelian nilpotent Lie algebra is a double extension of an (n−2)-dimensional flat metric nilpotent Lie algebra.
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BibTeX
@misc{an2026flat,
title = {{On flat metric nilpotent Lie algebras}},
author = {Huihui An and Zaili Yan and Hui Zhang and Shaoxiang Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23481},
eprint = {2610.23481},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23481},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On flat metric nilpotent Lie algebras
AU - Huihui An
AU - Zaili Yan
AU - Hui Zhang
AU - Shaoxiang Zhang
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.23481
UR - https://arxiv.org/abs/2610.23481
AB - We study flat left-invariant pseudo-Riemannian metrics on nilpotent Lie groups. Associated with a flat metric Lie algebra (g,⟨·,·⟩) there is a canonical affine action of the corresponding group on the vector space g, whose linear part is the Levi-Civita representation L. Combining this action with Scheuneman's theorem on pure translations in nilpotent affine groups, we prove that a flat metric nonabelian nilpotent Lie algebra has a nontrivial kernel of L, that is, ker L≠ 0. As a consequence, every n-dimensional flat metric nonabelian nilpotent Lie algebra is a double extension of an (n−2)-dimensional flat metric nilpotent Lie algebra.
ER -