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arXiv · 2610.23481

On flat metric nilpotent Lie algebras

Huihui An·Zaili Yan·Hui Zhang·Shaoxiang Zhang
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Abstract

We study flat left-invariant pseudo-Riemannian metrics on nilpotent Lie groups. Associated with a flat metric Lie algebra $(\mathfrak{g},\langle\cdot,\cdot\rangle)$ there is a canonical affine action of the corresponding group on the vector space $\mathfrak{g}$, 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, $\ker L\neq 0$. 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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Huihui An, Zaili Yan, Hui Zhang, Shaoxiang Zhang. 2026-10-16. On flat metric nilpotent Lie algebras. https://arxiv.org/abs/2610.23481

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