On flat pseudo-Euclidean solvable Malcev algebras
Abstract
A pseudo-Euclidean Malcev algebra is a Malcev algebra equipped with a non-degenerate symmetric bilinear form. In this paper, we introduce a curvature operator for pseudo-Euclidean Malcev algebras, generalizing the notion of curvature for pseudo-Euclidean Lie algebras. We define flat pseudo-Euclidean Malcev algebras and show that every flat Euclidean solvable Malcev algebra which is also a Lie algebra remains flat in the classical Lie algebra curvature sense. Furthermore, we develop the flat double extension construction for flat pseudo-Euclidean Malcev algebras and prove that every flat Lorentzian Malcev algebra with a degenerate center can be obtained via the flat double extension of a flat Euclidean Malcev algebra. Moreover, we demonstrate that all flat Lorentzian nilpotent Malcev algebras arise from the flat double extension of a Euclidean abelian Lie algebras. Finally, we establish that any flat Lorentzian nilpotent Malcev algebra is necessarily a Lie algebra and is flat in the classical Lie algebra curvature sense.
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BibTeX
@misc{boucetta2026flat,
title = {{On flat pseudo-Euclidean solvable Malcev algebras}},
author = {Mohamed Boucetta and Hamza El Ouali and Oumaima Tibssirte},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13950},
eprint = {2609.13950},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13950},
}
RIS
TY - UNPB
TI - On flat pseudo-Euclidean solvable Malcev algebras
AU - Mohamed Boucetta
AU - Hamza El Ouali
AU - Oumaima Tibssirte
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13950
UR - https://arxiv.org/abs/2609.13950
AB - A pseudo-Euclidean Malcev algebra is a Malcev algebra equipped with a non-degenerate symmetric bilinear form. In this paper, we introduce a curvature operator for pseudo-Euclidean Malcev algebras, generalizing the notion of curvature for pseudo-Euclidean Lie algebras. We define flat pseudo-Euclidean Malcev algebras and show that every flat Euclidean solvable Malcev algebra which is also a Lie algebra remains flat in the classical Lie algebra curvature sense. Furthermore, we develop the flat double extension construction for flat pseudo-Euclidean Malcev algebras and prove that every flat Lorentzian Malcev algebra with a degenerate center can be obtained via the flat double extension of a flat Euclidean Malcev algebra. Moreover, we demonstrate that all flat Lorentzian nilpotent Malcev algebras arise from the flat double extension of a Euclidean abelian Lie algebras. Finally, we establish that any flat Lorentzian nilpotent Malcev algebra is necessarily a Lie algebra and is flat in the classical Lie algebra curvature sense.
ER -