Can mathematics unlock the secrets of curved space-times? This paper presents a comprehensive classification of flat Lorentzian nilpotent Lie algebras, crucial for understanding pseudo-Euclidean Lie algebras associated with nilpotent Lie groups. The research focuses on Lie groups endowed with a left-invariant Lorentzian metric of vanishing curvature, providing a rigorous framework for mathematical analysis. The study simplifies the understanding of flat Lorentzian Lie algebras by demonstrating that every such algebra is the direct sum of an indecomposable flat Lorentzian Lie algebra and an abelian Euclidean summand. The paper delves into identifying the only non-abelian Lie algebras that support flat Lorentzian metrics and remain indecomposable, revealing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \begin{document}$\mathfrak{h}_3$\end{document} (the 3-dimensional Heisenberg Lie algebra) and semidirect products as key elements. It explores particular derivations, enhancing the current discussions in Lorentzian geometry and Lie theory. In conclusion, this classification provides a structured approach to understanding the intricate nature of flat Lorentzian Lie algebras. The study's findings contribute significantly to theoretical mathematics and offer potential applications in mathematical modeling and related fields, furthering the pursuit of pure knowledge.
Published in the Bulletin of the London Mathematical Society, a journal focusing on pure mathematics research, this paper aligns directly with the journal's scope. It contributes to the broader understanding of Lie algebras, a key area within mathematics. By classifying flat Lorentzian nilpotent Lie algebras, the paper expands knowledge in this specialized area, thus holding significance for the journal's readership.
| Category | Category Repetition |
|---|---|
| Science: Mathematics | 10 |
| Technology: Technology (General): Industrial engineering. Management engineering: Applied mathematics. Quantitative methods | 1 |