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arXiv · 1512.07944

Graphs and Metric 2-step Nilpotent Lie Algebras

Rachelle DeCoste·Lisa DeMeyer·Meera Mainkar
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Abstract

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra $\mathfrak{n}_G$ from a simple directed graph $G$ in 2005. There is a natural inner product on $\mathfrak{n}_G$ arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group $N$ with Lie algebra $\mathfrak{n}_G$. We classify singularity properties of the Lie algebra $\mathfrak{n}_G$ in terms of the graph $G$. A comprehensive description is given of graphs $G$ which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph $G$ and on a lattice $Γ\subseteq N$ for which the quotient $Γ\backslash N$, a compact nilmanifold, has a dense set of smoothly closed geodesics.

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Rachelle DeCoste, Lisa DeMeyer, Meera Mainkar. 2015-12-25. Graphs and Metric 2-step Nilpotent Lie Algebras. https://arxiv.org/abs/1512.07944

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