Graphs and metric 2-step nilpotent Lie algebras-Reference-Cited by-同舟云学术

Graphs and metric 2-step nilpotent Lie algebras

Published:2018-04-05 Issue:3 Volume:18 Page:265-284
ISSN:1615-7168
Container-title:Advances in Geometry
language:en
Short-container-title:

Author:

DeCoste Rachelle C.¹,DeMeyer Lisa²,Mainkar Meera G.²

Affiliation:

1. Mathematics Department , Wheaton College , Norton , MA 02766 , USA

2. Department of Mathematics , Central Michigan University , Mount Pleasant , MI 48859 , USA

Abstract

Abstract Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫 G from a simple directed graph G in 2005. There is a natural inner product on 𝔫 G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫 g . We classify singularity properties of the Lie algebra 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 ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

Publisher

Walter de Gruyter GmbH

Subject

Geometry and Topology

Link

https://www.degruyter.com/document/doi/10.1515/advgeom-2017-0052/pdf

Reference15 articles.

1. B. Bollobás, Modern graph theory. Springer 1998. MR1633290 Zbl 0902.05016

2. L. J. Corwin, F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I. Cambridge Univ. Press 1990. MR1070979 Zbl 0704.22007

3. S. G. Dani, M. G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs. Trans. Amer. Math. Soc. 357 (2005), 2235–2251. MR2140439 Zbl 1061.22008

4. R. C. DeCoste, Closed geodesics on compact nilmanifolds with Chevalley rational structure. Manuscripta Math. 127 (2008), 309–343. MR2448434 Zbl 1162.53032

5. R. C. DeCoste, L. DeMeyer, M. B. Mast, Characterizations of Heisenberg-like Lie algebras. J. Lie Theory21 (2011), 711–727. MR2858081 Zbl 1222.53055

Cited by 10 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Curvature exponent and geodesic dimension on Sard-regular Carnot groups;Analysis and Geometry in Metric Spaces;2024-01-01

2. Classification of K -forms in nilpotent Lie algebras associated to graphs;Communications in Algebra;2023-07-04

3. The Reidemeister spectrum of 2-step nilpotent groups determined by graphs;Communications in Algebra;2023-01-02

4. Abelian factors in 2-step nilpotent Lie algebras constructed from graphs;Communications in Algebra;2022-12-06

5. The geodesic flow on nilpotent Lie groups of steps two and three;Discrete & Continuous Dynamical Systems;2022