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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 18, Issue 3

Issues

Graphs and metric 2-step nilpotent Lie algebras

Rachelle C. DeCoste / Lisa DeMeyer / Meera G. Mainkar
Published Online: 2018-04-05 | DOI: https://doi.org/10.1515/advgeom-2017-0052

Abstract

Dani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫G from a simple directed graph G in 2005. There is a natural inner product on 𝔫G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫g. We classify singularity properties of the Lie algebra 𝔫g 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.

Keywords: Nilpotent Lie algebras; Heisenberg-like Lie algebra; closed geodesics; star graphs

MSC 2010: Primary: 22E25; Secondary: 53C30, 53C22

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About the article


Received: 2015-12-29

Revised: 2016-05-03

Published Online: 2018-04-05

Published in Print: 2018-07-26


Funding: Meera Mainkar was supported by the Central Michigan University ORSP Early Career Investigator (ECI) grant #C61940.


Citation Information: Advances in Geometry, Volume 18, Issue 3, Pages 265–284, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0052.

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