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Graphs and metric 2-step nilpotent Lie algebras

Graphs and metric 2-step nilpotent Lie algebras AbstractDani 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

Graphs and metric 2-step nilpotent Lie algebras

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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH Berlin/Boston
ISSN
1615-7168
eISSN
1615-7168
DOI
10.1515/advgeom-2017-0052
Publisher site
See Article on Publisher Site

Abstract

AbstractDani 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.

Journal

Advances in Geometry – de Gruyter

Published: Jul 26, 2018

References

  • The length spectrum of Riemannian two-step nilmanifolds
  • Riemannian nilmanifolds attached to Clifford modules
  • Graphs and two-step nilpotent Lie algebras
  • Closed geodesics on compact nilmanifolds with Chevalley rational structure
  • Smoothly closed geodesics in 2-step nilmanifolds
  • Graphs and two-step nilpotent Lie algebras