Graphs and metric 2-step nilpotent Lie algebras

Rachelle C. Decoste, Lisa Demeyer, Meera G. Mainkar

Research output: Contribution to journalArticlepeer-review

10 Scopus citations


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.

Original languageEnglish
Pages (from-to)265-284
Number of pages20
JournalAdvances in Geometry
Issue number3
StatePublished - Jul 26 2018


  • Heisenberg-like Lie algebra
  • Nilpotent Lie algebras
  • closed geodesics
  • star graphs


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