Graphs and metric 2-step nilpotent Lie algebras
| Autor: | Rachelle C. DeCoste, Lisa DeMeyer, Meera Mainkar |
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| Rok vydání: | 2018 |
| Předmět: |
Mathematics - Differential Geometry
Primary: 22E25 Secondary: 53C30 53C22 010102 general mathematics Lattice (group) Lie group 010103 numerical & computational mathematics Directed graph 01 natural sciences Combinatorics Nilpotent Lie algebra Nilpotent Differential Geometry (math.DG) Product (mathematics) Lie algebra FOS: Mathematics Geometry and Topology 0101 mathematics Nilmanifold Mathematics::Representation Theory Mathematics |
| Zdroj: | Advances in Geometry. 18:265-284 |
| ISSN: | 1615-7168 1615-715X |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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