TY - JOUR

T1 - Graphs and metric 2-step nilpotent Lie algebras

AU - Decoste, Rachelle C.

AU - Demeyer, Lisa

AU - Mainkar, Meera G.

N1 - Funding Information:
Funding: Meera Mainkar was supported by the Central Michigan University ORSP Early Career Investigator (ECI) grant #C61940.
Publisher Copyright:
© 2018 Walter de Gruyter GmbH Berlin/Boston 2018.

PY - 2018/7/26

Y1 - 2018/7/26

N2 - 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.

AB - 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.

KW - Heisenberg-like Lie algebra

KW - Nilpotent Lie algebras

KW - closed geodesics

KW - star graphs

UR - http://www.scopus.com/inward/record.url?scp=85045854090&partnerID=8YFLogxK

U2 - 10.1515/advgeom-2017-0052

DO - 10.1515/advgeom-2017-0052

M3 - Article

AN - SCOPUS:85045854090

VL - 18

SP - 265

EP - 284

JO - Advances in Geometry

JF - Advances in Geometry

SN - 1615-715X

IS - 3

ER -