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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


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


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


  • [1]

    B. Bollobás, Modern graph theory. Springer 1998. MR1633290 Zbl 0902.05016Google Scholar

  • [2]

    L. J. Corwin, F. P. Greenleaf, Representations of nilpotent Lie groups and their applications. Part I. Cambridge Univ. Press 1990. MR1070979 Zbl 0704.22007Google Scholar

  • [3]

    S. G. Dani, M. G. Mainkar, Anosov automorphisms on compact nilmanifolds associated with graphs. Trans. Amer. Math. Soc. 357 (2005), 2235–2251. MR2140439 Zbl 1061.22008CrossrefGoogle Scholar

  • [4]

    R. C. DeCoste, Closed geodesics on compact nilmanifolds with Chevalley rational structure. Manuscripta Math. 127 (2008), 309–343. MR2448434 Zbl 1162.53032Web of ScienceCrossrefGoogle Scholar

  • [5]

    R. C. DeCoste, L. DeMeyer, M. B. Mast, Characterizations of Heisenberg-like Lie algebras. J. Lie Theory 21 (2011), 711–727. MR2858081 Zbl 1222.53055Google Scholar

  • [6]

    L. DeMeyer, Closed geodesics in compact nilmanifolds. Manuscripta Math. 105 (2001), 283–310. MR1856612 Zbl 1076.53049CrossrefGoogle Scholar

  • [7]

    P. Eberlein, Geometry of 2-step nilpotent groups with a left invariant metric. Ann. Sci. École Norm. Sup. (4) 27 (1994), 611–660. MR1296558 Zbl 0820.53047CrossrefGoogle Scholar

  • [8]

    C. D. Godsil, Algebraic combinatorics. Chapman & Hall, New York 1993. MR1220704 Zbl 0784.05001Google Scholar

  • [9]

    R. Gornet, M. B. Mast, The length spectrum of Riemannian two-step nilmanifolds. Ann. Sci. École Norm. Sup. (4) 33 (2000), 181–209. MR1755115 Zbl 0968.53036CrossrefGoogle Scholar

  • [10]

    A. Kaplan, Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata 11 (1981), 127–136. MR621376 Zbl 0495.53046Google Scholar

  • [11]

    K. B. Lee, K. Park, Smoothly closed geodesics in 2-step nilmanifolds. Indiana Univ. Math. J. 45 (1996), 1–14. MR1406681 Zbl 0862.53037Google Scholar

  • [12]

    M. G. Mainkar, Graphs and two-step nilpotent Lie algebras. Groups Geom. Dyn. 9 (2015), 55–65. MR3343346 Zbl 1331.22010Web of ScienceCrossrefGoogle Scholar

  • [13]

    M. B. Mast, Closed geodesics in 2-step nilmanifolds. Indiana Univ. Math. J. 43 (1994), 885–911. MR1305951 Zbl 0818.53065CrossrefGoogle Scholar

  • [14]

    T. Politi, A formula for the exponential of a real skew-symmetric matrix of order 4. BIT 41 (2001), 842–845. MR1881220 Zbl 1021.65020CrossrefGoogle Scholar

  • [15]

    V. V. Prasolov, Problems and theorems in linear algebra, volume 134 of Translations of Mathematical Monographs. Amer. Math. Soc. 1994. MR1277174 Zbl 0803.15001Google Scholar

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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