Czechoslovak Mathematical Journal

, Volume 63, Issue 3, pp 847–863 | Cite as

The classification of two step nilpotent complex Lie algebras of dimension 8

Article

Abstract

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.

Keywords

two-step nilpotent Lie algebra base minimal system of generators related sets H-minimal system of generators 

MSC 2010

17B05 17B30 17B40 

References

  1. [1]
    J.M. Ancochea-Bermudez, M. Goze: Classification des algèbres de Lie nilpotentes complexes de dimension 7. (Classification of nilpotent complex Lie algebras of dimension 7). Arch. Math. 52 (1989), 175–185. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J.M. Ancochea-Bermudez, M. Goze: Classification des algèbres de Lie filiformes de dimension 8. (Classification of filiform Lie algebras in dimension 8). Arch.Math. 50 (1988), 511–525. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    R. Carles: Sur la structure des algèbres de Lie rigides. (On the structure of the rigid Lie algebras). Ann. Inst. Fourier 34 (1984), 65–82.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    G. Favre: Systeme de poids sur une algèbre de Lie nilpotente. Manuscr. Math. 9 (1973), 53–90.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    L.Y. Galitski, D.A. Timashev: On classification of metabelian Lie algebras. J. Lie Theory 9 (1999), 125–156.MathSciNetMATHGoogle Scholar
  6. [6]
    M.A. Gauger: On the classification of metabelian Lie algebras. Trans. Am. Math. Soc. 179 (1973), 293–329.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    M.P. Gong: Classification of Nilpotent Lie Algebras of Dimension 7 (Over Algebraically Closed Fields and R). Ph.D. Thesis, University of Waterloo, Waterloo, 1998.Google Scholar
  8. [8]
    M. Goze, Y. Khakimdjanov: Nilpotent Lie Algebras. Mathematics and its Applications 361. Kluwer Academic Publishers, Dordrecht, 1996.CrossRefGoogle Scholar
  9. [9]
    G. Leger, E. Luks: On derivations and holomorphs of nilpotent Lie algebras. Nagoya Math. J. 44 (1971), 39–50.MathSciNetMATHGoogle Scholar
  10. [10]
    B. Ren, D. Meng: Some 2-step nilpotent Lie algebras. I. Linear Algebra Appl. 338 (2001), 77–98.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    B. Ren, L. S. Zhu: Classification of 2-step nilpotent Lie algebras of dimension 8 with 2-dimensional center. Commun. Algebra 39 (2011), 2068–2081.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    P. Revoy: Algèbres de Lie metabeliennes. Ann. Fac. Sci. Toulouse, V. Ser., Math. 2 (1980), 93–100. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    L. J. Santharoubane: Kac-Moody Lie algebra and the classification of nilpotent Lie algebras of maximal rank. Can. J. Math. 34 (1982), 1215–1239.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    C. Seeley: 7-dimensional nilpotent Lie algebras. Trans. Am. Math. Soc. 335 (1993), 479–496.MathSciNetMATHGoogle Scholar
  15. [15]
    K.A. Umlauf: Ueber den Zusammenhang der endlichen continuirlichen Transformationsgruppen, insbesondere der Gruppen vom Range Null. Ph.D. Thesis, University of Leipzig, 1891. (In German.)Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2013

Authors and Affiliations

  1. 1.School of Mathematical Sciences and LPMCNankai UniversityTianjinP.R.China

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