The classification of two step nilpotent complex Lie algebras of dimension 8. (English) Zbl 1291.17013

The classification of complex nilpotent Lie algebras of small dimension has a long history, yet only for dimension less than or equal to seven has it been completed. Further advancements have been made by putting conditions on the algebra and that is the strategy in this paper. The authors classify complex Lie algebras that are two step nilpotent. Thus for these algebras, the derived algebra is contained in the center. The problem is reduced to the cases when the center has dimension 2, 3 or 4. R. Bin and L. S. Zhu solved the problem when the center is 2-dimensional [Commun. Algebra 39, No. 6, 2068–2081 (2011; Zbl 1290.17004)]. In this paper the authors complete the other two cases.


17B30 Solvable, nilpotent (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras
17B05 Structure theory for Lie algebras and superalgebras


Zbl 1290.17004
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[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.) · Zbl 0672.17005
[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.) · Zbl 0628.17005
[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. · Zbl 0519.17004
[4] G. Favre: Systeme de poids sur une algèbre de Lie nilpotente. Manuscr. Math. 9 (1973), 53–90. · Zbl 0253.17011
[5] L.Y. Galitski, D.A. Timashev: On classification of metabelian Lie algebras. J. Lie Theory 9 (1999), 125–156. · Zbl 0923.17015
[6] M.A. Gauger: On the classification of metabelian Lie algebras. Trans. Am. Math. Soc. 179 (1973), 293–329. · Zbl 0267.17015
[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.
[8] M. Goze, Y. Khakimdjanov: Nilpotent Lie Algebras. Mathematics and its Applications 361. Kluwer Academic Publishers, Dordrecht, 1996.
[9] G. Leger, E. Luks: On derivations and holomorphs of nilpotent Lie algebras. Nagoya Math. J. 44 (1971), 39–50. · Zbl 0264.17003
[10] B. Ren, D. Meng: Some 2-step nilpotent Lie algebras. I. Linear Algebra Appl. 338 (2001), 77–98. · Zbl 0992.17005
[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. · Zbl 1290.17004
[12] P. Revoy: Algèbres de Lie metabeliennes. Ann. Fac. Sci. Toulouse, V. Ser., Math. 2 (1980), 93–100. (In French.) · Zbl 0447.17007
[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. · Zbl 0495.17011
[14] C. Seeley: 7-dimensional nilpotent Lie algebras. Trans. Am. Math. Soc. 335 (1993), 479–496. · Zbl 0770.17003
[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.) · JFM 23.0406.01
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