Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ 7$-dimensional nilpotent Lie algebras


Author: Craig Seeley
Journal: Trans. Amer. Math. Soc. 335 (1993), 479-496
MSC: Primary 17B30
DOI: https://doi.org/10.1090/S0002-9947-1993-1068933-4
MathSciNet review: 1068933
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: All $ 7$-dimensional nilpotent Lie algebras over $ \mathbb{C}$ are determined by elementary methods. A multiplication table is given for each isomorphism class. Distinguishing features are given, proving that the algebras are pairwise nonisomorphic. Moduli are given for the infinite families which are indexed by the value of a complex parameter.


References [Enhancements On Off] (What's this?)

  • [1] J. Ancochea-Bermudez and M. Goze, Classification des algèbres de Lie nilpotentes complexes de dimension $ 7$, C.R. Acad. Sci. Paris Sér. I Math. 302 (1986), 611-613. MR 845651 (87g:17009)
  • [2] V. I. Arnold, Critical points of smooth functions, Proc. Vancouver ICM Conference, 1974, pp. 19-39. MR 0431217 (55:4218)
  • [3] R. Beck and B. Kolman, Construction of nilpotent Lie algebras over arbitrary fields Proc. of 1981 ACM Sympos. on Symbolic and Algebraic Computation (Paul S. Wang, ed.), ACM, New York, 1981, pp. 169-174.
  • [4] -, Algorithms for central extensions of Lie algebras, Proc. 1981 ACM Sympos. on Symbolic and Algebraic Computation (Paul S. Wang, ed.), ACM, New York, 1981, pp. 175-178.
  • [5] H. Bjar and O. Laudal, Deformation of Lie-algebras and Lie-algebras of deformations. Application to the study of hypersurface singularities, Preprint series 3, University of Oslo, Norway, 1987. MR 1059956 (91f:14002)
  • [6] R. Carles, Variétés d'algèbre de Lie: point de vue global et rigidité, Thesis, University of Poitiers, France, 1984.
  • [7] C. Chao, Uncountably many non-isomorphic nilpotent Lie algebras, Proc. Amer. Math. Soc. 13 (1962), 903-906. MR 0148715 (26:6221)
  • [8] M. Gauger, On the classification of metabelian Lie algebras, Trans. Amer. Math. Soc. 179 (1973), 293-329. MR 0325719 (48:4066)
  • [9] F. Grunewald and J. O'Halloran, Varieties of Lie algebras of dimensions less than six, J. Algebra 112 (1988), 315-325. MR 926608 (89c:17018)
  • [10] -, Nilpotent algebras and unipotent algebraic groups, J. Pure Appl. Algebra 37 (1985), 299-313. MR 797867 (87e:20069)
  • [11] J. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math., vol. 9, Springer-Verlag, New York, 1972. MR 0323842 (48:2197)
  • [12] N. Jacobson, Lie algebras, Tracts in Pure and Appl. Math., vol. 10, Interscience, New York, 1962. MR 0143793 (26:1345)
  • [13] O. Laudal and M. Pfister, The local moduli problem. Applications to isolated hypersurface singularities, Preprint series 11, University of Oslo, Norway, 1986.
  • [14] L. Magnin, Sur les algebres de Lie nilpotentes de dimension $ \leq 7$, J. Geom. Phys. 3 (1986), 119-144. MR 855573 (87k:17012)
  • [15] A. Malcev, Solvable Lie algebras, Amer. Math. Soc. Transl. (1) 9 (1962), 228-262. MR 0022217 (9:173f)
  • [16] J. Mather and S. Yau, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982), 243-251. MR 674404 (84c:32007)
  • [17] V. Morozov, Classification of nilpotent Lie algebras of $ 6$th order, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (1958), no. 5, 161-171. MR 0130326 (24:A190)
  • [18] M. Romdhani, Classification of real and complex nilpotent Lie algebras of dimension $ 7$, Linear and Multilinear Algebra 24 (1989), 167-189. MR 1007253 (90j:17021)
  • [19] È. N. Safiullina, Classification of nilpotent Lie algebras of order $ 7$ (Thesis), Candidates Works, no. 64, Math. Mech. Phys., Izdat. Kazan Univ., Kazan, 1964, pp. 66-69. MR 32:5797
  • [20] K. Saito, Einfache eliptische Singularitäten, Invent. Math. 23 (1974), 289-325.
  • [21] L. J. Santharoubane, Infinite families of nilpotent Lie algebras, J. Math. Soc. Japan 35 (1983), 515-519. MR 702774 (85b:17006)
  • [22] Craig Seeley, Degenerations of central quotients, Arch. Math. 56 (1991), 236-241. MR 1091876 (92b:17012)
  • [23] -, Degenerations of $ 6$-dimensional nilpotent Lie algebras, Comm. Algebra 18 (1990), 3493-3505. MR 1063991 (92a:17017)
  • [24] -, Some nilpotent Lie algebras of even dimension, Bull. Austral. Math. Soc. 45 (1992), 71-78. MR 1147246 (93b:17036)
  • [25] -, Seven-dimensional nilotent Lie algebras over the complex numbers, Thesis, University of Illinois at Chicago, 1988.
  • [26] Craig Seeley and S.-T. Yau, Variation of complex structures and variation of Lie algebras, Invent. Math. (1990), 545-565. MR 1032879 (90k:32067)
  • [27] T. Skjelbred and T. Sund, On the classification of nilpotent Lie algebras, C.R. Acad. Sci. Paris Sér A 286 (1978). MR 0498734 (58:16802)
  • [28] K. A. Umlauf, Über die Zusammensetzung der endlichen continuierlichen Transformationsgruppen insbesondere der Gruppen vom Range Null, Thesis, University of Leipzig, Germany, 1891.
  • [29] D. Wilkinson, Groups of exponent $ p$ and order $ {p^7}$ ($ p$ any prime), J. Algebra 118 (1988), 109-119. MR 961329 (89h:20034)
  • [30] S.-T. Yau, Singularities defined by $ sl(2,\mathbb{C})$ invariant polynomials and solvability of Lie algebras arising from isolated singularities, Amer. J. Math. 108 (1986), 1215-1240. MR 859777 (88d:32022)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B30

Retrieve articles in all journals with MSC: 17B30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1068933-4
Keywords: Nilpotent Lie algebra
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society