7-dimensional nilpotent Lie algebras

Seeley, Craig

doi:10.1090/S0002-9947-1993-1068933-4

$7$-dimensional nilpotent Lie algebras
HTML articles powered by AMS MathViewer

by Craig Seeley

Trans. Amer. Math. Soc. 335 (1993), 479-496

DOI: https://doi.org/10.1090/S0002-9947-1993-1068933-4

PDF | Request permission

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

José María Ancochéa-Bermúdez and Michel Goze, Sur la classification des algèbres de Lie nilpotentes de dimension $7$, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 17, 611–613 (French, with English summary). MR 845651
V. I. Arnol′d, Critical points of smooth functions, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 19–39. MR 0431217

Construction of nilpotent Lie algebras over arbitrary fields

Algorithms for central extensions of Lie algebras

Harald Bjar and Olav Arnfinn Laudal, Deformation of Lie algebras and Lie algebras of deformations, Compositio Math. 75 (1990), no. 1, 69–111. MR 1059956

Variétés d’algèbre de Lie

point de vue global et rigidité

Chong-yun Chao, Uncountably many nonisomorphic nilpotent Lie algebras, Proc. Amer. Math. Soc. 13 (1962), 903–906. MR 148715, DOI 10.1090/S0002-9939-1962-0148715-3
Michael A. Gauger, On the classification of metabelian Lie algebras, Trans. Amer. Math. Soc. 179 (1973), 293–329. MR 325719, DOI 10.1090/S0002-9947-1973-0325719-0
Fritz Grunewald and Joyce O’Halloran, Varieties of nilpotent Lie algebras of dimension less than six, J. Algebra 112 (1988), no. 2, 315–325. MR 926608, DOI 10.1016/0021-8693(88)90093-2
Fritz Grunewald and Joyce O’Halloran, Nilpotent groups and unipotent algebraic groups, J. Pure Appl. Algebra 37 (1985), no. 3, 299–313. MR 797867, DOI 10.1016/0022-4049(85)90103-3
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793

The local moduli problem. Applications to isolated hypersurface singularities

L. Magnin, Sur les algèbres de Lie nilpotentes de dimension $\leq 7$, J. Geom. Phys. 3 (1986), no. 1, 119–144 (French, with English summary). MR 855573, DOI 10.1016/0393-0440(86)90005-7
A. Malcev, On solvable Lie algebras, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 9 (1945), 329–356 (Russian, with English summary). MR 0022217
John N. Mather and Stephen S. T. Yau, Classification of isolated hypersurface singularities by their moduli algebras, Invent. Math. 69 (1982), no. 2, 243–251. MR 674404, DOI 10.1007/BF01399504
V. V. Morozov, Classification of nilpotent Lie algebras of sixth order, Izv. Vysš. Učebn. Zaved. Matematika 1958 (1958), no. 4 (5), 161–171 (Russian). MR 0130326
Mustapha Romdhani, Classification of real and complex nilpotent Lie algebras of dimension $7$, Linear and Multilinear Algebra 24 (1989), no. 3, 167–189. MR 1007253, DOI 10.1080/03081088908817910

Classification of nilpotent Lie algebras of order

32

Einfache eliptische Singularitäten

23

L. J. Santharoubane, Infinite families of nilpotent Lie algebras, J. Math. Soc. Japan 35 (1983), no. 3, 515–519. MR 702774, DOI 10.2969/jmsj/03530515
Craig Seeley, Degenerations of central quotients, Arch. Math. (Basel) 56 (1991), no. 3, 236–241. MR 1091876, DOI 10.1007/BF01190210
Craig Seeley, Degenerations of $6$-dimensional nilpotent Lie algebras over $\textbf {C}$, Comm. Algebra 18 (1990), no. 10, 3493–3505. MR 1063991, DOI 10.1080/00927879008824088
Craig Seeley, Some nilpotent Lie algebras of even dimension, Bull. Austral. Math. Soc. 45 (1992), no. 1, 71–77. MR 1147246, DOI 10.1017/S0004972700037023

Seven-dimensional nilotent Lie algebras over the complex numbers

Craig Seeley and Stephen S.-T. Yau, Variation of complex structures and variation of Lie algebras, Invent. Math. 99 (1990), no. 3, 545–565. MR 1032879, DOI 10.1007/BF01234430
Tor Skjelbred and Terje Sund, Sur la classification des algèbres de Lie nilpotentes, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 5, A241–A242 (French, with English summary). MR 498734

Über die Zusammensetzung der endlichen continuierlichen Transformationsgruppen insbesondere der Gruppen vom Range Null

David Wilkinson, The groups of exponent $p$ and order $p^7$ ($p$ any prime), J. Algebra 118 (1988), no. 1, 109–119. MR 961329, DOI 10.1016/0021-8693(88)90051-8
Stephen S.-T. Yau, Singularities defined by $\textrm {sl}(2,\textbf {C})$ invariant polynomials and solvability of Lie algebras arising from isolated singularities, Amer. J. Math. 108 (1986), no. 5, 1215–1239. MR 859777, DOI 10.2307/2374603