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Generators and relations for metabelian Lie algebras

Published online by Cambridge University Press:  24 October 2008

Jürgen Wisliceny  and
Rainer Zerck
Jürgen Wisliceny
Affiliation:
Universität Rostock, Aussenstelle Güstrow, Fachbereich Mathematik, Goldberger Str. 12, D-O-2600 Güstrow, Germany
Rainer Zerck
Affiliation:
Cottbuser Platz 7, D-O-1150 Berlin, Germany
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Extract

The aim of this paper is to prove an inequality of Golod-Shafarevich type for metabelian Lie algebras and to show that this inequality is best possible up to a constant factor. Investigations of this kind were started in [4] in connection with the solution of the class field tower problem. It was shown that if there is a finite p-group which may be presented as a pro-p-group with d ≥ 2 generators and r relations then the inequality r > ¼(d − 1)2 holds; and Vinberg [8] improved this result by showing

The inequality (1) also holds for presentations of nilpotent Lie algebras (see [6]) (with the exception (d, r) = (2, 1)) and nilpotent associative algebras (see [4, 8, 6]). If the relations have of degree at least m ≥ 3 then more relations are needed. More precisely, Koch [5] has shown that the inequality

holds for finite p-groups, and the corresponding results hold for associative algebras and Lie algebras.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 3 , November 1992 , pp. 449 - 453
Copyright
Copyright © Cambridge Philosophical Society 1992

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References

REFERENCES

[1]Amayo, R. K. and Stewart, I.. Infinite-dimensional Lie Algebras (Noordhoff, 1974).CrossRefGoogle Scholar
[2]Bahturin, Yu. A.. Identical Relations in Lie Algebras (VNU Science Press, 1987).Google Scholar
[3]Chen, K. T.. Integration in free groups. Ann. of Math. (2) 54 (1951), 147–162.CrossRefGoogle Scholar
[4]Golod, E. S. and Shafarevich, I. R.. On the class field tower. Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 261–272.Google Scholar
[5]Koch, H.. Galoissche Theorie der p-Erweiterungen (Deutscher Verlag der Wissenschaften, 1970).CrossRefGoogle Scholar
[6]Koch, H.. Erzeugenden- und Relationenrang für endlich dimensionale nilpotente Liesche Algebren. Algebra i Logika 16 (1977), 364–374.Google Scholar
[7]Sauerbier, G.. Zur Konstruktion nilpotenter Lie-Algebren. Wiss. Z. Pädagog. Hochsch. Güstrow 27 (1989), 237–246.Google Scholar
[8]Vinberg, E. B.. On the theorem concerning the infinite dimension of an associative algebra. Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 209–214.Google Scholar
[9]Wisliceny, J.. Zur Darstellung von pro-p-Gruppen und Lieschen Algebren durch Erzeugende und Relationen. Math. Nachr. 102 (1981), 57–78.CrossRefGoogle Scholar
[10]Wisliceny, J.. Konstruktion nilpotenter assoziativer Algebren mit wenig Relationen. Math. Nachr. 147 (1990), 53–60.CrossRefGoogle Scholar
[11]Wisliceny, J.. Konstruktion endlich graduierter assoziativer Algebren. In Beitr. Jahrestagung Algebra u. Grenzgebiete, pp. 105–110.Google Scholar