Generators and relations for metabelian Lie algebras

Abstract

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