Some examples of nil Lie algebras

Abstract

The first examples of infinite dimensional affine nil algebras were constructed by E.S. Golod and I.R. Shafarevich \cite{GS}. These algebras have a strong exponential growth. Later L. Bartholdi and R.I. Grigorchuk \cite{BG} showed that the Lie algebra associated to the ``self--similar Grigorchuk groups, is graded--nil and has Gelfand-Kirillov dimension 1. Using this result L. Bartholdi \cite{B} was able to construct an infinite dimensional affine graded--nil associative algebra of Gelfand--Kirillov dimension 2 over a finite field of characteristic 2. Recently T. Lenagan and A. Smoktunowicz \cite{LS} constructed a family of infinite dimensional affine nil algebras of finite Gelfand--Kirillov dimensions over an arbitrary countable field.