Narrow Lie Algebras: A Coclass Theory and a Characterization of the Witt Algebra

Abstract

In this paper we examine some narrowness conditions for Lie algebras over a fieldFof characteristic zero. In particular we show that the natural analogs of the main coclass conjectures forp-groups hold in the context of N-graded Lie algebrasLwhich are generated by their first homogeneous componentL1. While Lie algebras of finite coclass need not be soluble, we show that the positive part of the Witt algebra Der F[x] is the only non-soluble N-graded Lie algebraLof coclass 1 which is generated byL1andL2.