Diamonds of finite type in thin Lie algebras

Abstract

Borrowing some terminology from pro-p groups, thin Lie algebras are N-graded Lie algebras of width two and obliquity zero, generated in degree one. In particular, their homogeneous components have degree one or two, and they are termed diamonds in the latter case. In one of the two main subclasses of thin Lie algebras the earliest diamond after that in degree one occurs in degree 2q-1, where q is a power of the characteristic. This paper is a contribution to a classification project of this subclass of thin Lie algebras. Specifically, we prove that, under certain technical assumptions, the degree of the earliest diamond of finite type in such a Lie algebra can only have a certain form, which does occur in explicit examples constructed elsewhere.