Diamonds of finite type in thin Lie algebras

TL;DR

This paper investigates the structure of thin Lie algebras, focusing on the degrees of their earliest diamonds of finite type, and establishes constraints on these degrees under specific conditions, aiding classification efforts.

Contribution

It proves that, under certain assumptions, the degree of the earliest finite type diamond in a subclass of thin Lie algebras follows a specific form, aligning with known examples.

Findings

The degree of the earliest finite type diamond is constrained to a specific form.

The results support classification of a subclass of thin Lie algebras.

Explicit examples match the proven constraints.

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.