The structure of thin Lie algebras up to the second diamond

TL;DR

This paper characterizes the initial structure of thin Lie algebras up to the second diamond, revealing that the quotient by the second diamond is a graded Lie algebra of maximal class with specific properties depending on the characteristic.

Contribution

It provides a complete description of the structure of thin Lie algebras up to the second diamond, including the possible forms of the quotient algebra in different characteristics.

Findings

In characteristic not two, the quotient is metabelian and uniquely determined by its dimension.

In characteristic two, the quotient need not be metabelian, with all possibilities described.

The second diamond position plus one is always a power of two.

Abstract

Thin Lie algebras are Lie algebras L, graded over the positive integers, with all homogeneous components of dimension at most two, and satisfying a more stringent but natural narrowness condition modeled on an analogous one for pro-p groups. The two-dimensional homogeneous components of L, which include that of degree one, are named diamonds. Infinite-dimensional thin Lie algebras with various diamond patterns have been produced, over fields of positive characteristic, as loop algebras of suitable finite-dimensional simple Lie algebras, of classical or of Cartan type depending on the location of the second diamond. The goal of this paper is a description of the initial structure of a thin Lie algebra, up to the second diamond. Specifically, if L_k is the second diamond of L, then the quotient L/L^k is a graded Lie algebras of maximal class. In characteristic not two, L/L^k is known to be metabelian, and hence uniquely determined up to isomorphism by its dimension k, which ranges in an explicitly known set of possible values. The quotient L/L^k need not be metabelian in characteristic two. We describe here all the possibilities for L/L^k up to isomorphism. In particular, we prove that k+1 equals a power of two.