The structure of thin Lie algebras up to the second diamond
arXiv:0812.1250
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.
35 pages