Nottingham Lie algebras with diamonds of finite type

In her PhD thesis at Trento, Marina Avitabile had constructed certain infinite-dimensional, graded thin Lie algebras over finite fields, which share some common features with the graded Lie algebra of the celebrated Nottingham group. In this paper, we solve the uniqueness question for these algebra, as detailed below. The graded Lie algebra associated to the Nottingham group over the field with p elements, with respect to its lower central series, has been studied by the first author. This is a thin Lie algebra, with second diamond (homogeneous component of dimension 2) in weight p, and can be described as (the positive part of) a twisted loop algebra of the smallest Zassenhaus algebra W(1: 1). More generally, certain twisted loop algebras of the Zassenhaus algebrasW(1: n) over the field with p elements have been studied by the first author. These are thin Lie algebras, with diamonds occurring exactly in the weights congruent to 1 modulo q-1, where q = p^n. The goal of this paper is to give a complete description of a more general class of thin Lie algebras with diamonds (according a slightly more general definition) in these weights. These algebras we call Nottingham Lie algebras. We are naturally led to discuss some related features of thin Lie algebras with the second diamond in weight q.