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Nottingham algebras are a class of just-infinite-dimensional, modular, N-graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called diamonds. The first diamond occurs in degree 1, and the second occurs in degree q, a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a type, either belonging to the underlying field or equal to ∞. A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper, we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an arbitrary Nottingham algebra L can be assigned a type, in such a way that the degrees and types of the diamonds completely describe L. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals q - 1. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type ∞.

Thin Lie algebras are graded Lie algebras L = ⊕i = 1∞Li with dim Li ≤ 2 for all i, and satisfying a more stringent but natural narrowness condition modeled on an analogous condition for pro-p-groups. The two-dimensional homogeneous components of L, which include L1, 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 Lk is the second diamond of L, then the quotient L/Lk is a graded Lie algebras of maximal class. In odd characteristic p, the quotient L/Lk 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: 3, 5, a power of p, or one less than twice a power of p. However, the quotient L/Lk need not be metabelian in characteristic two. We describe here all the possibilities for L/Lk up to isomorphism. In particular, we prove that k + 1 equals a power of two. © World Scientific Publishing Company.

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.