The earliest diamond of finite type in Nottingham algebras

TL;DR

This paper investigates the structure of Nottingham algebras, focusing on the earliest occurrence of finite type diamonds and how they relate to the algebra's overall pattern, revealing new structural insights.

Contribution

It establishes the uniqueness of certain finite-dimensional quotients determining Nottingham algebras with mixed diamond types and characterizes the placement of infinite type diamonds.

Findings

The earliest finite type diamond occurs after a sequence of infinite type diamonds.

Each known example with mixed diamond types is uniquely determined by a finite-dimensional quotient.

The paper determines the maximum number of infinite type diamonds before the first finite type diamond.

Abstract

We prove several structural results on Nottingham algebras, a class of infinite-dimensional, modular, 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. Each diamond past the second is assigned a type, which either belongs to the underlying field or is $\infty$. Nottingham algebras with a variety of diamond patterns are known. In particular, some have diamonds of both finite and infinite type. We prove that each of those known examples is uniquely determined by a certain finite-dimensional quotient. Finally, we determine how many diamonds of type $\infty$ may precede the earliest diamond of finite type in an arbitrary Nottingham algebra.