论文标题
诺丁汉代数的钻石距离
Diamond distances in Nottingham algebras
论文作者
论文摘要
诺丁汉代数是一类仅仅耗时的，模块化的，$ \ mathbb {n} $ - 分级的lie代数，其中包括与诺丁汉集团相关的与诺丁汉集团相关的分级lie代数。诺丁汉代数的均匀组成部分具有一两个尺寸，在后一种情况下，它们被称为钻石。第一颗钻石以$ 1 $的速度出现，第二颗钻石以$ Q $（$ Q $）的特征发生。诺丁汉代数的许多例子都是已知的，其中每个钻石都可以分配一种类型，要么属于基础领域或等于$ \ infty $。 诺丁汉代数的前瞻性分类需要描述所有可能的钻石模式。在本文中，我们为该目标建立了一些至关重要的贡献。人们表明，所有诺丁汉代数$ l $的所有钻石都可以分配一种类型，以使钻石的学位和类型完全描述$ l $。同时，我们证明诺丁汉代数中任何两次钻石的学位差异等于$ q-1 $。作为调查的副产品，我们对所有钻石都有$ \ infty $的诺丁汉代数进行了分类。
Nottingham algebras are a class of just-infinite-dimensional, modular, $\mathbb{N}$-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 $\infty$. 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 $\infty$.
