Primitive ideals in semisimple affinoid enveloping algebras

非交换Iwasawa代数最早是由M.Lazard在1965年的论文[1]中定义的。本文给出了这类代数的基本性质。在接下来的二十年里，对这些代数的研究停滞不前。然而，与其他数学领域（例如数论）的联系的发现以及研究非交换诺特代数的新工具的开发表明，基于[1]的论文有所增加。在他们的综述[2]中，K.Ardakov和K.Brown描述了非交换岩泽代数的已知性质，并列出了该领域的一系列开放问题。自从该综述发表以来，人们对这些代数的性质的理解有了进展;然而，它们的结构和表示的许多方面仍然是未知的。非交换岩泽代数的数学兴趣之一是素理想的研究。以前在这方面的工作[3]，[4]对这类环的素理想的集合提出了限制。例如，[4，定理A]陈述了理想是完全素的一个充分条件。在[5]中，作者将特征p域上的Iwasawa代数的研究过渡到特征0域上的Iwasawa代数。主要结果是定理A，它指出半单有理Iwasawa代数的Verma模是忠实的。这项研究留下了开放的两个自然问题，共同构成一个计划，如果成功，将提供一个完整的分类素理想谱的半单理性岩泽代数。作为我的研究项目的一部分，我将解决两个问题中的第一个：问题[5，问题A]是否U\（g）n，K的每个具有K-有理无穷小中心特征标的本原理想都是简单仿射最高权模的零化子？有一些证据表明，对上述问题的答案是肯定的。在[6]中，M.Duflo证明了经典包络代数U（gK）的每一个具有K-有理无穷小中心特征标的本原理想都是最高权模的零化子。证明U\（g）n，K的任何本原理想都受U（gK）的控制就足够了，如果猜想成立，我们将朝着半单有理Iwasawa代数素理想谱的完全分类迈进一步.为证明而开发的工具可用于提供[5，问题B]的答案，并可被翻译为特征p中的字段的等价语句。为所得到的论文而构建的机器可潜在地应用于非交换代数和代数数论领域。该项目属于EPSRC代数研究领域的福尔斯。

Non-commutative Iwasawa algebras were first defined by M.Lazard in his 1965 paper [1]. In this paper, the basic properties of these algebras are derived. For the next two decades, the study of these algebras has stagnated. However, the discovery of connections with other mathematical areas such as number theory and the development of new tools for the study of non-commutative Noetherian algebras has shown an increase of papers build on [1]. In their survey[2], K.Ardakov and K.Brown provide a description of the known properties of non-commutative Iwasawa algebras and list a set of open questions in the field. Since the survey was published , there has been progress in understanding the properties of these algebras; however, many aspects of their structure and representation remain unknown.One thing of mathematical interest in the non-commutative Iwasawa algebras setting is the study of prime ideals. Previous work in this area [3], [4] puts constraints on the set of prime ideals for such rings. For example, [4, Theorem A] states a sufficient condition for an ideal to be completely prime. In [5], the authors make the transition fromstudying Iwasawa algebras over fields of characteristic p to Iwasawa algebras over fields of characteristic 0. The main result is TheoremA which states that Verma modules for semisimple rational Iwasawaalgebras are faithful. This research left open two natural questions, constituting together a program which, if successful, will provide a complete classification of the prime ideal spectrum of semisimple rational Iwasawa algebras. As part of my research project I will tackle the first of the two questions: Question [5, Question A] Is it the case that every primitive ideal of U\(g)n,K with K-rational infinitesimal central character is the annihilator of a simple affinoid highest weight module? There is some evidence that points toward a positive answer to the question above. In [6], M.Duflo proved that every primitive ideal of the classical enveloping algebra U(gK) with K-rational in- finitesimal central character is the annihilator of a highest weight module. To answer the question, it would be enough to prove that any primitive ideal of U\(g)n,K is controlled by U(gK).If the conjecture is proven to be true, we will take a step forward towards having a complete classification of the prime ideal spectrum of semisimple rational Iwasawa algebras. The tools developed for the proof may be used to provide an answer to [5, Question B] and could be translated to equivalent statements for fields in characteristic p. The machinery built for the resulting thesis could potentiallybe applied in the fields of non-commutative algebra and algebraicnumber theory.This project falls within the EPSRC Algebra research area.