Primitive ideals in semisimple affinoid enveloping algebras

非交换Iwasawa代数最初是由M.Lazard在他1965年的论文[1]中定义的。本文给出了这些代数的基本性质。在接下来的二十年里，对这些代数的研究停滞不前。然而，发现与其他数学领域的联系，如数论和研究非交换诺etherian代数的新工具的发展，表明建立在[1]上的论文有所增加。在他们的调查[2]中，K.Ardakov和K.Brown提供了对非交换Iwasawa代数的已知性质的描述，并列出了该领域的一组开放问题。自从这项调查发表以来，人们在理解这些代数的性质方面取得了进展；然而，它们的结构和表现的许多方面仍然未知。在非交换的伊wasawa代数集合中，有一件数学上有趣的事情是对素数理想的研究。先前在这一领域的工作[3]，[4]对这类环的素理想集施加了约束。例如，[4，定理A]陈述了理想是完全素数的充分条件。在[5]中，作者从研究特征为p的域上的Iwasawa代数过渡到研究特征为0的域上的Iwasawa代数。主要结果是定理，它表明半简单有理iwasawa代数的Verma模块是可靠的。这项研究留下了两个自然的问题，它们组成了一个程序，如果成功的话，将提供半单理性Iwasawa代数的素理想谱的完整分类。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：问题[5，问题A]是否每个具有K-有理无穷小中心特征的U\(g)n，K的原始理想都是简单仿射最高权重模块的湮灭子？有一些证据指向上述问题的积极答案。M.Duflo在1996年证明了具有k -有理有限元中心特征的经典包络代数U（gK）的每一个原始理想都是最高权模的湮灭子。要回答这个问题，只要证明U\(g)n，K的任何原始理想是由U（gK）控制的就足够了。如果这个猜想被证明是正确的，我们将朝着半单有理Iwasawa代数素数理想谱的完全分类迈进一步。为证明开发的工具可用于提供[5，问题B]的答案，并可转换为特征p域中的等价陈述。为所得论文构建的机制可能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。

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.