Primitive ideals in semisimple affinoid enveloping algebras

非交换岩泽代数是M.Lazard在1965年的论文[1]中首次定义的。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，随着与其他数学领域的联系的发现，如数论，以及研究非交换Noether代数的新工具的发展，在文献[1]的基础上发表的论文有所增加。在他们的综述[2]中，K.Ardakov和K.Brown描述了非交换Iwa awa代数的已知性质，并列出了该领域的一组公开问题。自从这篇综述发表以来，在理解这些代数的性质方面取得了进展，然而，关于它们的结构和表示的许多方面仍然是未知的。在非交换岩泽代数环境中，一件数学感兴趣的事情是对素理想的研究。前人在这方面的工作[3]、[4]对这类环的素数理想集提出了限制。例如，[4，定理A]给出了理想是完全素数的一个充分条件。在文[5]中，作者将特征为p的域上的岩泽代数的研究转化为特征为0的域上的岩泽代数的研究。主要结果是定理，它表明半单有理Iwa-代数的Verma模是忠实的。这项研究留下了两个自然的问题，共同组成了一个程序，如果成功的话，它将提供半单有理岩泽代数的素理想谱的完整分类。作为我的研究项目的一部分，我将回答两个问题中的第一个：问题[5，问题A]，是否每个具有K-有理无穷小中心特征标的U\(G)n，K的原始理想都是单仿射最高权模的零化子？有一些证据表明，上述问题的答案是肯定的。M.Duflo在文[6]中证明了具有K-有理无穷小中心特征标的经典包络代数U(Gk)的每个本原理想都是最高权模的零化子。为了回答这个问题，只要证明U(G)n，K的任一本原理想是由U(Gk)控制的就足够了。如果这个猜想被证明是正确的，我们就向对半单有理Iwa awa代数的素理想谱进行完全分类又迈进了一步。为证明而开发的工具可用于提供对[5，问题B]的答案，并可被转化为特征p中的域的等价陈述。为所得到的论文建立的机器可潜在地应用于非对易代数和代数数论领域。

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.