Primitive ideals in semisimple affinoid enveloping algebras

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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.