Classification of two-sided ideals in non-commutative Iwasawa algebra

可交换环k上的无限群G的完备群代数被构造为群环k[G/N]在N遍历G的开正规子群时的逆极限。它是表示论中的一个重要对象，因为研究完备群代数的模等价于研究群的连续k-表示。完备群代数的一个特别突出的例子是Iwasawa代数，其中G被取为紧的p进解析群，k被取为混合特征（0,p）中的完备离散赋值环。Iwasawa代数的研究起源于Iwasawa理论，它们在其中扮演着重要的角色。在20世纪50年代末，Iwasawa研究了环裂扩展的塔，并发现通过考虑它们的理想类群的逆极限是交换Iwasawa代数上的一个模，可以更实际地获得它们的并的理想类群——无限伽罗瓦扩展的信息。这项工作在数论中有许多应用，并且在1965年，Lazard能够将Iwasawa代数的概念扩展到非交换情况，这可以产生非常不同的结果。为了最大限度地发挥Iwasawa代数的效用，理解Iwasawa代数的结构是很重要的，我的研究主要关注非交换Iwasawa代数，并研究它们的双面理想分类，这与任何代数一样，在描述它们及其性质时都是必不可少的。在这个领域，一个特别重要的问题是理解Iwasawa代数的素谱——也就是说，对它的素理想进行分类，我将把我的大部分精力集中在这个领域。最近Konstantin Ardakov对这个问题作了一些发展，当G是一个幂零的有限秩完全p值群，且k具有特征p时，在这种情况下，G的非交换Iwasawa代数的素谱与交换Iwasawa代数的忠实素谱的不相交并之间存在一一对应。这个事实是非常有用的，因为它把简单的交换性情况和难以预测的非交换性情况联系起来。在我的研究中，我将以康斯坦丁·阿尔达科夫（Konstantin Ardakov）等人的工作为基础，并尝试将这一概念扩展到更一般的p进解析群类中。特别是，我将试图证明G是幂零的假设是不必要的，并且结果在G上较弱的条件下成立。我还将研究使用素数谱之间的对应关系，以提供k具有特征为零时的类似概念。特别地，我将考虑k是p进整数环的情况，这是表示理论中非常重要的情况。该项目属于EPSRC代数研究领域。

The completed group algebra of a profinite group G over a commutative ring k is constructed as the inverse limit of the group rings k[G/N] as N runs over the open normal subgroups of G. It is an important object in representation theory, because studying modules of the completed group algebra is equivalent to studying continuous k-representations of the group. A particularly prominent example of a completed group algebra is an Iwasawa algebra, where G is taken to be a compact p-adic analytic group, and k is taken to be a complete, discrete valuation ring in mixed characteristic (0,p).The study of Iwasawa algebras has its origins within Iwasawa theory, where they play an important role. In the late 1950s, Iwasawa studied towers of cyclotomic extensions, and found that by considering the inverse limit of their ideal class groups to be a module over a commutative Iwasawa algebra, it became more practical to obtain information about the ideal class group of their union - an infinite Galois extension. This work had numerous applications within number theory, and in 1965, Lazard was able to extend the notion of an Iwasawa algebra to the non-commutative case, which can produce very different results.It is important to understand the structure of Iwasawa algebras in order to maximise their usefulness, and my research primarily concerns non-commutative Iwasawa algebras and investigates classification of their two-sided ideals, which as with any algebra is essential in describing them and their properties.A particularly important problem in this area is understanding the prime spectrum of the Iwasawa algebra - that is, classifying its prime ideals, and I will concentrate much of my efforts in this area. Some developments of this problem have recently been made be Konstantin Ardakov, in the case where G is a nilpotent, complete p-valued group of finite rank, and k has characteristic p. In this case, there is a one to one correspondence between the prime spectrum of the non-commutative Iwasawa algebra of G, and the disjoint union of the faithful prime spectra of commutative Iwasawa algebras. This fact is highly useful because it relates the simpler commutative case to the less predictable non-commutative case.In my research, I will build on the work of Konstantin Ardakov among others, and try and extend this notion for more general classes of p-adic analytic groups. In particular, I will attempt to prove that the assumption that G is nilpotent is not necessary, and the result holds for weaker conditions on G. I will also work on using the correspondence between the prime spectra to provide a similar notion for when k has characteristic zero. In particular, I will consider the case when k is the ring of p-adic integers, a highly important case in representation theory.This project falls within the EPSRC Algebra research area.