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

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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.