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

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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.

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.

## People |
## ORCID iD |

Adam Jones (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509711/1 | 01/10/2016 | 30/09/2021 | |||

1789790 | Studentship | EP/N509711/1 | 01/10/2016 | 31/03/2020 | Adam Jones |

Description | My research concerns the representation theory of p-adic Lie groups, and my particular project is the classification of ideals in the Iwasawa algebra, whose module structure encodes the continuous representations of the group. My ultimate aim is to be able to completely describe all prime ideals, extending previous research done by Konstantin Ardakov, Simon Wadsley, Ken Brown, William Woods and others. Before I began, the best known result is this field was the complete classification of all prime ideals in the Iwasawa algebra of a nilpotent group over a field of characteristic p, achieved by Konstantin Ardakov in 2012. I have been able to extend this classification to certain small classes of non-nilpotent solvable groups, also in characteristic p, and I have submitted these results for publication. I have also begun to examine the picture in characteristic 0, where I have established that the same methods do not carry over well, and I have developed new methods of attacking the problem, concerning Lie theoretic representation theory. |

Exploitation Route | I am hoping that my work may have applications to the representation theory of compact p-adic Lie groups. Understanding the ideal structure of Iwasawa algebras is highly useful within this theory, particularly since any irreducible representation of such a group will correspond to a primitive ideal in the algebra. Most research within this branch of representation theory studies representations of semisimple algebraic groups, whereas my work concerns solvable groups, which are used far less. However, it is sometimes required to also consider representations of Borel subgroups, which are always solvable, so this work may have important applications there. |

Sectors | Education,Other |

URL | https://arxiv.org/pdf/1811.12842 |