# Primitive ideals in semisimple affinoid enveloping algebras

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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 Theorem

A 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 algebraic

number theory.

This project falls within the EPSRC Algebra research area.

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 Theorem

A 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 algebraic

number theory.

This project falls within the EPSRC Algebra research area.

## People |
## ORCID iD |

Ioan Stanciu (Student) |

### Studentship Projects

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

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

1789785 | Studentship | EP/N509711/1 | 01/10/2016 | 31/03/2020 | Ioan Stanciu |

Description | Representation theory is a key way of studying properties of abstract algebraic structures by studying properties of their modules. In my work, I am focusing on the representation theory of a certain family of rings, namely affinoid enveloping algebras of semisimple Lie algebras. These are topological completions of the enveloping algebras of classical semisimple Lie algebras, a family of rings that has been intensively studied since 1970's. One big step in understanding the ring structure of classical enveloping algebras is given by Duflo's theorem which states that any primitive ideal is the annihilator of a simple quotient of a Verma module. My primary objective is to obtain an analogue of this theorem for the affinoid enveloping algebra of a semisimple Lie algebra. The techniques used in Duflo's theorem do not extend to the affinoid enveloping algebra, so a new a technique had to be developed. In 2013, Ardakov and Wadsley proved an affinoid version of the classical Beilinson-Bernstein theorem, relating the representation theory of the affinoid enveloping algebra to the study of modules over affinoid sheaves of diffferential operators allowing for a geometric study of the module structure. I have managed to extend the affinoid Ardakov-Wadsley theorem to the equivariant setting and together with an affinoid version of the classical Borho-Brylinski theorem, I have managed to obtain a geometric proof of affinoid Duflo's theorem for primitive ideals with rational central character. |

Exploitation Route | The research answers a question posed by Ardakov and Wadsley. It opens new ways to study the representation theory of the affinoid enveloping algebra of a semisimple Lie algebra. The proof is mostly geometric, but there are still a lot of algebraic questions left open. In particular, one would like to define an affinoid category O, which should share most of properties of the classical BGG category O for classical enveloping algebra of a semisimple Lie algebra. Once the definition is made, one should check how many of the standard theorems/propositions carry from the classical setting to the affinoid setting. Another further path of research is the classification of primitive ideals in the Iwasawa algebra of semisimple compact p-adic Lie group. In one of their papers, Ardakov and Wadsley suggest that one may use the work of Schneider-Teitalbaum together with an affinoid version of Duflo's theorem to obtain this classification. Since the representation theory of the Iwasawa algebra controls the continous representation of the corresponding p-adic Lie group, we hope that further progress in this topic can be achieved. |

Sectors | Education,Other |

URL | https://sites.google.com/view/ioan-stanciu/research-projects |

Description | Co-organiser of Junior Algebra Seminar |

Form Of Engagement Activity | A formal working group, expert panel or dialogue |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Postgraduate students |

Results and Impact | Together with my colleague, I have organised the Junior Algebra Seminar; the goal is to encourage Ph.D students/ early career researchers in the field of algebra to give an one hour talk about their current research. |

Year(s) Of Engagement Activity | 2019,2020 |