A day and a half with Dixmier and Moeglin

30 June – 1 July 2022,  Department of Mathematics,  University of Manchester, UK

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The programme sometimes called the ÒDixmier-Moeglin equivalenceÓ started in the 80Õs when Dixmier and Moeglin characterised, in topological and algebraic terms, the primitive ideals of enveloping algebras of complex finite dimensional Lie algebras. This had interesting consequences in classifying their irreducible representations. Nowadays the programme focuses on determining which algebras share this same property; namely, those for which primitive ideals can be characterised topologically and algebraically. In the past 7 years, tools from model theory (branch of maths logic) have proven to be useful in this programme - sometimes giving positive answers and sometimes negative - .

In this meeting, we aim to bring together active researchers from algebra and logic to present recent developments on the subject, bring further awareness on the interactions between the groups, and initiate/stimulate collaborations. We expect this topic to be interesting to academics and PGR students in both algebra and/or logic groups (and further).

REGISTRATION: There is no registration fee but in order to estimate number of participants, please do register in the following link:  SURVEY_LINK

LOCATION: All talks will take place in the Frank Adams Room (FA) located in the 1st floor of the Alan Turing building, Department of Mathematics. Coffee breaks and reception will be in the kitchen/lounge area next to FA room.

ONLINE LINK: The meeting will be held in hybrid format. For the zoom link, email  omar.sanchez@manchester.ac.uk

FUNDING: Some modest funding is available to aid with travel expenses for PhD students to attend the meeting. Please email omar.sanchez@manchester.ac.uk to apply.

SPEAKERS: Adam Jones, Stephane Launois, Omar Leon Sanchez, Alexey Petukhov, Susan Sierra, Toby Stafford.  See the Schedule and Abstracts at the bottom of this site.

ADJACENT MEETING: Note that in the two days prior to the meeting, there will be a Model Theory meeting in Manchester. Further details in LYMOTSandSEEMOD

Queries and further details contact Omar Leon Sanchez at omar.sanchez@manchester.ac.uk.

SCAM WARNING: Please note that no party has been authorized to contact the participants to facilitate booking. Accordingly, treat any offer of such as a scam. If you are contacted by email or telephone with such an offer, do not engage in communication and report each incident to your IT department.

The meeting is supported by the Department of Mathematics, University of Manchester.

SCHEDULE:

Thursday 30 June

11-11.30.           Coffee

11.30-12.30      Alexey Petukhov

12.30- 2pm        Lunch

2-3pm.                 Susan Sierra

3-3.30.                 Coffee

3.30-4.30.           Omar Leon Sanchez

4.30-5.30            Reception

6.30 - ??              Dinner at Zouk Restaurant (The Quadrangle, Chester St, Manchester M1 5QS)

Friday 1 July

10-11am.            Stephane Launois

11-11.30.            Coffee

11.30-12.30       Toby Stafford

ABSTRACTS:

Title:  Affinoid envelopes and the deformed Dixmier-Moeglin equivalence.

Abstract: If g is a finite dimensional Lie algebra over a field K of characteristic 0, it was independently proved by Dixmier and Moeglin that U(g) satisfies the consequently named Dixmier-Moeglin equivalence. In the case where K is a complete, discretely valued field (e.g. a p-adic field), U(g) has a family of completions known as affinoid envelopes, which arise naturally from questions in representation theory, and we are interested in whether they too satisfy the DM-equivalence, or a particular topological generalisation known as the deformed DM-equivalence. We present results in this direction in the case where g is nilpotent, generalising the approach of Dixmier, and explore some avenues for future research.

Stephane Launois  (University of Kent)

Title: Is A_2 of type G_2?

Abstract: In the quest of identifying interesting quantum analogues of Weyl algebras, a few years ago I claimed in a talk that Ç A_1 is of type B_2 È. In this talk, I will consider the next step and explain why A_2 can be thought of as being of type G_2. This is joint work with Isaac Oppong.

Omar Leon Sanchez   (University of Manchester)

Title: How is the model-theoretic binding group useful in representation theory?

Abstract: I will attempt to give a gentle introduction to the notion of internality in the model-theoretic sense and state a classical result on witnessing groups of automorphisms as ÒdefinableÓ in a given structure. These are the so-called binding groups. I will then explain how this classical result can be used to deduce instances when  ÒrationalÓ implies Òlocally closedÓ. As a by-product, I will give an overview of the contributions of model-theoretic tools to the Dixmier-Moeglin equivalence.

Alexey Petukhov   (Kharkevich Institute=

Title: Topics in Lie-Dynkin nil-algebras

Abstract: I would like to talk about Lie-Dynkin nil-algebras (they are infinite generalizations of nilpotent radicals of simple Lie algebras) with universal enveloping algebras and Poisson algebras of these Lie algebras in focus. The primitive ideals for the corresponding finite-dimensional Lie algebras are studied up to some extent and one of the main known features is that Dixmier-Moeglin equivalence holds in this setting. In our joint paper with M. Ignatyev we checked that Dixmier-Moeglin equivalence holds in the setting of Lie-Dynkin nil-algebras. This fact is accompanied by some interesting features which seems to be quite common for infinite-dimensional Lie algebras - I would discuss these features (with hints of proofs) and after that I will try to convince the audience that very similar effects hold for other infinite-dimensional Lie algebras (which seems to be quite different from Lie-Dynkin nil-algebras).

Susan Sierra    (University of Edinburgh)

Title: The (almost) PDME for the symmetric algebras of the Witt and Virasoro Lie algebras

Abstract: Let $W = \mathbb C[t, t^{-1}] \partial$ be the Witt algebra of algebraic vector fields on the punctured affine line.  We classify Poisson primitive ideals in the symmetric algebra of $W$, and show that, although $W$ is infinite-dimensional, each such ideal corresponds to an orbit of an algebraic group acting on an affine variety.  As a consequence we show that $S(W)$ satisfied the Poisson Dixmier-Moeglin equivalence except for the zero ideal, which is Poisson rational and Poisson primitive but not Poisson locally closed.  We also establish a similar statement for the Virasoro algebra, the unique nontrivial central extension of $W$. This is joint work with Alexey Petukhov.

Toby Stafford   (University of Manchester)

Title: Invariant holonomic systems for symmetric spaces.

Abstract: Fix a complex reductive Lie group G with Lie algebra g and let V be a symmetric space over g with ring of differential operators D(V ). A fundamental class of D(V )- modules consists of the admissible modules (these are natural analogues of highest weight g-modules). In this lecture I will describe the structure of some important admissible modules. In particular, when V = g these results reduce to give Harish- ChandraÕs regularity theorem for G-equivariant eigendistributions and imply results of Hotta and Kashiwara on invariant holonomic systems. A key technique is relate (the admissible module over) invariant differential op- erators D(V )G on V to (highest weight modules over) Cherednik algebras. This research is joint with Bellamy, Levasseur and Nevins.