**A
day and a half with Dixmier and Moeglin**

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

** __________________________________________________________________________________________________________**

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**

10-11am.
Adam Jones

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:**

** Adam
Jones**
(University of Manchester)

**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.