# Welcome to the 5^{th} SNAG meeting!

### 28-29th of March 2023

We are happy to announce the 5th meeting of the Swedish Network for
Algebra and Geometry. The purpose of the network is to develop the
interaction between mathematicians working in the fields of algebra
and geometry at Swedish universities. In particular, we envisage an
active participation of PhD students and young researchers with the
aim to build networks and encourage collaboration.

## Organizers

Joakim Arnlind (Linköping University)

Sergei Silvestrov (Mälardalen University)

Johan Öinert (Blekinge Institute of Technology)

## Venue

The meeting will take place
at Mälardalen University from
Tuesday 28th of March to Wednesday 29th of March 2023. Participants are
encouraged to arrive on Monday 27th of March since the talks start
on Tuesday morning. Please note that participants are expected to
make their own arrangements for travel and accommodation.

## Registration

If you would like to participate, please send an
email to Joakim
Arnlind.

## Program

All talks take place in Room U2-016 on the second floor of the U-building.

(Click on the title below to see the abstract)

### Tuesday 28 March

09:05 - 09:15

Workshop opening

09:50 - 10:20

Kronecker algebras - a case study
(Joakim Arnlind)
Differential calculus in noncommutative geometry come in several
different flavors, and one of the more concrete versions go by
the name of derivation based differential calculus. This
calculus is built from a disinguished Lie algebra of
derivations, and lead to the formulation of differential forms,
cohomology and connections. A fundamental question in
noncommutative Riemannian geometry is the existence of a torsion
free and metric compatible connection; i.e a Levi-Civita
connection. For the moment, there are no general results
addressing this question, and I will present a case study based
on a simple quiver path algebra, and show how the existence of a
Levi-Civita connection depend on the choice of a Lie algebra of
derivations.

10:25 - 10:55

The noncommutative geometry of frame bundles
(Stefan Wagner)
Vector bundles in classical geometry typically arise as objects
associated with something more profound, a principal bundle. In
particular, each vector bundle E with fibre V is naturally
associated with a principal GL(V)-bundle, the frame bundle of E.
Frame bundles thus constitute a key tool for studying vector
bundles. Indeed, a connection on a frame bundle induces
covariant derivatives on all associated bundles in a coherent
way, leading to many important geometric constructions. This is
the situation in Riemannian geometry where, for a Riemannian
manifold M, the Levi-Civita connection on the frame bundle of M
induces a covariant derivative on the tensor fields, leading,
for instance, to the Riemannian curvature of M. The
noncommutative geometry of frame bundles, however, has not been
studied conclusively, although the notion of a noncommutative
principal bundle is certainly available. In this talk we present
our approach to the subject.

11:00 - 11:30

Paradoxicality in groups and rings
(Johan Öinert)
As a preparation for Karl's talk, we will recall some key
notions regarding paradoxicality in groups resp. rings. More
precisely, for groups, we will recall notions such as
paradoxical decompositions, amenability and supramenability. For
rings, we will recall notions such as IBN and UGN (unbounded
generating number) and introduce BGN (bounded generating
number). All concepts will be exemplified by plenty of
well-known and less well-known examples.

11:35 - 12:05

Group-graded rings with unbounded generating number
(Karl Lorensen)
For a ring $R=\oplus_{g\in G} R_g$ graded by a group
$G$, we identify three separate sets of conditions that
guarantee that $R$ has unbounded generating number if the base
ring $R_1$ does. These are:

(1). $G$ is amenable, and $R_g$
is a nonzero finitely generated free $R_1$-module for every
$g\in G$.

(2). $G$ is supramenable, and $R_g$ is a finitely
generated free $R_1$-module for every $g\in G$.

(3). $G$ is
locally finite, and $R_g$ is a finitely generated projective
$R_1$-module for every $g\in G$.

For each set (1)-(3), we
discuss whether all of the conditions are necessary. In the case
of (1), this gives rise to three ring-theoretic
characterizations of the property of amenability for a
group. This is joint work with Johan Öinert.

12:05 - 13:30

Lunch (Rosenhill Restaurant, U-building)

13:30 - 14:00

Finite-dimensional representations and Levi-Civita connections
(Axel Tiger Norkvist)
The framework of real calculi is a relatively direct
derivation-based approach to noncommutative geometry, where one
studies (right) modules over a *-algebra A in relation to a Lie
algebra g of derivations on A. Among other things one can
develop a notion of affine connections, and in this talk the
focus will lie on finite-dimensional modules over A and how the
existence of a Levi-Civita connection in this case depends on
the structure of g.

14:05 - 14:35

Lie algebra modules which are free over a subalgebra
(Jonathan Nilsson)
Given a subalgebra n of a Lie algebra g we can define the
category of U(g)-modules whose restriction to U(n) is free. In
this talk I will present some results about the classification
of such modules when g is of classical type and n is a
Cartan-subalgebra. I will also present some recent results for
the case where n is the nilradical of a maximal parabolic
subalgebra of sl(V) and discuss how these modules fit into
R. Block's classification of sl2-modules.

14:40 - 15:10

Graded von Neumann regularity of rings graded by semigroups
(Daniel Lännström)
In this talk, I discuss recent work on a complete
characterization of semigroup graded rings which are graded von
Neumann regular. We also demonstrate our results by applying
them to several classes of examples, including matrix rings and
groupoid graded rings. I also give some background to these
results. This talk is based on joint work with Johan inert.

15:10 - 15:30

Coffee break

15:30 - 16:00

The quantitative inverse Galois problem for Del Pezzo surfaces
(Olof Bergvall)
Over a finite field with q elements, a Del Pezzo surface X has
q^2+aq+1 points. Serre asked: Which values of a can occur? This
question is closely related to the more refined question of
which conjugacy classes (in an appropriate Galois group related
to X) can arise as the conjugacy class of the Frobenius
endomorphism of X. This is called the "inverse Galois problem
for Del Pezzo surfaces". In this talk, I will explain how to
answer the quantitative version of the inverse Galois problem:
given an element g in the Galois group, how many Del Pezzo
surfaces are there such that the Frobenius endomorphism acts as
g?

16:05 - 16:35

The trace property of ideals and modules
(Peder Thompsson)
In any ring, a trace ideal is an ideal which is the sum of
homomorphic images from some other fixed module to the
ring. Recent work has demonstrated the usefulness of trace
ideals in understanding ring and module structure. In this talk,
I will outline some of the theory of trace ideals, and trace
modules more generally. In particular, we will explore the
property of a module being trace in some envelope (such as the
injective envelope), and the types of rings that can be
characterized with this property. This talk is based on joint
work with Haydee Lindo.

16:25 - 16:55

Information theory via matroids and polymatroids
(Thomas Westerbäck)
Matroid theory and the theory of field-linear codes are closely
related since every matrix over a field defines a
matroid. Despite this fact, matroid theory has only rather
recently started to play an important role in the development of
coding and information theory, for example in areas such as
distributed storage, linear codes equipped with Hamming weights,
network coding, index coding, and caching. Polymatroids, a
generalization of matroids, can be used to capture properties of
Shannon entropy. This connection between entropy and
polymatroids enables polymatroids to be used to investigate
non-linear coding systems. In this talk I will present how
matroids and more generally polymatroids can be used for
applications in information theory.

### Wednesday 29 March

09:15 - 09:45

Non-associative algebras in an associative context
(Erik Darpö)
For any associative algebra $A$, the left regular representation
$\lambda_A:A\to \mathop{\rm End}(A)$ is an embedding of $A$ into
its linear endomorphism algebra $\mathop{\rm End}(A)$. In this
talk, I shall explain how this elementary observation can be
generalised to a (less elementary) structure theorem for general
non-associative algebras.
It turns out that, by adding some data to the left regular the
isomorphism type of a (not necessarily associative) algebra $A$
with identity element $e$ is encoded in the datum $(\mathop{\rm
im}(\lambda_A), \ker(\mathop{\rm ev}_e))$, where $\mathop{\rm
im}(\lambda_A)\subset \mathop{\rm End}(A)$ is the image of the
left regular representation, and $\mathop{\rm ev}_e:\mathop{\rm
End}(A)\to A,\:T\mapsto T(e)$ the evaluation map at the identity
element in $A$. This leads to a description of the category of
non-associative algebras in terms of associative algebras with
certain distinguished subspaces.

09:50 - 10:20

Other constructions of hom-associative algebras
(Lars Hellström)
A hom-algebra is an algebra that in addition to the ordinary
multiplication also has a linear unary operation, known as the
hom or twisting map. There exist a number of hom-variants of
ordinary classes of algebras, where axioms have been modified by
inserting homs in select positions: for example hom-Lie and
hom-associative algebras satisfy homified variants of the Jacobi
and associativity respectively axioms. A potentially troubling
matter is however that examples of hom-associative algebras
found in the literature are pretty much exclusively Yau twists
of associative algebras, which raises the worry that they might
all be just ordinary associative algebras presented in a silly
way. This talk puts that worry to rest by presenting two new
constructions - the homassocification of a general algebra, and
hom-associative algebras with truncated freedom - that can
produce non-Yau-twist hom-associative algebras. The latter
construction also provides the first explicit example of a
hom-associative algebra that is not strongly hom-associative.

10:25 - 10:55

On classification of (n+1)-dimensional n-Hom-Lie algebras
(Abdennour Kitouni)
In this work, we study the specific properties of
(n+1)-dimensional n-Hom-Lie algebras and give full lists of
algebras for some cases of the twisting map and n=3. We classify
up to isomorphism one subclass of the given lists by studying
solvability and nilpotency of these algebras and solving systems
of algebraic equations given by the isomorphism conditions, we
also present a few example and study some of their specific
properties related to derived series and central descending
series.

11:00 - 11:30

Non-associative skew Laurent polynomial rings
(Per Bäck)
I n this talk, I will introduce non-associative generalizations of
skew Laurent polynomial rings and some related rings, such as
skew power series rings and skew Laurent series rings. The focus
will mainly be on the former rings and results concerning their
ideals, such as when they are simple and generalizations of the
famous Hilbert’s basis theorem. In particular, we will see that
a right, but not left Noetherian version of the latter theorem
holds; a difference that does not exist in the associative
setting. This is based on joint work with Johan Richter.

12:00 - 13:30

Lunch (Rosenhill Restaurant, U-building)

13:30 - 14:00

Non-unital Ore extensions
(Johan Richter)
In this talk I will describe non-unital Ore extensions. I will
provide a characterization of simplicity for a certain class of
such Ore extensions. This result generalizes a result by Öinert,
Richter and Silvestrov from the unital setting. The talk is
based on joint work with Patrik Lundström and Johan Öinert.

14:05 - 14:35

Algebraically closed (σ,δ)-fields
(Masood Aryapoor)
In this talk, we will review the evaluation of a skew polynomial
as introduced by Lam and Leroy during the 80s. Using the
evaluation map, we will introduce some notions of algebraically
closed (σ,δ)-fields. It turns out that there are
different notions of “closedness” in the context of skew
polynomials. We will give some basic constructions and existence
results regarding algebraically closed
(σ,δ)-fields. If time allows, we will present an
application concerning the method of partial fraction
decomposition in the skew field of rational skew functions.

14:35 - 14:50

Coffee break

14:50 - 15:20

Symmetric (σ,τ)-derivations and (σ,τ)-Hochschild cohomology
(Kwalombota Ilwale)
Symmetric (σ,τ)-derivations are (σ,τ)-derivations
that are simultaneously (τ,σ)-derivations. In this
presentation, I will talk about symmetric
(σ,τ)-derivations together with some regularity conditions
and show that strongly regular (σ,τ)-derivations are
always inner. Moreover, I will introduce a
(σ,τ)-Hochschild cohomology theory which describes the
outer (σ,τ)-derivations in first degree, in analogy with
the classical Hochschild cohomology.

16:00 - 16:30

Divisibility in Hom-associative algebras
(German Garcia)
Divisibility is a transitive property in associative
algebras. This allows one to establish chains of divisibility
between elements of the algebra and gives stability to the
relation “x divides y” in the sense that one can multiply by a
third element at both sides of the relation. In Hom-associative
algebras, divisibility relations are altered by the twisting
map. Hom-associativity introduces more complicated relations
which involve, among others, the kernel of the twisting map and
zero-divisors within the algebra. We establish divisibility
relations between elements of arbitrary and (Hom)-unital
Hom-associative algebras.

16:30 - 16:40

Workshop closing (Sergei Silvestrov)

## Participants

Joakim Arnlind | (Linköping University) |

Masood Aryapoor | (Mälardalen University) |

Olof Bergvall | (Mälardalen University) |

Per Bäck | (Mälardalen University) |

Erik Darpö | (Linköping University) |

Domingos Djinja | (Mälardalen University) |

German Garcia | (Mälardalen University) |

Lars Hellström | (Mälardalen University) |

Kwalombota Ilwale | (Linköping University) |

Abdennour Kitouni | (Mälardalen University) |

Karl Lorensen | (Blekinge Institute of Technology/Pennsylvania State University) |

Patrik Lundström | (University West) |

Daniel Lännström | (Linköping University) |

Jonathan Nilsson | (Linköping University) |

Johan Richter | (Blekinge Institute of Technology) |

Sergei Silvestrov | (Mälardalen University) |

Peder Thompsson | (Mälardalen University) |

Axel Tiger Norkvist | (Linköping University) |

Stefan Wagner | (Blekinge Institute of Technology) |

Thomas Westerbäck | (Mälardalen University) |

Johan Öinert | (Blekinge Institute of Technology) |