# Welcome to the 6^{th} SNAG meeting!

### 21st - 22nd of March 2024

Participants of the 6th SNAG Workshop

We are happy to announce the 6th 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 Linköping University from
Thursday 21st of March to Friday 22nd of March 2024. Participants
are encouraged to arrive on Wednesday 20th of March since the talks
start on Thursday morning. Please note that participants are
expected to make their own arrangements for travel and
accommodation. Clarion
Hotel Slottsparken
and Best Western And
Hotel are two hotels that are located in the city center with a
convenient and quick bus connection to the university.

From "Linköping Resecentrum" you can take bus 12 (to
"Universitetet") or bus 4 (to "Nobeltorget") to get to the
university. If you are staying at Hotel Clarion Slottsparken or Best
Western And Hotel, you may take bus 12 from "Länsstyrelsen" to
"Universitetet".

## Registration

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

## Program

The lectures will be held in lecture room "Planck" (see map).

(Click on the title below to see the abstract)

### Thursday 21 March

09:20 - 09:30

Workshop opening

10:10 - 10:40

A new perspective on the Cayley-Dickson construction: flipped polynomial rings
(Per Bäck)
The Cayley—Dickson construction is a famous construction that
generates new *-algebras out of old ones. It is perhaps best
known for generating all the real, normed division algebras that
exist out of the real numbers themselves: the real numbers, the
complex numbers, the quaternions, and the octonions. However,
the construction is undoubtedly quite mysterious and seems to be
a patchwork created ad hoc to make algebras like the above fit
in a construction. In this talk, I will try to shed new light
on the Cayley—Dickson construction with the purpose of
illuminating the underlying patchwork. We introduce a new class
of polynomial rings with a “flipped” multiplication which all
Cayley—Dickson algebras naturally appear as quotients of. In
particular, this extends the classical construction of the
complex numbers as a quotient of a polynomial ring to the
quaternions, the octonions, and beyond. This is based on joint
work with Masood Aryapoor (MDU).
(pdf)

10:40 - 11:00

Coffee break

11:00 - 11:30

Hom-Lie structures of generalized sl(2)-type
(Stephen Mboya)
This work is devoted to certain properties and structures of
Hom-Lie algebras of generalized sl(2)-type. We construct
classes of linear twisting maps that turn a skew-symmetric
algebra of generalized sl(2)-type into a Hom-Lie algebra and
describe the subclasses of such linear maps which yield
multiplicative Hom-Lie algebras. We explore some properties of
these Hom-algebras related to their ideals, Hom-ideals,
subalgebras and Hom-subalgebras, with emphasis on their derived
series, and central descending series, as well as their
nilpotence and solvability properties. We investigate the
invariance of these sub-algebras under the linear twisting maps,
and determine whether these sub-algebras are weak subalgebras,
Hom-subalgebras, weak ideals, or Hom-ideals. In particular, we
investigate these sub-algebras and properties for the
subfamilies of non-multiplicative Hom-Lie algebras of
generalized sl(2)-type indicating the differences between
non-multiplicative and multiplicative cases. This is a joint
work with Abdennour Kitouni, Elvice Ongong’a, Jared Ongaro and
Sergei Silvestrov.
(pdf)

11:40 - 12:10

Strange algebra identities
(Vladimir G. Tkachev)
There are many distinguished algebras which are defined by
virtue of an algebra identity, classical examples are Jordan
algebras, some genetic algebras (including Bernstein and train
algebras), algebra of minimal cones. Several examples appeared
recently in the context of axial algebras. An identity is
traditionally used to infer a deeper information on the Peirce
spectrum and the corresponding fusion laws. We review some
recent results on the general Peirce decompositions for algebras
with identities with a special accent on the so-called “strange”
identities, i.e. those implying the trivial Peirce polynomial.
In the latter case, the Peirce spectrum becomes undetermined. We
explain how this class is related medial algebras.
(pdf)

14:10 - 14:40

Chain algebras of finite distributive lattices
(Lisa Nicklasson)
In this talk I will give a short introduction to toric algebras
defined by combinatorial objects. I will then introduce a new
family of toric algebras, namely those arising from maximal
chains of a finite distributive lattice. Applying results on
stable set polytopes, we can conclude that every such algebra is
normal and Cohen-Macaulay. We will also discuss how algebraic
invariants, such as Krull dimension and Hilbert series, can be
computed from the combinatorics of the underlying lattice.
(pdf)

14:40 - 15:10

Coffee break (+Workshop photo)

15:10 - 15:40

The Ext-algebra of Standard Modules of Twisted Doubles
(Mika Norlén Jäderberg)
Quasi-hereditary algebras are an important class of algebras
that arise in Lie-theory, representation theory and algebraic
geometry. Central to the study of a quasi-hereditary algebra is
a collection of modules called standard modules, and the
Ext-algebra of this collection is of particular interest. In
this talk, we discuss a special class of quasi-hereditary
algebras called twisted doubles, and we provide a formula for
the Ext-algebras of their standard modules in terms of the
Ext-algebra of the simple modules over a certain subalgebra. If
there is time, we will also discuss some possible
generalizations and other avenues of research.
(pdf)

15:50 - 16:20

Torsion free connections on g-connection modules
(Victor Hildebrandsson)
In noncommutative Riemannian geometry, as in its commutative
counterpart, one is interested in Levi-Civita connections. These
are connections which are torsion free and compatible with some
metric on the space. I will present the basic definitions
needed, introduce the notion of a g-connection module, and give
a coordinate free way of classifying all torsion free
connections on a g-connection module. This is part of an ongoing
project together with Joakim Arnlind.
(pdf)

### Friday 22 March

09:30 - 10:00

Primeness of groupoid graded rings
(Johan Öinert)
I will talk about a joint work with Paula S. E. Moreira (UFSC,
Florianopolis, Brazil) concerning the primeness of groupoid
graded rings. We have established a characterization of prime
nearly epsilon-strongly groupoid graded rings. By applying our
result to partial skew groupoid rings and to groupoid rings, in
particular, we obtain characterizations of primeness for those
types of rings. Our results generalize earlier results by
I.G. Connell and B. Steinberg.
(pdf)

10:10 - 10:40

Rudakov modules for Lie algebras of vector fields
(Jonathan Nilsson)
I will discuss some aspects of representations of Lie algebras
of polynomial vector fields on an affine variety. In particular
I will show how one can define Rudakov-modules over arbitrary
affine varieties, a generalization of Rudakov's work from
1974. The talk is based on joint work with Yuly Billig and
Vyacheslav Futorny.

10:40 - 11:10

Coffee break

11:10 - 11:40

A hom-associative Stafford’s theorem?
(Johan Richter)
A version of Hilbert’s basis theorem says that all one-sided
ideals in the Weyl algebras are finitely generated. A
strengthening by Stafford says that the number of generators
never need to be more than 2. A hom-associative version of the
first Weyl algebra was introduced by Bäck, Richter and
Silvestrov, and further studied by Bäck and Richter. In this
talk I will describe how Bäck and Richter tried to generalize
Stafford’s theorem to higher hom-associative Weyl algebras. It
turns out that in the hom-associative case generically all
one-sided ideals are principal, i.e. need only one generator.
(pdf)

11:50 - 12:20

On periodic algebras
(Erik Darpö)
An important open problem in the homological algebra of
self-injective algebras is to characterise periodic algebras. An
algebra B is said to be periodic if it has a periodic projective
resolution as a B-B-bimodule. I will speak about some old and
new results on periodic algebras including, in particular, our
recent characterisation of periodicity in trivial extension
algebras: the trivial extension T(A) of a finite-dimensional
algebra A is periodic if and only if A is fractionally
Calabi-Yau.
(pdf)

13:30 - 14:00

The structure of Fell bundles
(Stefan Wagner)
Fell invented Fell bundles to understand better and extend
Mackey’s pioneering works in the intersection of quantum logic,
the theory of infinite-dimensional unitary representations of
groups, the theory of operator algebras, and noncommutative
geometry. Nowadays, Fell bundles are an important topic both in
operator algebras and noncommutative geometry. Indeed, due to
their many applications these objects have found broad interest
and have been explored by many authors in recent years.
Noteworthly, to each Fell bundle one may naturally associate a
C*-algebra which generalizes crossed products.
In joint work with Nata Machado from UFSC, we try to better
understand the structure theory of Fell bundles, for instance,
to be able to provide a suitable classification theory. In this
talk I will give an introduction to Fell bundles and report on
the challenges that we have encountered in our investigation so
far.
(pdf)

14:10 - 14:40

Commutation relations for a class of noncommutative spheres
(Axel Tiger Norkvist)
In this talk, I will discuss some of the challenges inherent in
the initial phases of mathematical research projects. More
concretely, I will describe a family of noncommutative spheres
and showcase an unexpected challenge in what I had initially
assumed to be a routine verification task. Moreover, I will
present some basic attempts to gain a deeper understanding of
why the problems that arose exist in the first place.
(pdf)

14:40 - 15:10

Coffee break

15:10 - 15:40

A Lie-type construction based on twisted derivations
(German Garcia)
The Jacobian determinant, given n derivations over a commutative
associative algebra, gives an n-Lie algebra structure to said
algebra. In this talk, I will construct a generalized Jacobian
determinant using twisted derivations and study certain
properties of it with respect to twisted derivations over the
algebra. Under certain conditions on the derivations, it is
possible to obtain a series of generalized n-ary hom-Lie
structures and other more general structures, relying on
commutation relations between derivations and twisting maps. I
will discuss the construction and explore a particular case in
which a new generalization of n-ary hom-Lie algebras is found.
(pdf)

15:50 - 16:20

Chain conditions for rings with enough idempotents with applications to category graded rings
(Patrik Lundström)
We obtain criteria for when a ring with enough idempotents is
left/right artinian or noetherian in terms of local criteria
defined by the associated complete set of idempotents for the
ring. We apply these criteria to object unital category graded
rings in general and, in particular, to the class of skew
category algebras. Thereby, we generalize results by
Nastasescu-van Oystaeyen, Bell, Park and Zelmanov from the group
graded case to groupoid, and in some cases category, gradings.
(pdf)

## Participants

Joakim Arnlind | (Linköping University) |

Per Bäck | (Mälardalen University) |

Erik Darpö | (Linköping University) |

German Garcia | (Mälardalen University) |

Lars Hellström | (Mälardalen University) |

Victor Hildebrandsson | (Linköping University) |

Patrik Lundström | (University West) |

Stephen Mboya | (Mälardalen University) |

Lisa Nicklasson | (Mälardalen University) |

Jonathan Nilsson | (Linköping University) |

Mika Norlén Jäderberg | (Linköping University) |

Johan Richter | (Blekinge Institute of Technology) |

Sergei Silvestrov | (Mälardalen University) |

Axel Tiger Norkvist | (Linköping University) |

Vladimir G. Tkachev | (Linköping University) |

Stefan Wagner | (Blekinge Institute of Technology) |

Johan Öinert | (Blekinge Institute of Technology) |