# Welcome to the 4^{th} SNAG meeting!

Participants of the 2021 SNAG Workshop

We are happy to announce the 4th 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 be held via Zoom on the 2nd and 3rd of December
2021.

Link to Zoom meeting

Meeting ID: 615 5181 5251

Please email Joakim
Arnlind to get the passcode.

## Registration

If you would like to participate, please send an
email to Joakim
Arnlind. Then we can easily provide you with information as well
as the invitation and password for the Zoom meeting.

## Program

(Click on the title below to see the abstract and to download the slides.)

### Thursday 2 December

09:20 - 09:30

Workshop opening

10:10 - 10:40

Divisibility, twisted derivations and non-associative algebras
(German Garcia Butenegro)
Divisibility is not transitive in general algebras. Considering
one-sided divisibility, it holds in non-commutative algebras,
but fails on the non-associative not-associative. Considering
one-sided divisibility + Hom-associativity, some properties can
be stated related to one-sided factors and zero-divisors of
Hom-associative algebras. Composition of twisted derivations is
not another one in general. Given certain relations between
twisted derivations and twisting maps a higher-degree Leibniz
rule can be observed over the commutator of two such
operators. This can be used to establish a family of graded Lie
subalgebras of the algebra of linear maps over the
non-associative algebra A.

10:40 - 11:00

Coffee break

11:40 - 12:10

On solvability and nilpotency of (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras
(Abdennour Kitouni)
We study properties of (n+1)-Hom-Lie algebras induced by
n-Hom-Lie algebras. We investigate the relations between the
k-derived series and k-central descending series of a given
n-Hom-Lie algebra and those of an (n+1)-Hom-Lie algebra induced
by it using a generalized trace map. Some other properties as
well as a few examples are presented.
(pdf)

13:30 - 14:00

The universality of one half in commutative nonassociative algebras with identities
(Vladimir G. Tkachev)
The talk will discuss an interesting phenomenon which occurs in
general nonassociative algebras with identities. More precisely,
it will be shown that any finite-dimensional commutative
nonassociative algebra over a field satisfying an identity
always contains 1/2 in its Peirce spectrum. Another interesting
property of such algebras is that the corresponding 1/2-Peirce
module satisfies the Jordan type fusion laws. We also establish
an explicit representation of the Peirce polynomial for an
arbitrary algebra identity. Some further applications to genetic
algebras and algebra of minimal cones will be given.
(pdf)

14:10 - 14:40

Strong hom-associativity
(Lars Hellström)
Strong hom-associativity is a strengthening of ordinary
hom-associativity, which leads to a more straightforward rewrite
theory. In terms of axioms — the family of "Canyon identities" —
it may look extensive, but in fact the strengthening is quite
mild. Most examples of hom-associative algebras that to date
have been examined turn out to be strongly hom-associative.
(pdf)

### Friday 3 December

09:30 - 10:00

Twisted connections on commutative algebras
(Kwalombota Ilwale)
In classical geometry, all connections on one dimensional
manifolds are torsion free. In this work, we introduce an
analogy of vector fields that are twisted by endomorphisms called
(σ,τ)-derivations of an algebra and explore whether all
connections satisfying a twisted Leibniz rule in analogy with
the (σ,τ)-derivations are torsion free as in the classical case.
We begin by introducing a pair consisting of an associative
algebra and a set of (σ,τ)-derivations and consider a particular
model where the algebra is commutative and the set consist of
all possible (σ,τ)-derivations of the algebra. We show that all
torsion free connectionsdepends on the endomorphisms σ and τ.
(pdf)

10:10 - 10:40

Projective real calculi and the Levi Civita connection
(Axel Tiger Norkvist)
Real calculi is a derivation-based approach to noncommutative
geometry which makes it possible to generalize several notions
from classical differential geometry to a noncommutative
setting. One such notion is that of affine connections, and in
this talk we shall go over some of the current research that I
am involved in regarding real calculi over projective modules
and what can be said about the Levi Civita connection in this
case. As this talk is based on an ongoing research project, the
focus will be on exploring some questions I have at the moment,
rather than presenting finished results and conclusions.
(pdf)

10:40 - 11:00

Coffee break

11:00 - 11:30

Levi-Civita connections for a class of noncommutative minimal surfaces
(Joakim Arnlind)
We study connections on hermitian modules, and show that metric
connections exist on regular hermitian modules, i.e finitely
generated projective modules together with a non-singular
hermitian form. In addition, we develop an index calculus for
such modules, and provide a characterization in terms of the
existence of a pseudo-inverse of the matrix representing the
hermitian form with respect to a set of generators. The
framework is applied to a class of noncommutative minimal
surfaces, for which there is a natural concept of torsion, and
show that there exist metric and torsion-free connections for
every minimal surface in this class.
(pdf)

11:40 - 12:10

A new approach to noncommutative Riemannian spin geometry
(Stefan Wagner)
In the noncommutative setting the notion of a spectral triple
provides a natural framework for noncommutative Riemannian spin
manifolds. However, unlike in the classical setting, the
axiomatic description of a noncommutative Riemannian spin
manifold does neither incorporate noncommutative principal
bundles nor spin groups. In this talk, I will consider
noncommutative spin geometries from a noncommutative principal
bundle perspective and discuss the mathematical challenges that
come along with this new approach.
(pdf)

13:30 - 14:00

An extension of Hilbert’s basis theorem to non-assocciative Ore extensions
(Johan Richter)
The famous Hilbert’s basis theorem says that a polynomial ring
over a Noetherian ring is itself Noetherian. There is a
classical generalization of this theorem to Ore extensions. Per
Bäck and me have obtained a further generalization to
non-associative and hom-associative Ore extensions which I will
describe in this talk.
(pdf)

14:10 - 14:40

Prime Leavitt path algebras via nearly epsilon-strongly graded rings
(Daniel Lännström)
This talk is based on joint work with Lundström, Wagner and
Öinert. Recall that Connell’s Theorem characterizes when a group
ring is prime. Passman generalized this result to obtain a
characterization of algebraic crossed products during the
1980s. In a recent joint work, we characterized prime nearly
epsilonstrongly graded rings by extending Passman’s result. Our
result is especially interesting in connection with Leavitt path
algebras (algebraic graph C*-algebras). I talk about this
application to Leavitt path algebras and also discuss some ideas
for future work.
(pdf)

14:40 - 15:00

Coffee break

15:00 - 15:30

Homogeneity in commutative graded rings
(Johan Öinert)
In this talk I will tell you about some new results (joint work
with A. Tarizadeh) on commutative G-graded rings, where G is a
totally ordered abelian group. We will see that McCoy's theorem
and Armendariz' theorem for polynomial rings can be generalized
to the setting of G-graded rings. We will also see that
Bergman's famous theorem (which asserts that the Jacobson
radical of a Z-graded ring is a graded ideal) can be generalized
to the setting of G-graded rings. Using that generalization we
are able to establish several characterizations of totally
ordered abelian groups.
(pdf)

## Participants

Joakim Arnlind | (Linköping University) |

Masood Aryapoor | (Mälardalen University) |

Maxime Bury | (Mälardalen University) |

Per Bäck | (Mälardalen University) |

Domingos Djinja | (Mälardalen University) |

German Garcia | (Mälardalen University) |

Kwalombota Ilwale | (Linköping University) |

Abdennour Kitouni | (Mälardalen University) |

Lars Hellström | (Mälardalen University) |

Daniel Lännström | (Blekinge Institute of Technology) |

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) |