# Welcome to the 2^{nd} SNAG meeting!

Participants of the 2019 SNAG Workshop

We are happy to announce the second 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, Sergei Silvestrov and Johan Öinert.

## Venue

The meeting will take place at BTH (Karlskrona) on the 17th-18th of
October 2019
(map of the campus). The workshop starts on the 17th of October at
13.00, and end on the 18th of October at 16.00.

The following rooms are reserved for the workshop:

Thursday |
Room J1640 |

Friday (before lunch) |
Room J1640 |

Friday (after lunch) |
Room C216 |

Both rooms are on the ground floor. See the map of the campus to
find your way to the C-building and the J-building.

## Registration

If you would like to attend the meeting, please send an email
to Joakim Arnlind. Please note
that participants are expected to make their own arrangements for travel and accommodation.

## Program

(Click on the title below to see the abstract.)

### Thursday 17 October

13:00 - 13:10

Workshop opening

13:10 - 13:40

Object unital groupoid graded rings
(Patrik Nystedt)
We extend the classical construction by Noether of
crossed product algebras, defined by finite Galois field
extensions, to cover the case of separable (bot not necessarily
finite or normal) field extensions. This leads us naturally to
consider non-unital groupoid graded rings of a particular type
that we call object unital. We determine when such rings are
strongly graded, crossed products, skew groupoid rings and twisted
groupoid rings.
(pdf)

13:50 - 14:20

Natural generalizations of Kaplansky's conjectures
(Johan Öinert)
Kaplansky's conjectures for group algebras assert that
group algebras of torsion-free groups have only trivial units,
only trivial zero-divisors and only trivial idempotents. In this
talk we will show that these conjectures have natural
generalizations to general rings graded by torsion-free groups,
and that the generalized conjectures can be solved in important
special cases. We will also show that the generalized conjectures
exhibit the same hierarchy as the classical conjectures for group
algebras.
(pdf)

14:20 - 14:50

Coffee break

14:50 - 15:20

Graded von Neumann regular rings
(Daniel Lännström)
on Neumann regular rings are equipped with "weak
inverses". The notion of a graded von Neumann regular ring was
introduced and studied in the 1980s as a "graded version" of von
Neumann regular rings. In this talk, I relate graded von Neumann
regular rings to the class of nearly epsilon-strongly graded rings
(recently introduced by Nystedt and Öinert). We also discuss
applications to Leavitt path algebras with coefficients in a
general unital rings.
(pdf)

15:30 - 16:00

Realizations of free actions and their applications to lifting problems
(Stefan Wagner)
Given a compact group $G$, it follows from a seminal
paper by A. Wassermann that there is a bijective correspondence
between full multiplicity ergodic actions of $G$ and fixed point
algebras of equivariant coactions on $\mathcal{L}(L^2(G))$. In
this talk we will present an interesting generalization of
Wassermannn's result, namely that, in fact, each free action of
$G$ on a unital C$^*$\nobreakdash-algebra $\mathcal{A}$ can be
represented as the fixed point algebra of an equivariant coaction
on $\mathcal{A}^G \otimes \mathcal L(\mathfrak{H})$ for some
suitable Hilbert $G$-space $\mathfrak{H}$ with finite multiplicity
spaces. The applications we have in mind revolve around lifting
problems, for instance, permanence properties of free actions with
respect to spectral triples. This is joint work with
Kay~Schwieger.

16:10 - 16:40

Hilbert's basis theorem for hom-associative Ore extensions
(Johan Richter)
Ore extensions are a non-commutative generalization of
polynomial rings. Recently Ore extensions have further been
generalized in various directions. I will describe one such
generalization, Hom-associative Ore extensions, and describe a
version of the Hilbert Basis theorem for them obtained by Bäck and
Richter.
(pdf)

16:50 - 17:20

Semilattice Decomposition of Semigroups: from the Theory to
Applications
(Melanija Mitrovic)
A semigroup is an algebraic structure consisting of a
set with an associative binary operation defined on it. We can say
that most of the work within theory is done on semigroups with a
finiteness condition, i.e. a semi- groups possessing any property
which is valid for all finite semigroups - like, for example,
pi-regularity, completely pi-regularity, periodicity, finite
generation are. There are many different techniques for
describing various kinds of semigroups. Among the methods with
general applications is a semilattice decomposition of
semigroups. Certain types of semigroups being decomposable into a
semilattice of archimedean semigroups occurred in
semigroup-theoretic investigations of di- verse directions. Here,
we are interested, in particular, in the decomposability of a
certain type of semigroups with finiteness conditions into a
semilattice of archimedean semigroups. Roughly speaking this talk
will be mostly about the “games” which elements (special and/or
“ordinary”) of the semigroup can play and influence of that
“games” on semigroup structure, in first place in making
connection between semilattice decomposition, hereditarness and
periodicity. “Semigroups aren’t a barren, sterile flower on the
tree of algebra, they are a natural algebraic approach to some of
the most fundamental concepts of algebra (and mathematics in
general), this is why they have been in existence for more then
half a century, and this is why they are here to stay,”
(B. M. Schein in Semigroup Forum, 54, 1997,
264-268). Semigroup-theoretic approach becomes quite substantial
in more complex branches of mathematics, like theories of
automata, formal languages, codes, and combinatorics in
general. However, this talk does not pretend to mention all of the
existing applications of semigroup theory. Throughout this talk we
are going to list some of the applications of presented classes of
semigroups and their semilattice decompositions in certain types
of ring constructions. The results which will be presented are
mainly based on the ones given in [1], [2].
(pdf)

### Friday 18 October

09:30 - 10:00

Projectors on the noncommutative cylinder
(Joakim Arnlind)
The noncommutative cylinder is a simple example of a
noncompact noncommutative manifold, algebraically quite similiar
to the noncommuatitve torus. In this talk, I will recall basic
properties of the noncommutative cylinder as well as explicitly
constructing projectors representing all classes in
K-theory.
(pdf)

10:00 - 10:30

Coffee break

10:30 - 11:00

On K -invariant q-deformed Levi-Civita connections
(Kwalombota Ilwale)
In noncommutative geometry, free modules admits
connections. We define a q-deformed connection on a free module,
over the q-deformed 3-sphere, that is compatible with a metric
that is K-invariant. Torsionfreeness is also defined on the
module. And finally discuss the existence of a q-deformed
Levi-Civita connection.
(pdf)

11:10 - 11:40

Minimal embeddings in a noncommutative context
(Axel Tiger Norkvist)
In noncommutative geometry, one may use real calculi to
encode information about certain noncommutative manifolds by
pairing a *-algebra A with a set of derivations and a module
over A as analogues of vector fields. We introduce the concept
of homomorphisms of real calculi, which can be used to give a
general definition of embeddings in noncommutative
geometry. Several classical results in differential geometry can
then be shown to have analogues in the noncommutative context -
including Gauss' equations for the curvature of an embedding -
and we define the notion of mean curvature and minimal
embeddings. Using this framework, we show that the noncommutative
torus can be minimally embedded into the noncommutative 3-sphere
for a class of perturbed metrics.
(pdf)

11:50 - 12:20

Hom-algebra structures and quasi hom-Lie algebras
(Sergei Silvestrov)
In this talk, introduction and some open problems and
open direc- tions about hom-algebra structures will be
presented. These interesting and rich algebraic structures appear
for example when discretizing the differential calculus as well as
in constructions of differential calculus on non-commutative
spaces. Quasi Lie algebras encompass in a natural way the Lie
algebras, Lie superalgebras, color Lie algebras, hom-Lie algebras,
q-Lie algebras and various algebras of discrete and twisted vector
fields arising for example in connection to algebras of twisted
discretized derivations, Ore extension algebras, q-deformed vertex
operators structures and q-deferential calculus, multi-parameter
deformations of associative and non-associative algebras,
one-parameter and multi-parameter deformations of
infinite-dimensional Lie algebras of Witt and Virasoro type,
multi-parameter families of quadratic and almost quadratic
algebras that in- clude for special choices of parameters algebras
appearing in non-commutative algebraic geometry, universal
enveloping algebras of Lie algebras, Lie superalgebras and color
Lie algebras and their deformations. Common unifying feature for
all these algebras is appearance of some twisted generalizations
of Jacoby identities providing new structures of interest for
investigation from the side of associative algebras,
non-associative algebras, generalizations of Hopf algebras,
non-commutative differential calculi beyond usual differential
calculus and generalized quasi-Lie algebra central extensions and
Hom-algebra formal deformations and cohomology. Hom-algebra
generalizations of Nambu algebras, associative algebras and Lie
algebras to n-ary structures are also actively studied and some
constructions and results on n-ary hom-Lie algebras will be
presented in this talk.
(pdf)

14:00 - 14:30

Hom-Lie structures on 3-dimensional skew symmetric algebras
(Elvice Ongong'a)
We describe the dimension of the space of possible
linear endomorphisms that turn skew- symmetric three-dimensional
algebras into Hom-Lie algebras. We find a correspondence between
the rank of a matrix containing the structure constants of the
bilinear product and the dimension of the space of Hom-Lie
structures. Examples from classical complex Lie algebras are given
to demonstrate this correspondence.
(pdf)

14:40 - 15:10

Formal hom-associative deformations of Ore
extensions
(Per Bäck)
In this talk, I will give a brief introduction to the
non-associative and non-commutative polynomial rings known as
hom-associative Ore extensions. I will then show how one within
this framework can formally deform otherwise rigid algebras, such
as the Weyl algebras, and how one can prove an analogous famous
conjecture regarding them to hold true. I will also demonstrate
how these formal deformations give rise to formal deformations of
the corresponding commutator Lie algebras into so-called hom-Lie
algebras.
(pdf)

15:10 - 15:30

Coffee break

15:30 - 16:00

Centralizers in Skew PBW Extensions
(Alex Tumwesigye)
In this talk, I give a description for the centralizer
of the coefficient ring R in the skew PBW extension
\sigma(R)(x_1,x_2,...,x_n). An explicit description of the
centralizer is given, in the quasi-commutative case and a
necessary condition stated in the general case. Finally, I
consider the PBW extension \sigma(A)(x_1,x_2,...,x_n) of the
algebra of functions with finite support on a countable set,
describing the centralizer of A and the center of the skew PBW
extension.
(pdf)

## Participants

- Joakim Arnlind (Linköping University)
- Per Bäck (Mälardalen University)
- Kwalombota Ilwale (Linköping University)
- Daniel Lännström (Blekinge Institute of Technology)
- Melanija Mitrovic (University of Nis)
- Patrik Nystedt (University West)
- Elvice Ongonga (Mälardalen University)
- Johan Richter (Blekinge Institute of Technology)
- Sergei Silvestrov (Mälardalen University)
- Axel Tiger Norkvist (Linköping University)
- Alex Tumwesigye (Makerere University)
- Stefan Wagner (Blekinge Institute of Technology)
- Johan Öinert (Blekinge Institute of Technology)