# Welcome to the 2018 SNAG meeting!

Participants of the 2018 SNAG Workshop

We are happy to announce the first 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. The workshop is
supported by the Swedish Research Council (Vetenskapsrådet) and
the Royal Swedish Academy of Sciences.

Organizers: J. Arnlind, S. Silvestrov and J. Öinert.

## Venue

The meeting will take place
at Linköping University on the
27th -- 28th of September
2018.

Click here for a map of
the Campus.

On the 27th of September the lectures will be held in
our seminar room (Hopningspunkten), B-huset, entrance 23. Turn left
into the first corridor, and you will find Hopningspunkten on your
right hand side a few doors further down.

On the 28th of September the
lectures will be held in room BL34. Again, you enter B-huset via
entrance 23, walk up the stairs and continue straight
ahead. You will see the sign "BL34" hanging from the ceiling.

How to reach the university? By train you get off at the central
station "Linköping resecentrum". To reach the university you take
Bus 12 to "Universitetet Golfbanan" (which is also marked on the map
of the campus). For timetables see Östgötatrafiken.

Are you staying at Best Western Priceless Hotel (Storgatan 76)?
Then you get on bus 12 at the stop "Länsstyrelsen"
(map, marked "B") close to the hotel and get off at
the stop "Universitetet Golfbanan". For timetables
see Östgötatrafiken.

## 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

The program is also available in the SNAG Google calendar via this url.

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

### Thursday 27 September

13:20 - 13:30

Workshop opening

13:30 - 14:00

Commutative nonassociative algebras, representations of finite groups and minimal cones
(V. Tkachev, LiU)
In 1982, Robert Griess provided the first construction
of the Monster simple group M as a group of automorphisms of a
196884-dimensional commutative nonassociative algebra V. In
subsequent years, this construction was simplified and analyzed in
a number of ways. In particular, John Conway discovered an
association between a distinguished set of idempotents (called
axes) in V and a conjugacy class of involutions in M. In 1996,
Simon Norton studied the subalgebras in V generated by a few
axes. Alexander Ivanov axiomatized the Griess-Conway-Norton
algebra as a particular instance of the so-called Majorana
algebras. Hall, Shpectorov and Rehren extended the above results
for general axial algebras. On the other hand, it has been
recently realized that there is a class of axial algebra which
naturally appears in a completely different area: minimal
(i.e. zero mean curvature) cones. In my talk I try to explain some
remarkable connections between group theoretic, algebraic and
geometrical parts of the above picture.
(pdf)

14:10 - 14:40

Kähler-Poisson algebras
(A. Al-Shujary, LiU)
We introduce the concept of Kähler-Poisson algebras as
analogues of algebras of smooth functions on almost Kähler
manifolds. We first give a short review of Lie-Rinehart algebras,
and after that we give the definition and basic properties of
Kähler-Poisson algebras. It is then shown that the Kähler type
condition has consequences that allow for an identification of
geometric objects in the algebra which share several properties
with their classical counterparts; e.g. curvature and Levi-Civita
connections. Several examples are provided in order to illustrate
the novel concepts.
(pdf)

14:40 - 15:10

Coffee break

15:10 - 15:40

Skew inverse semigroup rings
(J. Öinert, BTH)
Skew inverse semigroup rings were introduced by Exel and
Vieira in 2010 and are generalizations of partial skew group
rings. In a joint work with Beuter, Goncalves and Royer, I have
obtained necessary and sufficient conditions for the simplicity of
certain skew inverse semigroup rings. During my talk I will review
the construction of a skew inverse semigroup ring as well as our
results. As an application, I will briefly mention how our results
can be used to give a new proof of the simplicity criterion for
Steinberg algebras. Steinberg algebras are algebraic versions of
Renault's C*-algebras of groupoids and include e.g. all Leavitt
path algebras.
(pdf)

15:50 - 16:20

Epsilon-strong systems, skew inverse semigroup rings and
Steinberg algebras
(P. Nystedt, HV)
We obtain sufficient criteria for simplicity of systems,
that is, rings $R$ that are equipped with a family of additive
subgroups $R_s$, for $s \in S$, where $S$ is a semigroup,
satisfying $R = \sum_{s \in S} R_s$ and $R_s R_t \subseteq
R_{st}$, for $s,t \in S$. These criteria are specialized to obtain
sufficient criteria for simplicity of, what we call, s-unital
epsilon-strong systems, that is systems where $S$ is an inverse
semigroup, $R$ is coherent, in the sense that for all $s,t \in S$
with $s \leq t$, the inclusion $R_s \subseteq R_t$ holds, and for
each $s \in S$, the $R_s R_{s^*}$-$R_{s^*} R_s$-bimodule $R_s$ is
s-unital. As an aplication of this, we obtain generalizations of
recent sufficient criteria for simplicity of skew inverse
semigroup rings, by Beuter, Goncalves, Öinert and Royer, and
then, in turn, for Steinberg algebras by Brown, Farthing, Sims,
Steinberg, Clark and Edie-Michel.
(pdf)

16:30 - 17:00

Chain conditions on epsilon-strongly graded rings with
applications to Leavitt path algebras
(D. Lännström, BTH)
The class of epsilon-strongly graded rings was recently
introduced by P. Nystedt, J. Öinert and H. Pinedo as a
generalization of strongly graded rings. In this talk, I will
present some background and examples and then discuss
characterizations of noetherian and artinian epsilon-strongly
graded rings. We will apply the general results to the important
special cases of unital partial crossed products and Leavitt path
algebras of a finite graph.
(pdf)

17:10 - 17:40

Geometric aspects of noncommutative principal bundles
(S. Wagner, BTH)
The notion of a free action of a quantum group on a
C∗-algebra provides a natural framework for noncommutative
principal bundles, which are not of purely mathematical interest
only. Indeed, besides their structure theory and relation with
K-theory, noncommutative principal bundles are becoming
increasingly prevalent in various applications of geometry and
mathematical physics. In this talk we explain how to
systematically study geometric aspects of noncommutative principal
bundles by means of Connes’ spectral triples, which allow to
extend many techniques from Riemannian spin geometry to the
noncommutative setting.
(pdf)

### Friday 28 September

09:00 - 09:30

Pseudo-Riemannian calculi and noncommutative Levi-Civita connections
(J. Arnlind, LiU)
In a series of papers we have developed a framework,
named Pseudo-Riemannian calculi, to discuss Levi-Civita
connections on vector bundles (projective modules) over
noncommutative algebras. It can be shown that at most one
torsion-free and metric connection exists, and under certain
reality conditions, the curvature operator has the same symmetries
as in the classical situation. In this talk, I will give an
overview of these results and in particular discuss their
application to the noncommutative torus and the noncommutative
3-sphere and a noncommutative version of the Chern-Gauss-Bonnet
theorem.
(pdf)

09:40 - 10:10

A q-derivation based Levi-Civita connection for the noncommutative three sphere
(K. Ilwale, LiU)
The q-derivations on a noncommutative three sphere are
given. In this paper we show that it is possible to construct a
Levi-Civita connection based on these q-derivations. Such a
connection is constructed on a finitely projective module in
general with a hermitian form. By specifying to the module of
differential forms over the noncommutative sphere, we can
find a metric and torsion free connection where the torsion
freeness is defined appropriately.
(pdf)

10:20 - 10:40

Coffee break

10:40 - 11:10

Hom-algebra structures
(S. Silvestrov, MDH)
n this talk an introductory overview on the subject of
Hom-algebra structures will be given with emphasize on hom-algebra
generalizations of Lie algebras and associative algebras. In
1990’th in the pioneering works of Curtright, Zachos, Ellinas,
Chaichian, Kulish, Lukierski, Presnajder, Popowicz, Isaev, Aizawa,
Sato, Hu quantum deformations of algebras, q-deformed oscillator
algebras, q-deformations of Witt and Virasoro algebras and related
families of algebras defined by generators and parameter
commutation relations have been constructed in connection to
quantum deformations and discretized models of mechanics and
quantum mechanics, q-deformations of vertex operators, q-deformed
conformal quantum field theory, q-deformed integrable systems,
q-deformed superstrings and central extensions. Also various
quantum n-ary extensions of Numbu mechanics and related n-ary
extensions of differential structures and of Lie algebras Jacobi
identities have been considered. It was noticed in particular that
many of quantum algebras and q-deformed Lie algebras obey certain
q-deformed versions of Jacobi identity generalizing Lie algebras
Jacobi identity. Motivated by these works Hartwig, Larsson and
Silvestrov in 2003 developed a general method of obtaining such
deformations and generalized Jacobi identities based on general
twisted derivations. This development, as well generalizations of
supersymmetry, lead to development of more general algebraic
structures such as quasi-Lie algebras and Hom-Lie algebras,
Hom-associative and Hom-Lie admissible algebras, Hom-Jordan
algebras, Hom-Poisson algebras, Hom-Yang-Baxter equations,
Hom-bialgebras, Hom-Hopf algebras, and other hom-algebra
structures, as well as Hom-Nambu and Hom-Nambu Lie algebras, n-ary
Hom-algebra generalizations of Nambu algebras, associative
algebras and Lie algebras and methods for their constructions and
classifications.
(pdf)

11:20 - 11:50

Formal languages, normal forms, and Hilbert series
(L. Hellström, MDH)
Rather than presenting new research (at least not
primarily), this talk is an introduction to some techniques from
the borderlands between Algebra, Discrete Mathematics, and
Computer Science — demonstrating how they can be used to gain
insights into questions of more traditional algebra. Topics that
may be mentioned are the power series formalism for formal
languages, regular languages (word and tree variants),
context-free languages, determinism, and Hilbert series. Examples
may include associative algebras (near-commutative vs. more
general), non-associative algebras (particularly hom-algebras),
and operads.
(pdf)

12:00 - 12:30

Undeformed commutators in the q-deformed Heisenberg algebra
(R. Cantuba, MDH)
Let F be a field, and fix an element q of F. The
q-deformed Heisenberg algebra H(q) is the unital associative
algebra over F with generators A, B and relation which asserts
that AB - qBA is the unity element. By the set of all Lie
polynomials or undeformed commutators in A, B, we mean the Lie
subalgebra L of H(q) generated by A, B. We present results from
several studies that describe such Lie polynomials. We fully
describe L in terms of a basis and a corresponding commutator
table, which vary according to cases based on the parameter q. If
F is the complex field, and if q is in the interval (0,1), then
H(q) is isomorphic to an algebra of operators on some sequence
space such that all the compact operators are in L, and the image
of H(q) in the Calkin algebra is an algebra of Laurent polynomials
in one indeterminate.
(pdf)

14:00 - 14:30

Non-associative Ore extensions
(J. Richter, MDH)
I will review the
definition of Ore extensions, a non-commutative (associative)
generalization of polynomial rings. I will present some results on
when Ore extensions are simple. I will then discuss
non-associative Ore extensions, which is a new concept, and
present some theorems on when they are simple, generalizing the
results for associative Ore extensions. The part about
non-associative Ore extensions is based on joint work by Patrik
Nystedt, Johan Öinert and myself.
(pdf)

14:40 - 15:10

On non-associative Weyl algebras and a Hilbert's basis theorem
(P. Bäck, MDH)
I will give a brief introduction to non-associative,
non-commutative polynomial rings known as hom-associative Ore
extensions. Within this framework, we will then see, for example,
how the first Weyl algebras, the quantum planes, and Hilbert's
basis theorem can be generalized into non-associative
versions.
(pdf)

15:10 - 15:30

Coffee break

15:30 - 15:45

Classification of low-dimensional hom-Lie algebras
(E. Ongonga, MDH)
For any n-dimensional hom-Lie algebra, a system of
polynomial equations is obtained from the hom-Jacobi identity,
having structure constants of both the skew-symmetric bilinear map
and twisting linear endomorphism. These equations are expressed as
linear system in structure constants of the twisting linear
endomorphism and thus represented as linear maps. We give a
description of realization of three-dimensional and
four-dimensional hom-Lie algebras, for kernel solutions of the
minimum dimension. We further give some examples of four-
dimensional Hom-Lie algebras constructed from a general nilpotent
linear endomorphism.
(pdf)

15:45 - 16:00

Reordering in a multi-parametric family of algebras
(J. Musonda, MDH)
The main object of this talk is the multi-parametric
family of algebras generated by $Q$ and the set $\{S_j\}$ satisfying
commutation relations of the form $S_jQ=\sigma_j(Q)S_j$ where $\sigma_j$ is some
function for which the expression $\sigma_j(Q)$ makes sense. Through a
linear transformation of the $S_j$ generators, we introduce and
consider another equivalent multi-parametric family of
algebras. We derive general reordering formulas for these algebras
and apply the results to the description of commutative
subalgebras. We give operator representations for particular cases
of $\sigma_j$, and in particular, we show that some multi-parameter
deformed symmetric difference and multiplication operators satisfy
defining relations of the corresponding algebras. Finally we
consider the algebras also in the context of $(\sigma,\tau)$-derivations and
Ore extensions.
(pdf)

## Participants

- J. Arnlind (LiU)
- S. Baghdari (BTH)
- P. Bäck (MDH)
- K. Ilwale (LiU)
- L. Hellström (MDH)
- D. Lännström (BTH)
- J. Musonda (MDH)
- P. Nystedt (HV)
- E. Ongonga (MDH)
- R. Cantuba (MDH)
- J. Richter (MDH)
- A. Al-Shujary (LiU)
- S. Silvestrov (MDH)
- V. Tkachev (LiU)
- S. Wagner (BTH)
- J. Öinert (BTH)