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)

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)

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)

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.

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)

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)

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)

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)

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)

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)

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)

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)

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)