Some constructions, examples and open problems on Hom-algebras and twisted derivations will be presented. (pdf)

Differential calculus in noncommutative geometry come in several different flavors, and one of the more concrete versions go by the name of derivation based differential calculus. This calculus is built from a disinguished Lie algebra of derivations, and lead to the formulation of differential forms, cohomology and connections. A fundamental question in noncommutative Riemannian geometry is the existence of a torsion free and metric compatible connection; i.e a Levi-Civita connection. For the moment, there are no general results addressing this question, and I will present a case study based on a simple quiver path algebra, and show how the existence of a Levi-Civita connection depend on the choice of a Lie algebra of derivations. (pdf)

Vector bundles in classical geometry typically arise as objects associated with something more profound, a principal bundle. In particular, each vector bundle E with fibre V is naturally associated with a principal GL(V)-bundle, the frame bundle of E. Frame bundles thus constitute a key tool for studying vector bundles. Indeed, a connection on a frame bundle induces covariant derivatives on all associated bundles in a coherent way, leading to many important geometric constructions. This is the situation in Riemannian geometry where, for a Riemannian manifold M, the Levi-Civita connection on the frame bundle of M induces a covariant derivative on the tensor fields, leading, for instance, to the Riemannian curvature of M. The noncommutative geometry of frame bundles, however, has not been studied conclusively, although the notion of a noncommutative principal bundle is certainly available. In this talk we present our approach to the subject. (pdf)

As a preparation for Karl's talk, we will recall some key notions regarding paradoxicality in groups resp. rings. More precisely, for groups, we will recall notions such as paradoxical decompositions, amenability and supramenability. For rings, we will recall notions such as IBN and UGN (unbounded generating number) and introduce BGN (bounded generating number). All concepts will be exemplified by plenty of well-known and less well-known examples. (pdf)

For a ring $R=\oplus_{g\in G} R_g$ graded by a group $G$, we identify three separate sets of conditions that guarantee that $R$ has unbounded generating number if the base ring $R_1$ does. These are:
(1). $G$ is amenable, and $R_g$ is a nonzero finitely generated free $R_1$-module for every $g\in G$.
(2). $G$ is supramenable, and $R_g$ is a finitely generated free $R_1$-module for every $g\in G$.
(3). $G$ is locally finite, and $R_g$ is a finitely generated projective $R_1$-module for every $g\in G$.
For each set (1)-(3), we discuss whether all of the conditions are necessary. In the case of (1), this gives rise to three ring-theoretic characterizations of the property of amenability for a group. This is joint work with Johan Öinert. (pdf)

The framework of real calculi is a relatively direct derivation-based approach to noncommutative geometry, where one studies (right) modules over a *-algebra A in relation to a Lie algebra g of derivations on A. Among other things one can develop a notion of affine connections, and in this talk the focus will lie on finite-dimensional modules over A and how the existence of a Levi-Civita connection in this case depends on the structure of g. (pdf)

Given a subalgebra n of a Lie algebra g we can define the category of U(g)-modules whose restriction to U(n) is free. In this talk I will present some results about the classification of such modules when g is of classical type and n is a Cartan-subalgebra. I will also present some recent results for the case where n is the nilradical of a maximal parabolic subalgebra of sl(V) and discuss how these modules fit into R. Block's classification of sl2-modules. (pdf)

In this talk, I discuss recent work on a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded rings. I also give some background to these results. This talk is based on joint work with Johan inert. (pdf)

Over a finite field with q elements, a Del Pezzo surface X has q^2+aq+1 points. Serre asked: Which values of a can occur? This question is closely related to the more refined question of which conjugacy classes (in an appropriate Galois group related to X) can arise as the conjugacy class of the Frobenius endomorphism of X. This is called the "inverse Galois problem for Del Pezzo surfaces". In this talk, I will explain how to answer the quantitative version of the inverse Galois problem: given an element g in the Galois group, how many Del Pezzo surfaces are there such that the Frobenius endomorphism acts as g?

In any ring, a trace ideal is an ideal which is the sum of homomorphic images from some other fixed module to the ring. Recent work has demonstrated the usefulness of trace ideals in understanding ring and module structure. In this talk, I will outline some of the theory of trace ideals, and trace modules more generally. In particular, we will explore the property of a module being trace in some envelope (such as the injective envelope), and the types of rings that can be characterized with this property. This talk is based on joint work with Haydee Lindo.

Matroid theory and the theory of field-linear codes are closely related since every matrix over a field defines a matroid. Despite this fact, matroid theory has only rather recently started to play an important role in the development of coding and information theory, for example in areas such as distributed storage, linear codes equipped with Hamming weights, network coding, index coding, and caching. Polymatroids, a generalization of matroids, can be used to capture properties of Shannon entropy. This connection between entropy and polymatroids enables polymatroids to be used to investigate non-linear coding systems. In this talk I will present how matroids and more generally polymatroids can be used for applications in information theory. (pdf)

For any associative algebra $A$, the left regular representation $\lambda_A:A\to \mathop{\rm End}(A)$ is an embedding of $A$ into its linear endomorphism algebra $\mathop{\rm End}(A)$. In this talk, I shall explain how this elementary observation can be generalised to a (less elementary) structure theorem for general non-associative algebras. It turns out that, by adding some data to the left regular the isomorphism type of a (not necessarily associative) algebra $A$ with identity element $e$ is encoded in the datum $(\mathop{\rm im}(\lambda_A), \ker(\mathop{\rm ev}_e))$, where $\mathop{\rm im}(\lambda_A)\subset \mathop{\rm End}(A)$ is the image of the left regular representation, and $\mathop{\rm ev}_e:\mathop{\rm End}(A)\to A,\:T\mapsto T(e)$ the evaluation map at the identity element in $A$. This leads to a description of the category of non-associative algebras in terms of associative algebras with certain distinguished subspaces.

A hom-algebra is an algebra that in addition to the ordinary multiplication also has a linear unary operation, known as the hom or twisting map. There exist a number of hom-variants of ordinary classes of algebras, where axioms have been modified by inserting homs in select positions: for example hom-Lie and hom-associative algebras satisfy homified variants of the Jacobi and associativity respectively axioms. A potentially troubling matter is however that examples of hom-associative algebras found in the literature are pretty much exclusively Yau twists of associative algebras, which raises the worry that they might all be just ordinary associative algebras presented in a silly way. This talk puts that worry to rest by presenting two new constructions - the homassocification of a general algebra, and hom-associative algebras with truncated freedom - that can produce non-Yau-twist hom-associative algebras. The latter construction also provides the first explicit example of a hom-associative algebra that is not strongly hom-associative. (pdf)

In this work, we study the specific properties of (n+1)-dimensional n-Hom-Lie algebras and give full lists of algebras for some cases of the twisting map and n=3. We classify up to isomorphism one subclass of the given lists by studying solvability and nilpotency of these algebras and solving systems of algebraic equations given by the isomorphism conditions, we also present a few example and study some of their specific properties related to derived series and central descending series. (pdf)

I n this talk, I will introduce non-associative generalizations of skew Laurent polynomial rings and some related rings, such as skew power series rings and skew Laurent series rings. The focus will mainly be on the former rings and results concerning their ideals, such as when they are simple and generalizations of the famous Hilbert’s basis theorem. In particular, we will see that a right, but not left Noetherian version of the latter theorem holds; a difference that does not exist in the associative setting. This is based on joint work with Johan Richter. (pdf)

In this talk I will describe non-unital Ore extensions. I will provide a characterization of simplicity for a certain class of such Ore extensions. This result generalizes a result by Öinert, Richter and Silvestrov from the unital setting. The talk is based on joint work with Patrik Lundström and Johan Öinert. (pdf)

In this talk, we will review the evaluation of a skew polynomial as introduced by Lam and Leroy during the 80s. Using the evaluation map, we will introduce some notions of algebraically closed (σ,δ)-fields. It turns out that there are different notions of “closedness” in the context of skew polynomials. We will give some basic constructions and existence results regarding algebraically closed (σ,δ)-fields. If time allows, we will present an application concerning the method of partial fraction decomposition in the skew field of rational skew functions.

Symmetric (σ,τ)-derivations are (σ,τ)-derivations that are simultaneously (τ,σ)-derivations. In this presentation, I will talk about symmetric (σ,τ)-derivations together with some regularity conditions and show that strongly regular (σ,τ)-derivations are always inner. Moreover, I will introduce a (σ,τ)-Hochschild cohomology theory which describes the outer (σ,τ)-derivations in first degree, in analogy with the classical Hochschild cohomology. (pdf)

In this talk, some constructions of representations of covariance-type commutation relations by operators on Banach spaces will be presented. (pdf)

Divisibility is a transitive property in associative algebras. This allows one to establish chains of divisibility between elements of the algebra and gives stability to the relation “x divides y” in the sense that one can multiply by a third element at both sides of the relation. In Hom-associative algebras, divisibility relations are altered by the twisting map. Hom-associativity introduces more complicated relations which involve, among others, the kernel of the twisting map and zero-divisors within the algebra. We establish divisibility relations between elements of arbitrary and (Hom)-unital Hom-associative algebras. (pdf)