In this work, representations of covariant type commutation relations are constructed. Representations involve pairs of linear integral and multiplication operator. (pdf)

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.

I will give an introduction to some notions, constructions and examples of hom-algebra structures and n-ary hom-algebra structures and mention some open directions for future investigations. (pdf)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)