This talk will be devoted to Hom-algebras and color Hom-algebras, including some constructions, examples and open problems. (pdf)

The Cayley—Dickson construction is a famous construction that generates new *-algebras out of old ones. It is perhaps best known for generating all the real, normed division algebras that exist out of the real numbers themselves: the real numbers, the complex numbers, the quaternions, and the octonions. However, the construction is undoubtedly quite mysterious and seems to be a patchwork created ad hoc to make algebras like the above fit in a construction. In this talk, I will try to shed new light on the Cayley—Dickson construction with the purpose of illuminating the underlying patchwork. We introduce a new class of polynomial rings with a “flipped” multiplication which all Cayley—Dickson algebras naturally appear as quotients of. In particular, this extends the classical construction of the complex numbers as a quotient of a polynomial ring to the quaternions, the octonions, and beyond. This is based on joint work with Masood Aryapoor (MDU). (pdf)

This work is devoted to certain properties and structures of Hom-Lie algebras of generalized sl(2)-type. We construct classes of linear twisting maps that turn a skew-symmetric algebra of generalized sl(2)-type into a Hom-Lie algebra and describe the subclasses of such linear maps which yield multiplicative Hom-Lie algebras. We explore some properties of these Hom-algebras related to their ideals, Hom-ideals, subalgebras and Hom-subalgebras, with emphasis on their derived series, and central descending series, as well as their nilpotence and solvability properties. We investigate the invariance of these sub-algebras under the linear twisting maps, and determine whether these sub-algebras are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. In particular, we investigate these sub-algebras and properties for the subfamilies of non-multiplicative Hom-Lie algebras of generalized sl(2)-type indicating the differences between non-multiplicative and multiplicative cases. This is a joint work with Abdennour Kitouni, Elvice Ongong’a, Jared Ongaro and Sergei Silvestrov. (pdf)

There are many distinguished algebras which are defined by virtue of an algebra identity, classical examples are Jordan algebras, some genetic algebras (including Bernstein and train algebras), algebra of minimal cones. Several examples appeared recently in the context of axial algebras. An identity is traditionally used to infer a deeper information on the Peirce spectrum and the corresponding fusion laws. We review some recent results on the general Peirce decompositions for algebras with identities with a special accent on the so-called “strange” identities, i.e. those implying the trivial Peirce polynomial. In the latter case, the Peirce spectrum becomes undetermined. We explain how this class is related medial algebras. (pdf)

I will talk about examples as an important but under-appreciated component in the mathematics literature, and discuss how micropublishing can both lower the threshold for providing them and simplify finding examples with relevant properties. The Catalogue of Mathematical Objects is a design for how this aspect of mathematics may be wikified, without sacrificing the principles of rigour, quality, and peer review. (pdf)

In this talk I will give a short introduction to toric algebras defined by combinatorial objects. I will then introduce a new family of toric algebras, namely those arising from maximal chains of a finite distributive lattice. Applying results on stable set polytopes, we can conclude that every such algebra is normal and Cohen-Macaulay. We will also discuss how algebraic invariants, such as Krull dimension and Hilbert series, can be computed from the combinatorics of the underlying lattice. (pdf)

Quasi-hereditary algebras are an important class of algebras that arise in Lie-theory, representation theory and algebraic geometry. Central to the study of a quasi-hereditary algebra is a collection of modules called standard modules, and the Ext-algebra of this collection is of particular interest. In this talk, we discuss a special class of quasi-hereditary algebras called twisted doubles, and we provide a formula for the Ext-algebras of their standard modules in terms of the Ext-algebra of the simple modules over a certain subalgebra. If there is time, we will also discuss some possible generalizations and other avenues of research. (pdf)

In noncommutative Riemannian geometry, as in its commutative counterpart, one is interested in Levi-Civita connections. These are connections which are torsion free and compatible with some metric on the space. I will present the basic definitions needed, introduce the notion of a g-connection module, and give a coordinate free way of classifying all torsion free connections on a g-connection module. This is part of an ongoing project together with Joakim Arnlind. (pdf)

I will talk about a joint work with Paula S. E. Moreira (UFSC, Florianopolis, Brazil) concerning the primeness of groupoid graded rings. We have established a characterization of prime nearly epsilon-strongly groupoid graded rings. By applying our result to partial skew groupoid rings and to groupoid rings, in particular, we obtain characterizations of primeness for those types of rings. Our results generalize earlier results by I.G. Connell and B. Steinberg. (pdf)

I will discuss some aspects of representations of Lie algebras of polynomial vector fields on an affine variety. In particular I will show how one can define Rudakov-modules over arbitrary affine varieties, a generalization of Rudakov's work from 1974. The talk is based on joint work with Yuly Billig and Vyacheslav Futorny.

A version of Hilbert’s basis theorem says that all one-sided ideals in the Weyl algebras are finitely generated. A strengthening by Stafford says that the number of generators never need to be more than 2. A hom-associative version of the first Weyl algebra was introduced by Bäck, Richter and Silvestrov, and further studied by Bäck and Richter. In this talk I will describe how Bäck and Richter tried to generalize Stafford’s theorem to higher hom-associative Weyl algebras. It turns out that in the hom-associative case generically all one-sided ideals are principal, i.e. need only one generator. (pdf)

An important open problem in the homological algebra of self-injective algebras is to characterise periodic algebras. An algebra B is said to be periodic if it has a periodic projective resolution as a B-B-bimodule. I will speak about some old and new results on periodic algebras including, in particular, our recent characterisation of periodicity in trivial extension algebras: the trivial extension T(A) of a finite-dimensional algebra A is periodic if and only if A is fractionally Calabi-Yau. (pdf)

Fell invented Fell bundles to understand better and extend Mackey’s pioneering works in the intersection of quantum logic, the theory of infinite-dimensional unitary representations of groups, the theory of operator algebras, and noncommutative geometry. Nowadays, Fell bundles are an important topic both in operator algebras and noncommutative geometry. Indeed, due to their many applications these objects have found broad interest and have been explored by many authors in recent years. Noteworthly, to each Fell bundle one may naturally associate a C*-algebra which generalizes crossed products. In joint work with Nata Machado from UFSC, we try to better understand the structure theory of Fell bundles, for instance, to be able to provide a suitable classification theory. In this talk I will give an introduction to Fell bundles and report on the challenges that we have encountered in our investigation so far. (pdf)

In this talk, I will discuss some of the challenges inherent in the initial phases of mathematical research projects. More concretely, I will describe a family of noncommutative spheres and showcase an unexpected challenge in what I had initially assumed to be a routine verification task. Moreover, I will present some basic attempts to gain a deeper understanding of why the problems that arose exist in the first place. (pdf)

The Jacobian determinant, given n derivations over a commutative associative algebra, gives an n-Lie algebra structure to said algebra. In this talk, I will construct a generalized Jacobian determinant using twisted derivations and study certain properties of it with respect to twisted derivations over the algebra. Under certain conditions on the derivations, it is possible to obtain a series of generalized n-ary hom-Lie structures and other more general structures, relying on commutation relations between derivations and twisting maps. I will discuss the construction and explore a particular case in which a new generalization of n-ary hom-Lie algebras is found. (pdf)

We obtain criteria for when a ring with enough idempotents is left/right artinian or noetherian in terms of local criteria defined by the associated complete set of idempotents for the ring. We apply these criteria to object unital category graded rings in general and, in particular, to the class of skew category algebras. Thereby, we generalize results by Nastasescu-van Oystaeyen, Bell, Park and Zelmanov from the group graded case to groupoid, and in some cases category, gradings. (pdf)