Abstract: Koppinen introduced bi-Frobenius algebras as vector spaces with two Frobenius algebra structures with a certain compatibility. Double Frobenius algebras as introduced by Doi and Takeuchi have a Frobenius algebra and a Frobenius coalgebra structure. Applying the equivalence between Frobenius algebras and coalgebras, both structures can be compared. We will discuss the axioms, and present them in a natural way, based on the approach to Frobenius algebras by Lowell Abrams.

Abstract: Based on C. Calinescu, S. Dascalescu, A. Masuoka, C. Menini, Quantum Lines over Non-cocommutative Cosemisimple Hopf Algebras.

Abstract: The aim of this talk is to review some relevant results by Dragos Stefan about Hochschild cohomology on Hopf Galois extensions and discuss more recent ones obtained in a joint work.

Abstract: In the talk I will give a quick survey on structural results on Nichols algebras and related open problems. Then I will discuss how Hopf orders, Hopf filtrations and number theory can help to solve some of these problems.

Abstract: One of the tools used in the models of non-commutative geometry is the construction of non-commutative algebras from commutative algebras. This construction is modeled on the ring of quantum polynomials and others. In this talk we will give a brief introduction to the construction of twisted tensor products and the associated algebraic and geometric elements.

Abstract: I will present some results that use closure properties of left perpendicular classes to obtain characterizations for (co)silting objects in various triangulated categories. In particular, we will see that a weak cogenerator in a triangulated category with products is cosilting if and only if it belongs to its left perpendicular class, and this class is closed with respect to products.

Abstract: We take a closer look at V.I. Arnold's claim that the Jacobi identity forces the heights of a triangle to cross at one point.

Abstract: We study the category of Doi-Koppinen modules for semiperfect coalgebras, several adjoint pairs functors are constructed  and information on the injective objects is obtained. Some applications will be presented for injective objects and integrals. (joint work with Dascalescu and Nastasescu).

Abstract: Koszul modules, introduced by Papadima and Suciu are multi-linear algebra objects associated to an arbitrary subspace in a second exterior power. They naturally appear in the Geometric Group Theory, in relations with Alexander invariants of groups, and their support, resonance schemes, have a very interesting geometry. We survey some recent developments on this topic (joint project with Gavril Farkas, Claudiu Raicu, Alex Suciu, Jerzy Weyman).

Abstract: We prove in a very general framework some versions of the classical Poincar\'e-Birkhoff-Witt Theorem, which extend results known in the literature. 

This talk is based on a joint work with P. Saracco and D. Stefan published in the "Israel J. Math", 249 (2022).

Abstract: In this talk, we introduce a class of finite dimensional quasi-Hopf algebras, called smal quasi-quantum groups, and compute their  representation rings.  It turns out that their stable representation rings are isomorphic to the ones of the classic small quntum groups. 

Abstract: We give an overview of some classification results, of which stem from work in collaboration with S. Dascalescu and C. Nastasescu. Together, we obtained several results classifying and characterizing pointed co-Frobenius coalgebras and Hopf algebras which are either path coalgebras or more generally, monomial. Based on these ideas, I will explain further research classifying infinite dimensional co-serial coalgebras and Hopf algebras, and more recently, results classifying pointed Hopf algebras which have discrete representation type, in the sense that any finite subcategory of that of comodules has finite representation type. Time permitting, we might mention some current questions; the presentation will focus on a general line of work and will mention results spanning several joint publications of the author and Sorin Dascalescu.

Abstract: In this talk we introduce a family of determinant-like maps $det^{S^r}:V_d^{\otimes\binom{rd}{r}}\to k$ and discuss their properties. We show that the map $det^{S^2}$ detects if a $d$-partition of the complete graph $K_{2d}$ is cycle-free or not. We present an application to geometry as well as a relation between the map $det^{S^2}$ and the equilibrium problem in physics. We also show that the map $det^{S^r}$ can be used to detect if a $d$-partition of the complete $r$-uniform hypergraph $K_{rd}^r$ has zero Betti numbers or not. These results are joint work with Matthew Fyfe, Steven Lippold, Alin Stancu, and Jacob Van Grinsven.

Abstract: We describe the congruences characterizing the bicrossproduct by automorphisms of two cyclic groups, and the necessary and sufficient conditions such that these congruences have non-trivial solutions in both symmetric and general cases. We then discuss some primality conditions that arise from this group-theoretic construction, and study the properties of some related families of pairs of prime numbers. The talk is based on joint work with M. Cipu, M.-T. Tsai and A. Zaharescu.

Abstract: A BiHom-associative algebra is a (nonassociative) algebra A endowed with two commuting multiplicative linear maps α, β : A → A such that α(a)(bc) = (ab)β(c), for all a, b, c ∈ A. Rota-Baxter and Nijenhuis operators are usually considered on associative and Lie algebras, but they can be defined for any class of algebras, in particular for BiHom-associative algebras, where they behave similarly to the associative case. However, the aim of the talk is to show that in the BiHom-associative context one can (and has to) consider some generalized versions of them. 

The talk is based on joint work with Ling Liu, Abdenacer Makhlouf and Claudia Menini.

Abstract: There are many interesting connections between Frobenius algebras and Hopf algebras (such as the famous Larson-Sweedler theorem). Our aim to to prove a new connection between “Frobenius” and “Hopf". To arrive there, recall that Sweedler introduced the notion of a “(universal) measuring coagebra” between two algebras. Considering such a coalgebra for every pair of algebras, this leads to an enrichment of the category of algebras over the category of coalgebras, turning the category of algebras in a so-called "semi-Hopf category”. Universal measuring coalgebras can however be considered between any type of algebraic structures, in particular between Frobenius algebras. Our main result states that in case of Frobenius algebras, the associated semi-Hopf category admits an antipode, that is, it is a Hopf category. This implies for example that the universal acting bialgebra on a Frobenius algebra is a Hopf algebra. By restricting to grouplike elements of this Hopf algebra we recover in this way the well-known fact that any morphism of Frobenius algebras is invertible. If time permits, we also will discuss some duality results and present some small examples.

This talk is based on a joint work with Paul Grosskopf (ULB).

Abstract: The study of partial symmetries (e.g. partial dynamical systems, partial (co)actions, partial (co)representations) is a relatively recent research area in continuous expansion, whose origins can be traced back to the study of $C^*$-algebras generated by partial isometries.

One of the central questions in the field is the existence and uniqueness of a so-called globalization or enveloping (co)action. 

In the framework of partial actions of groups, any global action of a group on a set induces a partial action of the group on any subset by restriction. 

The idea behind the concept of globalization of a given partial action is to find a (universal) global action such that the initial partial action can be realized as the restriction of this global one. The importance of this procedure is testified by the numerous globalization results already existing in the literature which, however, are based on some ad hoc constructions, depending on the nature of the objects carrying the partial action.

In this presentation I will report on some recent developments in partial representation theory, focusing in particular on a novel and unified approach to globalization, and on some future directions we are working on.

(based on a joint project with E. Batista, M. D'Adderio, W. Hautekiet, J. Vercruysse)