Skew braces and the YBE
March 18
Janis Gathot
TBA
TBA
March 4
Quantum Yang–Baxter equation, Yetter–Drinfeld braces, and semi-abelian categories
In this talk, we show how to construct solutions of the quantum Yang–Baxter equation from Yetter–Drinfeld braces, algebraic structures that generalize cocommutative Hopf braces and are equivalent to matched pairs of actions on Hopf algebras. We further prove that cocommutative Yetter–Drinfeld braces (equivalently, cocommutative Hopf braces) form a semi-abelian category that is also strongly protomodular, as is the case for the categories of groups, skew braces, and cocommutative Hopf algebras. Moreover, under suitable assumptions on the base field, we show that skew braces and post-Lie algebras can be used to construct a torsion theory for this category. This talk is partially based on a joint work with D. Ferri and on a joint work with M. Gran.
February 25
Lukas Simons
Different definitions of Nichols algebras
Nichols algebras are certain braided Hopf algebras that play an important role in the theory of Hopf algebras and quantum groups. Originally introduced by Nichols, they have since revealed themselves as rich objects with applications far beyond Hopf algebra theory. In this talk, I will briefly outline the different definitions of Nichols algebras and explain why these definitions are equivalent.
February 13
Constructing coset geometries with Magma
We will discuss briefly what is available currently in the Incidence Geometry and Coset Geometry package to work with incidence geometries and subclasses of them as abstract regular polytopes, abstract chiral polytopes, hypertopes, etc. We will present some usual problems that one encounters when developing algorithms in Magma.
February 12
Francqui VUB-Leerstoel Inaugural Lecture. On some beautiful children of abstract algebra
From time to time, abstract algebraic structures like Nichols algebras produce during their studies interesting, rather elementary but non-trivial mathematical structures. In the lecture I will demonstrate two of such structures, frieze patterns of integers and the hyperplane geometry of Weyl groupoids.
February 11
On the total degree of a finite group
For a finite group $G$, the total degree $T(G)$ is defined as the sum of the degrees of the complex irreducible characters of $G$. We classify the groups with $T(G) \leq 100$, using computational methods, and investigate special cases such as groups of prime power order and groups with total degree the square of a prime. (This is joint work with László Héthelyi, Burkhard Külshammer, and Magdolna Szöke.)