Insalata di Matematica "Non-existence of integral Hopf orders for twists of several simple groups of Lie type."

Per il ciclo di seminari "Insalate di Matematica", il 17 maggio 2023 alle ore 16:00, Elisabetta Masut (Padova) parlerà di

Title: "Non-existence of integral Hopf orders for twists of several simple groups of Lie type."

Abstract:  In 1975 Kaplansky listed 10 conjectures on Hopf algebras, which have been the focus of a great deal of research. Some of these conjectures are still unanswered. In particular, we will be interested on the sixth one, that is a generalization of Frobenius theorem in the framework of representation theory for Hopf algebras.
It states that, given a complex finite-dimensional semisimple Hopf algebra , the dimension of every irreducible representation of  divides the dimension of . 
Larson proved a weaker version of it: if the complex finite-dimensional semisimple Hopf algebra admits a Hopf order over a number ring, then Kaplansky' sixth conjecture is satisfied. 
A natural question now arises: does every complex semisimple finite-dimensional Hopf algebra which satisfies Kaplansky' sixth conjecture admit a Hopf order over a number ring?
The answer is negative. 
In this talk, after a brief excursus on Hopf algebras, we present families of Hopf algebras, which satisfy Kamplansky' sixth conjecture, but they do not admit a Hopf order over a number ring. These Hopf algebras are constructed as Drinfel'd twist of the group algebras , with  a number field. We will prove that for every finite simple group , there is always a deformation, such that the twisted Hopf algebra does not admit a Hopf order over a number ring. Moreover, we will show that for two families of groups, this non-existence result holds for any twist.
This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra.

Il seminario si svolgerà in aula 3014 (U5) e sarà fruibile online.