Tuesday, June 18, 14:00, 7.527

Christian Lomp (Porto), Constructing semisimple Hopf algebras using skew polynomial rings

Abstract:
The aim of my talk is to illustrate the use of skew polynomial rings in the construction of Hopf algebras (finite or infinite dimensional ones). I will quickly revise how skew polynomial ring were used to disprove Kaplansky's 10th conjecture. Then I will focus on the action of semisimple Hopf algebras and their actions on rings. A Theorem by Etingof and Walton says that any action of a semisimple Hopf algebra on a commutative domain factors through a group algebra action. The same is true for actions on enveloping algebras of finite dimensional Lie algebras or on iterated skew polynomial rings of derivation type (L-Pansera, 2017). An action is said to be inner-faithful, if it does not factor through a proper quotient Hopf algebra. And in the search for inner-faithful actions, D. Pansera, in his PhD thesis (Porto, 2017), constructed inner-faithful actions of a new family of semisimple Hopf algebras of dimension 2n² (generalizing the 8-dimensional Kac-Paljutkin algebra) on quantum planes. I will present some ongoing work, extending this construction to a new class of semisimple Hopf algebras of dimension 6n³.