Tuesday, May 27, 14:00, 7.527

Christian Lomp (Porto), Generalized Kac-Paljutkin algebras

Abstract:
The aim of this talk is to show how to construct a two-parameter family of non-trivial semisimple Hopf algebras H_n,m of dimension n^mm! over a field K containing a primitive n^th root of unity, for integers n, m > 1. The well-known 8-dimensional Kac-Paljutkin algebra arises as H_2,2, while the Hopf algebras constructed by Pansera correspond to the cases H_n,2. Each algebra H_n,m is an extension of the symmetric group algebra KΣ_m by the m-fold tensor product Kℤ_n^⊗m of the group algebra of the cyclic group of order n, and can be realized as a crossed product H_n,m = Kℤ_n^⊗m #_γ Σ_m. I will show how to construct a family of irreducible m-dimensional representations of H_n,m, that are inner faithful as Kℤ_n^⊗m -modules, and exhibit a nontrivial inner-faithful action of a certain n^mm-dimensional subalgebra of H_n,m on a quantum polynomial algebra in m generators.