On the algebra structure of some bismash products

Abstract

We study several families of semisimple Hopf algebras, arising as bismash products, which are constructed from finite groups with a certain specified factorization. First we associate a bismash product H q of dimension q(q − 1)(q + 1) to each of the finite groups PGL 2 (q) and show that these H q do not have the structure (as algebras) of group algebras (except when q = 2, 3). As a corollary, all Hopf algebras constructed from them by a comultiplication twist also have this property and are thus non-trivial. We also show that bismash products constructed from Frobenius groups do have the structure (as algebras) of group algebras.

On Quasitriangular Structures in Hopf Algebras Arising from Exact Group Factorizations

“…We have shown, as part of the proof of Proposition 3.2 and Theorem 4.2, that the bicrossed product Hopf algebra H is not twist equivalent to the group algebra of G. For the case where G = S n , n = p + 1 or p + 2, p > 3 a prime number, and H corresponds to the exact factorization considered in Section 3, this fact follows from the main result of [5] that says that H is not isomorphic as an algebra to any group algebra. The analogous result is true for a bismash product (split extension) Hopf algebra associated to the groups PGL 2 (q), q = 2, 3, as shown in [4].…”