Weak Hopf algebras, smash products and applications to adjoint-stable algebras

Abstract

For a semisimple quasi-triangular Hopf algebra [Formula: see text] over a field [Formula: see text] of characteristic zero, and a strongly separable quantum commutative [Formula: see text]-module algebra [Formula: see text], we show that [Formula: see text] is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra [Formula: see text]. With these structures, [Formula: see text] is the monoidal category introduced by Cohen and Westreich, and [Formula: see text] is tensor equivalent to [Formula: see text]. If [Formula: see text] is in the Müger center of [Formula: see text], then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.