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

For a semisimple quasi-triangular Hopf algebra $(H,R)$ over a field $k$ of characteristic zero, and a strongly separable quantum commutative $H$-module algebra $A$, we show that $A\#H$ is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra $End{A}^{\ast }\otimes H$. With these structures, ${}_{A\#H}Mod$ is the monoidal category introduced by Cohen and Westreich, and ${}_{End{A}^{\ast }\otimes H}{ℳ}$ is tensor equivalent to ${}_H{ℳ}$. If $A$ is in the Müger center of ${}_H{ℳ}$, 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.