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