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 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 EndAββH. With these structure,
A#HβMod is the monoidal category introduced by Cohen and
Westreich, and EndAββHβM is tensor
equivalent to HβM. If A is in the M{\"{u}}ger center of
HβM, 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