  We investigate when a weak Hopf algebra H is Frobenius; we show this is not
always true, but it is true if the semisimple base algebra A has all its matrix
blocks of the same dimension. However, if A is a semisimple algebra not having
this property, there is a weak Hopf algebra H with base A which is not
Frobenius (and consequently, it is not Frobenius "over" A either). We give,
moreover, a categorical counterpart of the result that a Hopf algebra is a
Frobenius algebra for a noncoassociative generalization of weak Hopf algebra.
