  A Frobenius algebra is a finite-dimensional algebra $A$ which comes equipped
with a coassociative, counital comultiplication map $\Delta$ that is an
$A$-bimodule map. Here, we examine comultiplication maps for generalizations of
Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius)
algebras. We show that large classes of such algebras, including
finite-dimensional weak Hopf algebras, come equipped with a nonzero map
$\Delta$ as above that is not necessarily counital. We also conjecture that
this comultiplicative structure holds for self-injective algebras in general.
