  We show that the diagonal complex computing the Gerstenhaber-Schack
cohomology of a bialgebra (that is, the cohomology theory governing bialgebra
deformations) can be given the structure of an operad with multiplication if
the bialgebra is a (not necessarily finite dimensional) Hopf algebra with
invertible antipode; if the antipode is involutive, the operad is even cyclic.
Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries
a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one
obtains a cup product and a cyclic boundary B that generate the Gerstenhaber
bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In
case the Hopf algebra in question is finite dimensional, the Gerstenhaber
bracket turns out to be zero in cohomology and hence the interesting structure
is not given by this e_2-algebra structure but rather by the resulting
e_3-algebra structure, which is expressed in terms of the cup product and B.
