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We show that for any bimodule  M  and bicomodule  C  of a Hopf algebra  H  with involutive antipode, Hochschild complexandare cocyclic-modules. Their para-cocyclic operators are compatible in Gerstenhaber-Schack bicomplex of a Hopf algebra morphismwhich is a combination of Hochschild complexand coHochschild complexfor anyand make the bicomplex into a cylindrical-module under appropriate conditions. From this bicomplex, we get a cocyclic structure of the diagonal complex. With these cocyclic operators, the operadic structure of the diagonal complex is cyclic, so that the Gerstenhaber algebra structure on the diagonal cohomology evolves into a Batalin-Vilkovisky algebra. Hence we obtain the structure of Gerstenhaber-Schack cohomology ofwhich is isomorphic to the diagonal cohomology. Our results can be applied to group algebraand arbitrary  G -graded Hopf algebra  A . Letbe  ϵ A(resp.  u B,  id A). Then we can get the Gerstenhaber structure on Hochschild cohomology(resp. Adams Cobar construction of  B , Gerstenhaber-Schack cohomology of  A ). Moreover, the Gerstenhaber structure operators of  ϵ Aand  u Bcommute with composition.

Cocyclic complexes of Hopf algebras with special antipodes