Cyclic Gerstenhaber–Schack cohomology

Abstract

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 $B$.