# Cyclic Gerstenhaber-Schack cohomology

## TL;DR

This paper demonstrates that the Gerstenhaber-Schack cohomology of certain Hopf algebras can be endowed with rich algebraic structures like operads, cyclic operads, and Gerstenhaber-Batalin-Vilkovisky algebras, revealing new insights into their deformation theory.

## Contribution

It establishes that the diagonal complex computing Gerstenhaber-Schack cohomology forms an operad with multiplication and cyclic structure for Hopf algebras with invertible antipode, introducing new algebraic structures on cohomology.

## Key findings

- Gerstenhaber-Schack cohomology has an operad with multiplication structure.
- Cyclic structure appears when the antipode is involutive.
- Finite-dimensional case yields a trivial Gerstenhaber bracket in 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.