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An interesting open question is to determine the group of monoidalautoequivalences of the category of Yetter-Drinfeld modules over a finite group$G$, or equivalently the group of Bigalois objects over the dual of theDrinfeld double $DG$. In particular one would hope to decompose this group intoterms related to monoidal autoequivalences for the group algebra, the dualgroup algebra and interaction terms.  We report on our progress in this question: We first prove a decomposition ofthe group of Hopf algebra automorphisms of the Drinfeld double into threesubgroups, which reduces in the case $G=\mathbb{Z}_p^n$ to a Bruhatdecomposition of $\mathrm{GL}_{2n}(\mathbb{Z}_p)$. Secondly, we propose aKuenneth-like formula for the Hopf algebra cohomology of $DG^*$ into threeterms and prove partial results in the case of lazy cohomology. We use theseresults for the calculation of the Brauer-Picard group in the lazy case in[LP15].

On monoidal autoequivalences of the category of Yetter-Drinfeld modules over a group: The lazy case