G-structure on the cohomology of Hopf algebras

We prove that $\mathrm{Ext} ^{\bullet}_A(k,k)$ is a Gerstenhaber algebra, where $A$ is a Hopf algebra. In case $A=D(H)$ is the Drinfeld double of a finite-dimensional Hopf algebra $H$, our results imply the existence of a Gerstenhaber bracket on $H^{\bullet}_{GS}(H,H)$. This fact was conjectured by R. Taillefer. The method consists of identifying $H^{\bullet}_{GS}(H,H)\cong {\mathrm{Ext}}^{\bullet}_A(k,k)$ as a Gerstenhaber subalgebra of $H^{\bullet}(A,A)$ (the Hochschild cohomology of $A$).