Gauge Deformations for Hopf Algebras with the Dual Chevalley Property

Let $A$ be a Hopf algebra over a field $K$ of characteristic zero such that its coradical $H$ is a finite dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation $\zeta $ on $A$ such that $ A^{\zeta }\cong Q\#H$ where $A^\zeta$ is the dual quasi-bialgebra obtained from $A$ by twisting its multiplication by $\zeta$, $Q$ is a connected dual quasi-bialgebra in $^H_H\mathcal{YD}$ and $Q \#H $ is a dual quasi-bialgebra called the bosonization of $Q$ by $H $.