Twisted quantum doubles (Q1774767)

Summary: Using diagrammatic pictures of tensor contractions, we consider a Hopf algebra \((A_{op} \otimes^{\mathcal{R}_\lambda}A^*)^*\) twisted by an element \(\mathcal{R}_\lambda \in A^*\otimes A_{op}\) corresponding to a Hopf algebra morphism \(\lambda : A \rightarrow A\). We show that this Hopf algebra is quasitriangular with the universal \(R\)-matrix coming from \(\mathcal{R}_\lambda\) when \(\lambda^2=\text{id}_A\), generalizing the quantum double construction which corresponds to the case \(\lambda = \text{id}_A\).