Twisted quantum doubles

Abstract

Using diagrammatic pictures of tensor contractions, we consider a Hopf algebra (A op ⊗ λ A * ) * twisted by an element λ ∈ A * ⊗ A op corresponding to a Hopf algebra morphism λ : A → A. We show that this Hopf algebra is quasitriangular with the universal R-matrix coming from λ when λ 2 = id A , generalizing the quantum double construction which corresponds to the case λ = id A .