Yetter–Drinfeld modules under cocycle twists

Abstract

We give an explicit formula for the correspondence between simple Yetter–Drinfeld modules for certain finite-dimensional pointed Hopf algebras H and those for cocycle twists H σ of H. This implies an equivalence between modules for their Drinfeld doubles. To illustrate our results, we consider the restricted two-parameter quantum groups u r , s ( sl n ) under conditions on the parameters guaranteeing that u r , s ( sl n ) is a Drinfeld double of its Borel subalgebra. We determine explicit correspondences between u r , s ( sl n ) -modules for different values of r and s and provide examples where no such correspondence can exist. Our examples were obtained via the computer algebra system Singular::Plural.