Graded twisting of comodule algebras and module categories

Continuing our previous work on graded twisting of Hopf algebras and monoidal categories, we introduce a graded twisting construction for equivariant comodule algebras and module categories. As an example we study actions of quantum subgroups of G⊂SL−1(2) on K−1[x,y] and show that in most cases the corresponding invariant rings K−1[x,y]G are invariant rings K[x,y]G′ for the action of a classical subgroup G′⊂SL(2). As another example we study Poisson boundaries of graded twisted categories and show that under the assumption of weak amenability they are graded twistings of the Poisson boundaries