Graded twisting of comodule algebras and module categories

Abstract

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 $G\subset {\mathrm{SL}}_{-1}(2)$ on $K_{-1}[x,y]$ and show that in most cases the corresponding invariant rings $K_{-1}[x,y{]}^G$ are invariant rings $K[x,y{]}^{G^{\prime }}$ for the action of a classical subgroup $G^{\prime }\subset \mathrm{SL}(2)$. 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.