Graded twisting of comodule algebras and module categories

TL;DR

This paper develops a graded twisting method for equivariant comodule algebras and module categories, providing new insights into quantum group actions, invariant rings, and Poisson boundaries within the framework of monoidal categories.

Contribution

It introduces a graded twisting construction for equivariant comodule algebras and module categories, extending previous work on Hopf algebras and categories.

Findings

Invariant rings under quantum subgroup actions relate to classical subgroup invariants.

Poisson boundaries of graded twisted categories are graded twistings of original boundaries under weak amenability.

The construction applies to actions of quantum subgroups on polynomial algebras.

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 $G\subset SL_{-1}(2)$ 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 $G'\subset SL(2)$. 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.