Weak Hopf algebra actions as fusion category actions
Alexander Betz
TL;DR
The paper establishes a precise bridge between fusion-category actions on algebras and actions of finite-dimensional semisimple weak Hopf algebras, showing that based actions of $coRep(H)$ on an algebra $A$ over the weak fiber functor $F_H$ correspond to left $H$-actions on $A$ up to $J$-twisted isomorphism. Using Eilenberg–Watts, it recasts fusion-category actions as monoidal functors into bimodule categories and proves an equivalence between based actions and weak Hopf algebra actions, enabling a systematic classification of quantum symmetries of path algebras. The framework is then applied to graded, separable idempotent-split actions, demonstrating that such actions on path algebras arise from weak Hopf algebra actions via the weak fiber functor, and showing that for $PSU(2)_{p-2}$ these are all weak Hopf actions up to conjugacy. The work provides concrete pathways to realize and classify weak Hopf algebra actions on quivers and highlights the role of module-category data, the $J$-twisted isomorphism, and the $6j$-symbol structure in encoding the action.
Abstract
This article develops the theory of fusion categories acting on algebras. We will demonstrate that weak Hopf algebra actions on algebras correspond to specific actions of fusion categories. As an application of this theory, we introduce a family of filtered actions of weak Hopf algebras on the path algebra, and for weak Hopf algebras whose representation categories are equivalent to $\text{PSU(2)}_{p-2}$, we describe all of the separable based fusion category actions in terms of weak Hopf algebra actions.