Weak Hopf algebra actions as fusion category actions

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.