Partially Dualized Quasi-Hopf Algebras Reconstructed from Dual Tensor Categories to Finite-Dimensional Hopf Algebras

Let $\mathsf{R}\mathsf{e}\mathsf{p}(H)$ be the category of finite-dimensional representations of a finite-dimensional Hopf algebra $H$. Andruskiewitsch and Mombelli proved in 2007 that each indecomposable exact $\mathsf{R}\mathsf{e}\mathsf{p}(H)$-module category has form $\mathsf{R}\mathsf{e}\mathsf{p}(B)$ for some indecomposable exact left $H$-comodule algebra $B$. This paper reconstructs and determines a quasi-Hopf algebra structure from the dual tensor category of $\mathsf{R}\mathsf{e}\mathsf{p}(H)$ with respect to $\mathsf{R}\mathsf{e}\mathsf{p}(B)$, when $B$ is a left coideal subalgebra of $H$. Consequently, it is categorically Morita equivalent to $H$, and some other elementary properties are also studied. As applications, our construction could be applied to imply some classical results on bismash products of matched pair of groups and bosonizations of dually paired Hopf algebras.