Partially Dualized Quasi-Hopf Algebras Reconstructed from Dual Tensor Categories to Finite-Dimensional Hopf Algebras
Let $H$ be a finite-dimensional Hopf algebra with a left coideal subalgebra $B$. It is known that $\mathsf{Rep}(B)$, the category of finite-dimensional representations of $B$, is an indecomposable exact left $\mathsf{Rep}(H)$-module category. This paper determines and systematically studies a quasi-Hopf algebra structure $(H/B^+H)^\ast\#B$, called a (left) partial dual of $H$, which is reconstructed from the dual tensor category of $\mathsf{Rep}(H)$ with respect to $\mathsf{Rep}(B)$. Consequently, $\mathsf{Rep}((H/B^+H)^\ast\#B)$ is categorically Morita equivalent to $\mathsf{Rep}(H)$. As applications: 1) Our construction of partial duals unifies some classical results in the literature, such as bismash products of matched pair of groups given by Takeuchi, bosonizations of dually paired Hopf algebras given by Heckenberger and Schneider, etc. 2) We show that any finite-dimensional Hopf algebra with coradical being an abelian extension is categorically Morita equivalent to a basic quasi-Hopf algebra. 3) We provide a process for constructing genuine quasi-Hopf algebras with an example.