2023/09/09 by Kangqiao Li, Li, Kangqiao
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.CT #math.QA #math.RA
paper · pdf · doi:10.48550/arxiv.2309.04886
openalex publication_date 2023/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a finite-dimensional Hopf algebra with a left coideal subalgebra B. It is known that Rep(B), the category of finite-dimensional representations of B, is an indecomposable exact left Rep(H)-module category. This paper determines and systematically studies a quasi-Hopf algebra structure (H/B+H)^∗#B, called a (left) partial dual of H, which is reconstructed from the dual tensor category of Rep(H) with respect to Rep(B). Consequently, Rep((H/B+H)^∗#B) is categorically Morita equivalent to 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.