Let H be a finite-dimensional Hopf algebra with left coideal subalgebra B. There is a structure of quasi-Hopf algebra on the smash product (H/B+H)*#B, which is called the left partially dualized quasi-Hopf algebra (or partial dual for short) of H for B. It reconstructs the dual tensor category of Rep(H) with respect to its left module category Rep(B), where Rep(H) and Rep(B) denote categories of finite-dimensional representations of respective algebras. In this talk, we introduce the construction of partial duals, as well as:
1) Conditions when partial duals are particular;
2) Formulations of partial duals for some classical structures, such as bismash products and (generalized) quantum doubles;
3) Comparison for gauge invariants of semisimple Hopf algebra and its partial duals, including Frobenius-Schur indicators and exponent.
