Fusion category and Hopf algebra

Question

Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed from $K$ by some compatible data [1].

I want to know if there exists a similar result on Rep($H$) and Rep($K$), where Rep($H$) is the fusion category of finite-dimensional representations of $H$? If not, what can be said about Rep($H$) and Rep($K$)?

Thank you!

[1]N. Andruskiewitsch, Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996), 3-42

Answer 1

The following article may interest you: C. Pinzari and J. Roberts, A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C *-Algebras, Communications in Mathematical Physics 277 (2008) no 2, 385-421.

However, this article is written in the language of C*-tensor categories and compact quantum groups.

I hope this helps...