In the paper http://arxiv.org/abs/1509.01548, section 1.3, I found the following definitions:

Two fusion categories $\mathcal{C}$ and $\mathcal{D}$ are Morita-equivalent if there exists an indecomposable $\mathcal{C}$-module category $\mathcal{M}$ such that $\mathcal{D}$ is tensor equivalent to $End_{\mathcal{C}}(\mathcal{M})$ or equivalently if their centers are equivalent as braided tensor categories.

Two semisimple Hopf-algebras $H$ and $K$ are Morita-equivalent if Rep$K$ is Morita-equivalent to Rep$H$, with Rep$H$ being the representation category of $H$.

While it is obvious that isomorphic semisimple Hopf-algebras are also Morita-equivalent, I can`t think of any example of non-isomorphic, Morita-equivalent semisimple Hopf-algebras. Does anyone know an (easy) example?

Extra question: Why are both definitions restricted to fusion categories respectively semisimple Hopf-algebras?

Thanks for your help!


1 Answer 1


A few weeks ago, a user posted a good hint for the solution, but deleted his answer after only one day (I couldn't even award him with that 50 bounty he deserved). I did the calculations and the following is a good example for my question:

Let $K$ be a field of characteristic zero and $G$ be any finite non-abelian group and consider the Hopf-algebras $K[G]$ (group algebra over $G$) and its dual $K^G$. Then $K[G]$ is not commutative, but co-commutative and $K^G$ is commutative, but not co-commutative. Therefore these two Hopf-algebras cannot be isomorphic. The group algebra $K[G]$ is semisimple (Maschke-theorem) and obviously co-semisimple, so $K^G$ is also semisimple. One then shows with a few calculations that $K[G]$ and $K^G$ are indeed Morita-equivalent, but that is a bit too long for posting this here.

I also have found out that the definition of Morita-equivalence is restricted to semisimple Hopf-algebras to make sure $RepH$ is a fusion category. And in terms of categories, Morita-equivalence is restricted to fusion categories to make sure the given conditions are equivalent.


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