In the paper [1], it is shown that there exist finite dimensional, semisimple Hopf algebras $H$ where the antipode $S:H \to H$ is traceless.

Unfortunately, the method of proof in [1] seems rather inexplicit to me. My understanding is that the authors use the fact that every fusion category $\mathcal{C}$ satisfying some conditions is the representation category of some Hopf algebra. Then they convert the trace condition $\text{Tr}(S) = 0$ into a condition on the fusion category $\mathcal{C}$ and show that they can easily construct categories satisfying this new condition, in addition to the conditions necessary for $\mathcal{C}$ to be a representation category of some $H$.

The paper [1] is eight years old now, so I'm hoping that some more explicit examples might be in the literature at this point. So my question is:

Question: What are some explicit examples of a finite dimensional, involutory Hopf algebra (over a field $k$ of characteristic $0$) where the antipode is traceless? By explicit, I mean given by a generator/relation presentation or as some structure tensors in a specific basis.

Here I weakened the hypothesis to assume that $H$ is just involutory, i.e. that $S^2 = \text{Id}$. A theorem of Larson and Radford [2] says that semisimple Hopf algebras are involutory when $\text{char}(k) = 0$.

Alternatively, I would appreciate some advice on what I need to know in order to extract an explicit example from [1], if that is what's necessary.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.