Does there exist a fusion category with an object $X$ such that $XX^\*\ncong X^\*X$ (where the isomorphism need not be natural in any way)?


Feel free to add adjectives such as pivotal, spherical, unitary, etc.



Is there a fusion category with an object which does not commute with its dual?
The principal even part of extended Haagerup gives a counterexample. Look at the table in the appendix to our paper <http://arxiv.org/pdf/0909.4099> (joint with Stephen Bigelow, Scott Morrison, and Emily Peters) to see that the objects labelled A and B are dual to each other but AB=1+P while BA=1+Q (or maybe the other way around, I'm having trouble remembering our conventions for whether the principal graph is left multiplication or right multiplification).


