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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.

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If you allow "infinite depth" things then the universal example (oriented Temperley-Lieb) is a counterexample. – Noah Snyder May 26 '11 at 19:12
up vote 10 down vote accepted

The principal even part of extended Haagerup gives a counterexample. Look at the table in the appendix to our paper (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).

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