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.
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
6
|
||||||
|
|
7
|
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). |
||
|
|
