# Nonbraided rigid monoidal category where left and right duals coincide

In a braided rigid monoidal category $$(\mathcal{M},\otimes)$$ left and right duals coincide. What is an example of a rigid monoidal category where left and right duals coincide but there exist no braiding for the category?

• Modules over an involutive Hopf algebra, which is not quasitriangular. – Bugs Bunny Feb 22 at 16:12
• It may be useful to know that the standard term for when left and right duals coincide in a coherent way is "pivotal category". Any pivotal category which is braided is automatically spherical. So any non-spherical pivotal category will give an example which does not have any braiding. – Tobias Fritz Feb 22 at 16:27
• @Bugs Bunny: Does left and right duals coinciding in $_H-mod$ imply the Hopf algebra $H$ s quasi-triangular? – Fofi Konstantopoulou Feb 22 at 16:56
• @Fofi Konstantopoulou No way. – Bugs Bunny Feb 22 at 17:31