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. Feb 22, 2020 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. Feb 22, 2020 at 16:27
• @Bugs Bunny: Does left and right duals coinciding in $_H-mod$ imply the Hopf algebra $H$ s quasi-triangular? Feb 22, 2020 at 16:56
• @Fofi Konstantopoulou No way. Feb 22, 2020 at 17:31
• Adding to Tobias Fritz' comment, it's worth mentioning that a braided rigid category is not pivotal in general. Left and right duals a priori only coincide as functors, but not as monoidal functors, which is really the compatibility condition one would like. Jan 21 at 12:07

1 Answer

The simplest example is G-graded vector spaces where G is a non-abelian group.

• Is it clear that this category does not admit a braiding? Feb 22, 2020 at 22:50
• The fusion rules aren't commutative! Feb 22, 2020 at 23:18
• Yes, of course! Feb 23, 2020 at 2:27