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?
$\begingroup$
$\endgroup$

1$\begingroup$ Modules over an involutive Hopf algebra, which is not quasitriangular. $\endgroup$ – Bugs Bunny Feb 22 at 16:12

3$\begingroup$ 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 nonspherical pivotal category will give an example which does not have any braiding. $\endgroup$ – Tobias Fritz Feb 22 at 16:27

$\begingroup$ @Bugs Bunny: Does left and right duals coinciding in $_Hmod$ imply the Hopf algebra $H$ s quasitriangular? $\endgroup$ – Fofi Konstantopoulou Feb 22 at 16:56

$\begingroup$ @Fofi Konstantopoulou No way. $\endgroup$ – Bugs Bunny Feb 22 at 17:31
$\begingroup$
$\endgroup$
The simplest example is Ggraded vector spaces where G is a nonabelian group.

$\begingroup$ Is it clear that this category does not admit a braiding? $\endgroup$ – Fofi Konstantopoulou Feb 22 at 22:50

1
