4
$\begingroup$

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?

$\endgroup$
  • 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 non-spherical 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 $_H-mod$ imply the Hopf algebra $H$ s quasi-triangular? $\endgroup$ – Fofi Konstantopoulou Feb 22 at 16:56
  • $\begingroup$ @Fofi Konstantopoulou No way. $\endgroup$ – Bugs Bunny Feb 22 at 17:31
6
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Is it clear that this category does not admit a braiding? $\endgroup$ – Fofi Konstantopoulou Feb 22 at 22:50
  • 1
    $\begingroup$ The fusion rules aren't commutative! $\endgroup$ – Noah Snyder Feb 22 at 23:18
  • $\begingroup$ Yes, of course! $\endgroup$ – Fofi Konstantopoulou Feb 23 at 2:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.