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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?

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    $\begingroup$ Modules over an involutive Hopf algebra, which is not quasitriangular. $\endgroup$
    – Bugs Bunny
    Feb 22, 2020 at 16:12
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    $\begingroup$ 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. $\endgroup$ Feb 22, 2020 at 16:27
  • $\begingroup$ @Bugs Bunny: Does left and right duals coinciding in $_H-mod$ imply the Hopf algebra $H$ s quasi-triangular? $\endgroup$ Feb 22, 2020 at 16:56
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    $\begingroup$ @Fofi Konstantopoulou No way. $\endgroup$
    – Bugs Bunny
    Feb 22, 2020 at 17:31
  • $\begingroup$ 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. $\endgroup$
    – Jo Mo
    Jan 21 at 12:07

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

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

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