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A fusion category is called noncommutative if its Grothendieck ring is noncommutative. Let us call a fusion category strongly noncommutative if every fusion category Morita equivalent to it (i.e. same Drinfeld center up to equiv.) is noncommutative.

Question: Is there a strongly noncommutative fusion category (say over $\mathbb{C}$)?
If so, what are the known examples?

Note that if $G$ is a finite group then $Vec(G)$ is not strongly noncommutative (even if $G$ is noncommutative) because it is Morita equivalent to $Rep(G)$ which has a commutative Grothendieck ring. Moreover, the Extended Haagerup fusion categories are also not strongly noncommutative because they form a Morita equivalent class and one of them is commutative.

This post is in the same spirit than this one about strongly simple fusion categories. The main difference is that I know examples of strongly simple fusion categories whereas I do not know a single strongly noncommutative fusion category.

The next step would be about fusion categories which are both strongly simple and strongly noncommutative.

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  • $\begingroup$ Is not having same Drinfeld center too strong? E. g. for all I know there might be tons of very different fusion categories with "trivial" Drinfeld center (equivalent to the category of fd vector spaces), no? In any case, could you please refer to some introductory material about Morita equivalence in this context? $\endgroup$ May 27 at 10:06
  • $\begingroup$ @მამუკაჯიბლაძე: see Section 7.13 in this book. My guess is that if (Drinfeld center) $\mathcal{Z}(\mathcal{C}) \simeq Vec$ then $\mathcal{C} \simeq Vec$ (at least over $\mathbb{C}$), but I may be wrong, I need to check. Maybe you had in mind general tensor categories. $\endgroup$ May 27 at 10:36
  • $\begingroup$ Thank you for the link, but I could not find there that Morita equivalence in their sense (7.12.17, that one is equivalent to the category of bimodules over an algebra in the other) follows from having the same Drinfeld center? $\endgroup$ May 27 at 11:28
  • $\begingroup$ Also now I noticed that my "too strong" looks wrong. I meant not the condition but the resulting equivalence notion. Writing "too coarse" would be probably better. $\endgroup$ May 27 at 11:35
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    $\begingroup$ @მამუკაჯიბლაძე I can at least provides a definite answer to your first comment. Theorem 7.16.6 in the book mentioned above (usually called EGNO book) states that for any finite tensor category $FPdim(\mathcal{Z}(\mathcal{C})) = FPdim(\mathcal{C})^2$. So if $\mathcal{Z}(\mathcal{C}) \simeq Vec$ then $FPdim(\mathcal{Z}(\mathcal{C})) =1$, so $FPdim(\mathcal{C}) = 1$, which means that $\mathcal{C} \simeq Vec$, as I guessed. $\endgroup$ May 28 at 18:07

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Consider the symmetric group group $G = S_3$ of order $6$. Then $\mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1))$. Then $\omega$ restricts nontrivially to every nontrivial subgroup of $G$.

It follows that $\mathbf{Vec}^\omega[G]$ is not Morita-equivalent to any other fusion category. (Recall that, for any $G,\omega$, fusion categories equivalent to $\mathbf{Vec}^\omega[G]$ are indexed by pairs consisting of a subgroup $H \subset G$ together with a 2-cochain $\psi$ on $H$ solving $\mathrm{d}\psi = \omega|_H$.)

But $G$ is noncommutative.

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  • $\begingroup$ What is the reference for what you recalled? $\endgroup$ May 27 at 13:47
  • $\begingroup$ I'm sure it's in EGNO somewhere. And I should didn't quite quote the indexing correctly --- the correct statement is that pairs (fusion category $\mathcal F$, morita equivalence $M : \mathcal F \simeq Vec^\omega[G]$) are indexed by pairs $(H,\psi)$, and that if you want to work with just sets (and not spaces) then you should take $(\mathcal F, M)$ up to appropriate equivalence, and you have to complement this by taking $\psi$ modulo $\psi \leadsto \psi+d\phi$. Anyway, if $\omega|_H$ is always nontrivial (except $H = \{1\}$), then there are no nontrivial Morita equivalents. $\endgroup$ May 27 at 14:19
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    $\begingroup$ One proof is roughly: Given fusion category $\mathcal C$, to give a Morita equivalence $(\mathcal{F},M)$ is to give the simple $\mathcal C$-module $M$ (at which point $\mathcal F = End_{\mathcal C}(M)$), and by Ostrik these are the same as simple algebra objects, and then you have to convince yourself that in $Vec^\omega[G]$ any simple algebra must be $A = \bigoplus_{h \in H} h$ for some subgroup $H \subset G$, at which point $\psi$ gives you the multiplication rule on $A$. $\endgroup$ May 27 at 14:22
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    $\begingroup$ Technically no --- that corollary only does the case of trivial $\omega$. You want EGNO 9.7.2. $\endgroup$ May 27 at 15:59
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    $\begingroup$ @მამუკაჯიბლაძე The trivial subgroup is the identity morita equivalence $Vec^\omega[G] \simeq Vec^\omega[G]$. $\endgroup$ May 29 at 15:11

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