# Symmetries of modular categories coming from quantum groups

This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided autoequivalences for $SU(2)_N$.

• Computing all braided autoequivalences (up to equivalence) is an ongoing project even in the simplest possible cases. It amounts to trying to compute a Brauer-Picard group, and these are notoriously difficult to determine. Even the Brauer-Picard group for (f.d.) $G$-graded $\mathbb C$-vector spaces (with trivial associator) remains an object of ongoing study, with only a handful of examples completely understood. Although quantum groups have their own particular interest and handy properties, I'm doubtful such a description exists (beyond the usual abstract characterizations). – zibadawa timmy Jun 27 '16 at 17:51
• See earlier question, mathoverflow.net/questions/225512/… and the note I have because of that – AHusain Jun 27 '16 at 19:08
• @Zibadawa and AHusain: Thanks for the comments! The Picard group and the braided autoequivalence group of modular categories are isomorphics, but they are not the same. For example an explicit description of invertible bimodule categories of pointed fusion categories is easy, but an explicit description of braided autoequivalence of associated twisted Drinfeld double is hard. – César Galindo Jun 27 '16 at 19:40
• I am not interested in the group structure of the full braided autoequivalence, only in known and/or explicit examples. For example, I believe it is possible to define braided equivalence of the $\mathcal{C}(\mathfrak(g),q)$ from some suitable Hopf algebra automorphism of $U_q(\mathfrak{g})$. – César Galindo Jun 27 '16 at 19:44
• Adding the right adjectives every algebra object up to Morita equivalence in $\mathcal C$ with trivial left and right center gives a braided autoequivalence and the converse is also true. For $SU(2)_k$ these are classified and exactly the $D_{2n+1}$ give non-trivial autoequivalences. – Marcel Bischoff Jun 28 '16 at 3:08