Auto-equivalences of the modular tensor categories of type A, B, C and G

Type 𝐼𝐼 quantum subgroups of 𝔰𝔩_{𝔑}. ℑ: Symmetries of local modules

“…The planar algebra P Λ 1 +Λ𝑟 is well understood [18,39]. It is generated by two trivalent vertices satisfying the Thurston relations (see [39,Lemma 3.2]).…”

“…The coefficients 𝑐 1 , 𝑐 2 , 𝑐 3 , 𝑐 4 for which 𝜙 preserves the Thurston relations are solved for in [18,Lemma 3.1]. With the condition that 𝜙 is braided, there are two solutions, which we denote 𝜙 id and 𝜙 cc .…”

“…To construct the remaining auto-equivalences, we observe that the chargeconjugation auto-equivalences exist for C(𝔰𝔩 𝑟+1 , 𝑘) ad when 𝑘 ≥ 3 and 𝑟 ≥ 2 [18]. These auto-equivalences preserve the Rep(ℤ 𝑚 ′ ) subcategory, and hence descend to autoequivalences of C(𝔰𝔩 𝑟+1 , 𝑘) ad…”

“…For the four truly exceptional examples found by Gannon we can quickly compute that the auto-equivalence groups are all trivial, except for the Galois conjugate of C(𝔰𝔩 2 , 10) which has auto-equivalence group ℤ 2 [15,Theorem 1.2]. For the type 𝐼 quantum subgroups coming from conformal inclusions of Lie groups, the group of braided auto-equivalences has been computed in earlier works of the author [18,Theorem 1.1]. For completeness, we collect the results here.…”

“…we show all non-exceptional braided auto-equivalences are charge conjugation, simple current auto-equivalence, or come from the canonical ℤ 𝑚 -action. Here a simple current auto-equivalence is a symmetry of the category constructed via the action of invertible elements, see [18,Lemma 2.4] for additional details. Proving this result is highly technical, and requires several powerful techniques.…”