The Brauer-Picard groups of the fusion categories coming from the $ADE$ subfactors

Abstract

We compute the group of Morita auto-equivalences of the even parts of the $ADE$ subfactors, and Galois conjugates. To achieve this we study the braided auto-equivalences of the Drinfeld centres of these categories. We give planar algebra presentations for each of these Drinfeld centres, which we leverage to obtain information about the braided auto-equivalences of the corresponding categories. We also perform the same calculations for the fusion categories constructed from the full $ADE$ subfactors. Of particular interest, the even part of the $D_{10}$ subfactor is shown to have Brauer-Picard group $S_3\times S_3$. We develop combinatorial arguments to compute the underlying algebra objects of these invertible bimodules.

Version 2: Major changes in section 4 and minor changes elsewhere, change of title

References in corpus (1)

• G-Extensions of Quantum Group Categories and Functorial SPT

Cited by in corpus (3)

• Equivalences of Graded Categories
• A field guide to categories with $A_n$ fusion rules
• Classification of Module Categories for $SO(3{)}_{2m}$