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

TL;DR

This paper computes the Brauer-Picard groups of fusion categories derived from ADE subfactors by analyzing their Drinfeld centres and braided auto-equivalences, revealing new algebraic structures and symmetries.

Contribution

It provides the first detailed calculations of the Brauer-Picard groups for categories from ADE subfactors, including explicit descriptions and planar algebra presentations.

Findings

The Brauer-Picard group of the even part of the D10 subfactor is S3 x S3.

Planar algebra presentations facilitate the study of braided auto-equivalences.

Explicit algebra objects of invertible bimodules are computed.

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.