Cyclic extensions of fusion categories via the Brauer-Picard groupoid

Understand

We construct a long exact sequence computing the obstruction space, pi_1(BrPic(C_0)), to G-graded extensions of a fusion category C_0.

• The other terms in the sequence can be computed directly from the fusion ring of C_0.
• We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as G-extensions.
• The most striking of these is a Z/2Z-extension of one of the Asaeda-Haagerup fusion categories, which is one of only two known 3-supertransitive fusion categories outside the ADE series.