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arXiv · 1211.6414

Cyclic extensions of fusion categories via the Brauer-Picard groupoid

Pinhas Grossman·David Jordan·Noah Snyder
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Abstract

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. In another direction, we show that our long exact sequence appears in exactly the way one expects: it is part of a long exact sequence of homotopy groups associated to a naturally occuring fibration. This motivates our constructions, and gives another example of the increasing interplay between fusion categories and algebraic topology.

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Pinhas Grossman, David Jordan, Noah Snyder. 2013-04-11. Cyclic extensions of fusion categories via the Brauer-Picard groupoid. https://doi.org/10.4171/qt%2F64

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