Cyclic extensions of fusion categories via the Brauer–Picard groupoid

  • Pinhas Grossman

    University of New South Wales, Sydney, Australia
  • David Jordan

    University of Texas at Austin, USA
  • Noah Snyder

    Indiana University, Bloomington, USA

Abstract

We construct a long exact sequence computing the obstruction space, , to -graded extensions of a fusion category . The other terms in the\linebreak sequence can be computed directly from the fusion ring of . We apply our result to several examples coming from small index subfactors, thereby constructing several new fusion categories as -extensions. The most striking of these is a -extension of one of the Asaeda–Haagerup fusion categories, which is one of only two known -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.

Cite this article

Pinhas Grossman, David Jordan, Noah Snyder, Cyclic extensions of fusion categories via the Brauer–Picard groupoid. Quantum Topol. 6 (2015), no. 2, pp. 313–331

DOI 10.4171/QT/64