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Quotients of A 2 * T 2

  • Masaki Izumi (a1), Scott Morrison (a2) and David Penneys (a3)


We study unitary quotients of the free product unitary pivotal category ${{A}_{2}}\,*\,{{T}_{2}}$ . We show that such quotients are parametrized by an integer $n\,\ge \,1$ and an $2n$ –th root of unity $\omega $ . We show that for $n\,=\,1,\,2,\,3$ , there is exactly one quotient and $\omega \,=\,1$ . For $4\,\le \,n\,\le \,10$ , we show that there are no such quotients. Our methods also apply to quotients of ${{T}_{2}}\,*\,{{T}_{2}}$ , where we have a similar result.

The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of ${{A}_{2}}\,*\,{{T}_{2}}$ and ${{T}_{2}}\,*\,{{T}_{2}}$ , we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.

During the preparation of this manuscript, we learnt of Liu's independent result on composites of ${{A}_{3}}$ and ${{A}_{4}}$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch–Haagerup showed that the principal graph of a composite of ${{A}_{3}}$ and ${{A}_{4}}$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\,\ge \,4$ .

This is an abridged version of arxiv:1308.5723.



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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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