Quotients of A 2 * T 2

Izumi, Masaki ; Morrison, Scott ; Penneys, David

Canadian journal of mathematics, Tome 68 (2016) no. 5, pp. 999-1022

Résumé

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.

Détail

DOI : 10.4153/CJM-2015-017-4

Mots-clés : 46L37, pivotal category, free product, quotient, subfactor, intermediate subfactor