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Does there exist a fusion category with an object $X$ such that $XX^*\ncong X^*X$ (where the isomorphism need not be natural in any way)?

Feel free to add adjectives such as pivotal, spherical, unitary, etc.

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    $\begingroup$ If you allow "infinite depth" things then the universal example (oriented Temperley-Lieb) is a counterexample. $\endgroup$ May 26, 2011 at 19:12

2 Answers 2

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The principal even part of extended Haagerup gives a counterexample. Look at the table in the appendix to our paper http://arxiv.org/pdf/0909.4099 (joint with Stephen Bigelow, Scott Morrison, and Emily Peters) to see that the objects labelled A and B are dual to each other but AB=1+P while BA=1+Q (or maybe the other way around, I'm having trouble remembering our conventions for whether the principal graph is left multiplication or right multiplification).

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I wrote a piece of Mathematica code to find fusion rings (that are available in the Anyonica package) that satisfy this constraint:

hasNonComAP[ ring_ ] := 
  With[ {
    particleLabels = Range @ Rank @ ring,
    d              = ConjugateCharge @ ring,
    mt             = MultiplicationTable @ ring
    },

    MemberQ[ 
      particleLabels,
      a_ /;  mt[[a, d[a]]] =!= mt[[d[a], a]]
    ]

  ];

interestingCases = Select[ FRL, hasNonComAP ];

It found 19 multiplicity-free fusion rings and 12 rings with multiplicity. (All with rank <= 9)

The multiplicity free rings are the following:

$ \mathrm{FR}^{8, 1, 2}_{6} $, $ \mathrm{FR}^{8, 1, 2}_{11} $, $ \mathrm{FR}^{8, 1, 2}_{29} $, $ \mathrm{FR}^{8, 1, 2}_{30} $, $ \mathrm{FR}^{8, 1, 4}_{8} $, $ \mathrm{FR}^{9, 1, 2}_{7} $, $ \mathrm{FR}^{9, 1, 2}_{22} $, $ \mathrm{FR}^{9, 1, 2}_{32} $, $ \mathrm{FR}^{9, 1, 2}_{38} $, $ \mathrm{FR}^{9, 1, 2}_{41} $, $ \mathrm{FR}^{9, 1, 2}_{44} $, $ \mathrm{FR}^{9, 1, 4}_{4} $, $ \mathrm{FR}^{9, 1, 4}_{8} $, $ \mathrm{FR}^{9, 1, 4}_{9} $, $ \mathrm{FR}^{9, 1, 4}_{17} $, $ \mathrm{FR}^{9, 1, 4}_{20} $, $ \mathrm{FR}^{9, 1, 4}_{22} $, $ \mathrm{FR}^{9, 1, 4}_{31} $, $ \mathrm{FR}^{9, 1, 4}_{33} $, $ \mathrm{FR}^{6, 2, 2}_{33} $, $ \mathrm{FR}^{8, 2, 2}_{105} $,$ \mathrm{FR}^{8, 2, 2}_{106} $, $ \mathrm{FR}^{8, 2, 2}_{107} $, $ \mathrm{FR}^{8, 2, 2}_{144} $,$ \mathrm{FR}^{8, 2, 2}_{145} $, $ \mathrm{FR}^{8, 2, 2}_{146} $, $ \mathrm{FR}^{8, 2, 2}_{258} $, $ \mathrm{FR}^{8, 2, 2}_{263} $,$ \mathrm{FR}^{8, 2, 4}_{112} $, $ \mathrm{FR}^{6, 6, 2}_{312} $, $ \mathrm{FR}^{6, 7, 2}_{115} $

The rings with multiplicity have not yet been uploaded to the anyonwiki (but I might do that soon under a page of non-commutative fusion rings). One of them is a Hecke algebra:

\begin{array}{cccccc} ๐Ÿญ & ๐Ÿฎ & ๐Ÿฏ & ๐Ÿฐ & ๐Ÿฑ & ๐Ÿฒ \\ ๐Ÿฎ & ๐Ÿญ+๐Ÿฎ & ๐Ÿฒ & ๐Ÿฐ+๐Ÿฑ & ๐Ÿฐ & ๐Ÿฏ+๐Ÿฒ \\ ๐Ÿฏ & ๐Ÿฑ & ๐Ÿญ+๐Ÿฏ & ๐Ÿฐ+๐Ÿฒ & ๐Ÿฎ+๐Ÿฑ & ๐Ÿฐ \\ ๐Ÿฐ & ๐Ÿฐ+๐Ÿฒ & ๐Ÿฐ+๐Ÿฑ & ๐Ÿญ+๐Ÿฎ+๐Ÿฏ+2 ๐Ÿฐ+๐Ÿฑ+๐Ÿฒ & ๐Ÿฏ+๐Ÿฐ+๐Ÿฑ+๐Ÿฒ & ๐Ÿฎ+๐Ÿฐ+๐Ÿฑ+๐Ÿฒ \\ ๐Ÿฑ & ๐Ÿฏ+๐Ÿฑ & ๐Ÿฐ & ๐Ÿฎ+๐Ÿฐ+๐Ÿฑ+๐Ÿฒ & ๐Ÿฐ+๐Ÿฒ & ๐Ÿญ+๐Ÿฏ+๐Ÿฐ \\ ๐Ÿฒ & ๐Ÿฐ & ๐Ÿฎ+๐Ÿฒ & ๐Ÿฏ+๐Ÿฐ+๐Ÿฑ+๐Ÿฒ & ๐Ÿญ+๐Ÿฎ+๐Ÿฐ & ๐Ÿฐ+๐Ÿฑ \\ \end{array}

The other 11 rings (which includes the one mentioned by Noah Snyder) have multiplication tables that are too big to display properly on this site.

The tables can be found, however, as an attachment to our paper (which has been published here, but I'm not sure whether the data can be accessed freely at the moment) in the file FusionRingMultiplicationTables where they appear at positions 546, 1417, 1418, 1419, 1456, 1457, 1458, 1570, 1575, 1700, 7183, 8513. These are also the positions in the FusionRingList of the Anyonica package.

The paper also has a section on some other general results about non-commutative fusion rings that might be of interest.

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