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Consider the Fibonacci category F: $A\bigotimes{A}=E\bigoplus{A}$.
A study by Study (SCNR :-) already 1890 listed all unital associative algebras (with rank<=4). (http://en.wikipedia.org/wiki/Algebra_over_a_field#Classification_of_low-dimensional_algebras). Now where is F? Not there in this form, because by a basis change it's isomorph to $A\bigotimes{A}=E$ (and that is listed). But the two are different fusion rings!

Thus: Are the "simple objects" of a fusion ring "marked" somehow and basis change is strictly forbidden?

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That's right, fusion rings have more data than just being a ring, they also have a preferred basis corresponding to the simple objects. So two fusion rings can be isomorphic as rings but not as fusion rings.

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    $\begingroup$ There are lots of other rings out there, e.g. $H^*(G/P)$, that are pretty boring as rings but interesting as rings-with-bases. $\endgroup$
    – Allen Knutson
    Oct 9, 2014 at 17:55

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