5. Objects in fusion categories
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Problem 5.1.
[Rowell] What values can \dim(X) take in [2,3] for X\in\mathcal{C}, a braided fusion category?-
Remark. [Scott Morrison] These two dimensions come from Cyclotomic integers, fusion categories, and subfactors Frank Calegari, Scott Morrison and Noah Snyder, Communications in Mathematical Physics Volume 303, Issue 3 (2011), pp. 845-896 [arXiv:1004.0665].
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Problem 5.2.
[Snyder] What are all the \mathcal{C} generated by X with \text{FPdim}(X)\leq2, and X not self-dual? Are they group theoretical if \dim(X)=2?
Snyder: I think I can do it if X\otimes X^*\cong X^*\otimes X,
Rowell: enough if X is self-dual and the Grotheneick ring is commutative. -
A fusion category version of supertransitivity
Problem 5.3.
[Jones] In a fusion category, is there an upper bound on the N such that X^{\otimes N} is a simple object (where \dim(X)>1)? -
Problem 5.4.
There are accumulation points from below for \text{FPdim}(X) for an object in a fusion category or [M\colon N] for finite depth subfactors. Are there any accumulation points from above?-
Remark. [Scott Morrison] Are there non-integer accumulation points?
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Cite this as: AimPL: Classifying fusion categories, available at http://aimpl.org/fusioncat.