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### Is every premodular category the *full* subcategory of a modular category?

In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore ...
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### What's the relation between fusion and coproduct?

For an irreducible finite depth finite index subfactor $(N \subset M)$, there is a structure of fusion category given by the even part of its principal graph. The simple objects $(X_i)_{i \in I}$ of ...
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Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual. Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{... 0answers 153 views ### Are there workable numerical approaches for the pentagon equation? Warning: this post is the "numerical" analog of Are there workable algebraic geometry approaches for the pentagon equation? I've replaced "algebraic geometry" by "numerical" in its content, ... 1answer 713 views ### Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects Let$\mathcal{C}$be a monoidal category and Mon$_{\mathcal{C}}$the category of monoids (also called algebra objects) on$\mathcal{C}$. Questions: are there definitions of image and kernel for a ... 0answers 135 views ### Is there an integral simple fusion ring rank<6, FPdim>60 and Frobenius type? A fusion ring is a finite dimensional complex space$\mathbb{C}\mathcal{B}$together with a distinguished basis$\mathcal{B} = \{ h_1,...,h_r\}$and fusion rules$ h_i \cdot h_j = \sum_k n_{ij}^kh_k $... 0answers 279 views ### Is there a non-trivial Hopf algebra without left coideal subalgebra? Let$H$be a Hopf algebra over an algebraically closed field$\mathbb{K}$of characteristic$0$. A subalgebra$I$of$H$is called a left coideal subalgebra if$\Delta(I) \subset H \otimes I$.$H$is ... 1answer 194 views ### Brauer-Picard for a fusion category coming from a quantum group In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group$O(A \...
My understanding is that the principal graphs are a pair of undirected bipartite graphs, $\Gamma_+,\Gamma_-$. They can be calculated from a fusion ring with a given simple object $X$. How to calculate ...