12
votes
Accepted
Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?
In the non-unitary setting ENO proved that if $Z(C)$ and $Z(D)$ are equivalent as braided tensor categories, then C and D are Morita equivalent. This is Theorem 3.1 of this paper. Note that they ...
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11
votes
Accepted
Is there a strongly noncommutative fusion category?
Consider the symmetric group group $G = S_3$ of order $6$. Then $\mathrm{H}^3_{\mathrm{gp}}(G;\mathrm{U}(1)) \cong \mathbb Z/6\mathbb Z$. Choose a generator $\omega \in \mathrm{H}^3_{\mathrm{gp}}(G;\...
- 54.6k
8
votes
Is the central charge of a Drinfeld center always 0?
The Drinfeld center of a spherical fusion category has topological central charge $0\pmod 8$ see Remark 5.19 in
Müger, Michael:
From subfactors to categories and topology. II. The quantum double ...
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6
votes
Accepted
Generalization of Drinfeld double to comodule algebras
Such an algebra exists. As far as I am aware, the algebra was first described in chapter 6 of The blob complex by Morrison and Walker. In this paper, the algebra is construct from a diagrammatic ...
- 3,830
4
votes
Accepted
Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
1)2) is standard for an arbitrary f.d. Hopf algebra $H$, as you say it's not hard to idenfity $D(H)$-modules with Yetter-Drinfeld modules. Then, given two of those, say $V,W$ you can define a braiding ...
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3
votes
Accepted
Relationship between fusion category and its Drinfel'd center
The short answer is no.
Suppose you have a fully faithful monoidal functor $(F,J):\mathcal C\to\mathcal B$, where $\mathcal C$ and $\mathcal B$ are fusion and $J_{X,Y}:F(X)\otimes F(Y)\to F(X\otimes Y)...
- 556
3
votes
Generalization of Drinfeld double to comodule algebras
I think Davydov's papers Centre of an algebra and Full centre of an H-module algebra might be what you are looking for.
- 54.6k
3
votes
Is there a definition of Heisenberg double for quasi-Hopf algebras?
$\newcommand{\dmod}{\text{-}\mathrm{mod}}$I've been asking myself that same question for a while, and I'm fairly certain the answer is "no". Let me expand a bit:
First of all, in the finite ...
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2
votes
Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$
This is my study note that spells out @Adrien 's answer to 1) and 2). As suggested by @Adrien, we will follow Kassel's Quantum Groups, mainly chapter XIII.5. It is a very detailed account.
Explicit ...
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2
votes
Fusion category and induction matrix to its Drinfeld center: combinatorial properties
BQ1 has the following counter-example (inspired by a discussion with Corey Jones):
For $\mathcal{C} = Rep(D_9)$, the fusion rules are:
$$ \begin{smallmatrix}1&0&0&0&0&0\\
0&...
2
votes
Generalization of Drinfeld double to comodule algebras
Section 1 (in particular Prop 1.23) of On module categories over finite-dimensional Hopf algebras by Andruskiewitsch-Mombelli come close to an answer to your question: namely, they show that if $H$ is ...
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1
vote
Relationship between fusion category and its Drinfel'd center
I assume we are under the assumption the category is braided.
This is mentioned in
Drinfeld, Vladimir; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, On braided fusion categories. I, Sel. Math., ...
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