Questions tagged [drinfeld-center]

The Drinfeld center of a monoidal category, which is a braided monoidal category.

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Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

Let $G$ be a finite group and $D(G)$ its quantum double. As in my previous question, a typical irreducible representation (finite dimensional over $\mathbb{C}$) is labeled by $(\theta,\pi)$, where $\...
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1answer
107 views

Fusion category and induction matrix to its Drinfeld center: combinatorial properties

This question is inspired by this paper of Scott Morrison and Kevin Walker. Consider a fusion category $\mathcal{C}$ of rank $r$, and its Drinfeld center $Z(\mathcal{C})$ of rank $s$. Let $N_i = (n_{...
10
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3answers
262 views

Generalization of Drinfeld double to comodule algebras

Let $ \mathcal C $ be a monoidal category. Then $ \mathcal C $ is both a left and right module category over itself. Moreover, the Drinfeld centre of $ \mathcal C $ can be defined as the category of ...
7
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0answers
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Why are Levin-Wen/Turaev-Viro models said to be non-chiral?

I'd like to bring together the following two notions of "non-chiral": On the abstract algebraic side, a modular fusion category describing the anyon content of some physical system is said to be non-...
7
votes
1answer
193 views

Is there a simple argument that shows that two unitary fusion categories are Morita equivalent if their Drinfeld centers are equal?

By Morita equivalent I mean that there is an invertible bi-module between the two fusion categories. [Feel free to replace the Drinfeld centers being "equal" by an appropriate categorial notion of "...
4
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0answers
119 views

Can non-chiral 3D TQFTs be extended to non-orientable manifolds whereas chiral ones cannot?

As far as I know, when talking about TQFT, one usually means TQFTs on oriented manifolds with boundary (cobordisms) It appears to me that the Turaev-Viro-Barrett-Westbury state-sum construction can ...
3
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0answers
189 views

How does the relative Drinfeld center interact with the relative Deligne tensor product?

Let $\mathcal{C}$ be a fusion category, and $\mathcal{M}, \mathcal{N}$ semisimple $(\mathcal{C}, \mathcal{C})$-bimodule categories. The left $\mathcal{C}$-action is denoted as $- \triangleright - \...
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Categorical interpretation of quantum double $D(A,B,\eta)$

It is known that the Drinfel'd double $D(A)$ of a Hopf algebra $A$ is characterized by the following two properties: The category of left $D(A)$-modules $_{D(A)}\mathcal{M}$ is equivalent to the ...
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1answer
236 views

Is the central charge of a Drinfeld center always 0?

(If yes, is there a reference for this statement?)
8
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0answers
296 views

Which Drinfeld centers are balanced monoidal, i.e. have a twist?

A twist is an automorphism $\theta$ of the identity functor of a monoidal category with braiding $c$, such that $\theta_{X \otimes Y} = c_{Y,X} c_{X,Y} (\theta_X \otimes \theta_Y)$. A braided monoidal ...
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2answers
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Relationship between Hochschild cohomology and Drinfeld centers

Let $HH_*(A,N)$ (or $HH^*(A,N)$) be the Hochschild homology (or cohomology) of an associative algebra $A$ with coefficients in an $A$-bimodule $N$. I was reading nlab's entry on Hochschild cohomology ...