# Questions tagged [drinfeld-center]

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

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### Quantum dimension in the Drinfeld center

Let $\mathcal{C}$ be a spherical tensor category. It is known that the Drinfeld center of $\mathcal{C}$ is modular (and therefore also spherical), see for example, Corollary 8.20.14 in [1]. Recall the ...

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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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### 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_{...

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### 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 ...

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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-...

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### 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 "...

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### 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 ...

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### 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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### Is the central charge of a Drinfeld center always 0?

(If yes, is there a reference for this statement?)

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### 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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### 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 ...