nLab Drinfeld center

Idea

The notion of center of a monoidal category or Drinfeld center is the categorification of the notion of center of a monoid(associative algebra, group, etc.) from monoids to monoidal categories.

Where the center of a monoid is just a sub-monoid with the property that it commutes with everything else, under categorification this becomes a structure, since we have to specify how the objects in the Drinfeld center commute (braid) with everything else.

Definition

We first give the general-abstract definition

• Abstract definition

of Drinfeld centers. Then we spell out what this means in components in

• Definition in components.

Abstractly

Definition

For $(\mathcal{C}, \otimes)$ a monoidal category, write $\mathbf{B}_\otimes \mathcal{C}$ for its delooping, the pointed 2-category with a single object $*$ such that $Hom_{\mathbf{B}_\otimes \mathcal{C}}(*, *) \simeq \mathcal{C}$.

The Drinfeld center $Z(\mathcal{C}, \otimes)$ of $(\mathcal{C}, \otimes)$ is the monoidal category of endo-pseudonatural transformations of the identity-2-functor on $\mathbf{B}_\otimes \mathcal{C}$:

$$Z(\mathcal{C}, \otimes) \coloneqq End_{\mathbf{B}_\otimes \mathcal{C}}(id_{\mathbf{B}_\otimes \mathcal{C}}) \,.$$

Remark

Unwinding the definitions, we find that an object of $Z(\mathcal{C}, \otimes)$, $\Phi \colon id_{\mathbf{B}_\otimes \mathcal{C}} \to id_{\mathbf{B}_\otimes \mathcal{C}}$, has for components pseudonaturality squares

$$\array{ \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Y}}\downarrow &\swArrow_{\Phi_Y}& \downarrow^{\mathrlap{Y}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast }$$

for each $Y \in Obj(\mathcal{C})$. As shown, these consist of a choice of an object $X \in \mathcal{C}$ together with a natural isomorphism

$$\Phi_{(-)} \colon X \otimes (-) \to (-) \otimes X$$

in $\mathcal{C}$.

The transfor-property of $\Phi$ says that

$$\array{ \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Y}}\downarrow &\swArrow_{\Phi_Y}& \downarrow^{\mathrlap{Y}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Z}}\downarrow &\swArrow_{\Phi_Z}& \downarrow^{\mathrlap{Z}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast } \;\;\;\; \simeq \;\;\;\; \;\;\;\; \array{ \ast &\stackrel{X \coloneqq \Phi(\ast)}{\to}& \ast \\ {}^{\mathllap{Y \otimes Z}}\downarrow &\swArrow_{\Phi_{Y \otimes Z}}& \downarrow^{\mathrlap{Y \otimes Z}} \\ \ast &\underset{X = \Phi(\ast)}{\to}& \ast } \,.$$

And so forth. Writing this out in terms of $(\mathcal{C}, \otimes)$ yields the following component characterization of Drinfeld centers, def. .

In components

Definition

Let $(\mathcal{C}, \otimes)$ be a monoidal category. Its Drinfeld center is a monoidal category $Z(\mathcal{C})$ whose

• objects are pairs $(X, \Phi)$ of an object $X \in \mathcal{C}$ and a natural isomorphism (braiding morphism)

  $$\Phi \colon X \otimes (-) \to (-) \otimes X$$

  such that for all $Y \in \mathcal{C}$ we have

  $$\Phi_{Y \otimes Z} = (id \otimes \Phi_Z) \circ (\Phi_Y \otimes id)$$

• morphisms are given by

  $$Hom((X, \Phi), (Y,\Psi)) = \left\{ f \in Hom_{\mathcal{C}}(X,Y) \;|\; (id \otimes f) \circ \Phi_Z = \Psi_Z \circ (f \otimes id), \; \forall Z \in \mathcal{C} \right\} \,.$$

• the tensor product is given by

  $$(X, \Phi) \otimes (Y, \Psi) = (X \otimes Y, (\Phi \otimes id) \circ (id \otimes \Psi)) \,.$$

Properties

Extra structure on the Drinfeld center

Proposition

The Drinfeld center $Z(\mathcal{C})$ is naturally a braided monoidal category.

Proposition

If $\mathcal{C}$ is a fusion category over an algebraically closed field of characteristic zero, then the Drinfeld center $Z(\mathcal{C})$ is also naturally a fusion category.

(Etingof, Nikshych & Ostrik 2005, Thm. 2.15, review in Davydov, Mueger, Nikshych & Ostrik 2003, Sec. 2.3)

See also Drinfeld, Gelaki, Nikshych & Ostrik 2010, Cor. 3.9, Mueger 2003.

Relation to Drinfeld double under Tannaka duality

Under Tannaka duality, forming the Drinfeld center of a category of modules of some Hopf algebra corresponds to forming the category of modules over the corresponding Drinfeld double algebra. See there for more.

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

Examples

$Vec_G$

For $G$ a group, let $Vec_G$ denote the monoidal category of $G$-graded vector spaces. Then the objects in $\mathcal{Z}(Vec_G )$ consist of pairs $(C,V)$ for $C$ a finite conjugacy class of $G$, along with $V$ a finite finite irreducible representation of the centralizer of $g\in C$. This is Example 8.5.4. in EGNO 2010.

Related concepts

• center of a category, center of an additive category

• Drinfeld double

References

Original articles:

• Michael Mueger, From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors, J. Pure Appl. Alg. 180, 159-219 (2003) (arXiv:math/0111205)

• Pavel Etingof, Dmitri Nikshych, Victor Ostrik, On fusion categories, Annals of Mathematics Second Series, Vol. 162, No. 2 (Sep., 2005), pp. 581-642 (arXiv:math/0203060, jstor:20159926)

• Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, On braided fusion categories I, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 (doi:10.1007/s00029-010-0017-z)

Review:

• Alexei Davydov, Michael Mueger, Dmitri Nikshych, Victor Ostrik, Sec. 2.3 of: The Witt group of non-degenerate braided fusion categories, Journal fuer reine und angewandte Mathematik, 677 (2013. 135-177), de Gruyter 2003 (arXiv:1009.2117, doi:10.1515/crelle.2012.014)

Textbook accounts:

• Shahn Majid, Foundations of quantum group theory, Cambridge Univ. Press

• Christian Kassel, Quantum groups, Graduate Texts in Mathematics 155, Springer 1995 (doi:10.1007/978-1-4612-0783-2, webpage, errata pdf)

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik. (2010). Tensor Categories (Vol. 205). American Mathematical Soc.

A general discussion of centers of monoid objects in braided monoidal 2-categories (which reduces to the above for the 2-category Cat with its cartesian product) is in

• Ross Street, The monoidal center as a limit, pdf

An application to character sheaves is in

• George Lusztig, Non-unipotent character sheaves as a categorical centre, arxiv/1506.04598

In relation to spectra of tensor triangulated categories:

• Kent B. Vashaw, Balmer spectra and Drinfeld centers (arXiv:2010.11287)

Relation to Frobenius monoidal functors:

• Johannes Flake, Robert Laugwitz, Sebastian Posur: Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers [arXiv:2410.08702]