nLab
Drinfeld center

Contents

Context

Monoidal categories

2-Category theory

Contents

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 stuff, structure, property, 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

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

Abstractly

Definition

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

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

Z(𝒞,)End B 𝒞(id B 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(𝒞,)Z(\mathcal{C}, \otimes), Φ:id B 𝒞id B 𝒞\Phi \colon id_{\mathbf{B}_\otimes \mathcal{C}} \to id_{\mathbf{B}_\otimes \mathcal{C}}, has for components pseudonaturality squares

* XΦ(*) * Y Φ Y Y * X=Φ(*) * \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 YObj(𝒞)Y \in Obj(\mathcal{C}). As shown, these consist of a choice of an object X𝒞X \in \mathcal{C} together with a natural isomorphism

Φ ():X()()X \Phi_{(-)} \colon X \otimes (-) \to (-) \otimes X

in 𝒞\mathcal{C}.

The transfor-property of Φ\Phi says that

* XΦ(*) * Y Φ Y Y * X=Φ(*) * Z Φ Z Z * X=Φ(*) ** XΦ(*) * YZ Φ YZ YZ * X=Φ(*) *. \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 (𝒞,)(\mathcal{C}, \otimes) yields the following component characterization of Drinfeld centers, def. .

In components

Definition

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

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

    Φ:X()()X \Phi \colon X \otimes (-) \to (-) \otimes X

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

    Φ YZ=(idΦ Z)(Φ Yid) \Phi_{Y \otimes Z} = (id \otimes \Phi_Z) \circ (\Phi_Y \otimes id)
  • morphisms are given by

    Hom((X,Φ),(Y,Ψ))={fHom 𝒞(X,Y)|(idf)Φ Z=Ψ Z(fid),Z𝒞}. 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,Φ)(Y,Ψ)=(XY,(Φid)(idΨ)). (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(𝒞)Z(\mathcal{C}) is naturally a braided monoidal category.

Proposition

If 𝒞\mathcal{C} is a fusion category then the Drinfeld center Z(𝒞)Z(\mathcal{C}) is also naturally a fusion category.

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 algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

References

A standard textbook references are

  • Shahn Majid, Foundations of quantum group theory, Cambridge Univ. Press
  • C. Kassel, Quantum groups

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

An application to character sheaves is in

Last revised on June 12, 2016 at 07:16:13. See the history of this page for a list of all contributions to it.