nLab Drinfeld center

Idea

The notion of the center of a monoidal category or the Drinfeld center is a categorification of the notion of the 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 (cf. EGNO15 §7.13) the braided monoidal category $Z(\mathcal{C})$ whose

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

  $$\Phi_{(-)} \;\colon\; X \otimes (-) \longrightarrow (-) \otimes X$$

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

  $$\Phi_{Y \otimes Z} = \big(id_Y \otimes \Phi_Z\big) \circ \big(\Phi_Y \otimes id_Z \big)$$

• morphisms are given by

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

• the tensor product is given by

  $$(X, \Phi) \otimes (Y, \Psi) \;=\; \big( X \otimes Y, (\Phi \otimes id_Y) \circ (id_X \otimes \Psi) \big) \,.$$

• the braiding is given (cf. EGNO15 (8.15)) by:

  $$b_{ (X,\Phi), (Y,\Psi) } = \Psi_X \mathrlap{\,.}$$

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, EGNO 2015, Thm. 9.3.2).

Relation to Drinfeld double

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 (cf. EGNO15, §7.14).

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

Of $G$-graded vector spaces

The archetypical example of Drinfeld centers is that of the of the category of finite-dimensional $G$-graded vector spaces for a group $G$.

This is also known as the representation category of the Drinfeld double of $G$.

We spell out (following SS26 §A.3) the characterization of the simple objects (Prop. , which is classical, cf. EGNO 2015 Example 8.5.4. in) and also the fusion coefficients (Prop. , which, by other means, were given only recently in Li 2026).

The fusion category

We start by recalling definitions and establishing notation:

By $\mathrm{Vec} \coloneqq \big({\mathrm{FD}\mathbb{C}\mathrm{Vec}, \otimes, \mathbb{C}}\big)$ we denote the monoidal category of finite-dimensional complex vector spaces with the usual tensor product of vector spaces. For $G$ a group, the monoidal category $\mathrm{Vec}_G$ of $G$-graded vector spaces has:

1. as objects vector spaces $V \in \mathrm{Vec}$ equipped with $G$-indexed direct sum decomposition

   $$V \equiv \bigoplus_{g \in G} V_g \mathrlap{\,,}$$

2. as morphisms linear maps that respect the $G$-grading,

3. as monoidal structure $\widetilde \otimes$ the ordinary tensor product but equipped with the convolution grading:

   (1)$$\big(V \widetilde\otimes W\big)_g \coloneqq \bigoplus_{k \in G} \big({ V_{k} \otimes W_{k^{-1} g} }\big) \mathrlap{\,.}$$

We will be referring to $\widetilde\otimes$, and its incarnation (6) in the Drinfeld center, as the fusion product, to be distinguished from the degreewise tensor product (cf. also Rem. below).

Equivalently, with

$$C(G) \in HopfAlg$$

denoting the Hopf algebra of functions $G \to \mathbb{C}$ (with product the pointwise product of functions and with coproduct induced by precomposition of functions with the group multiplication), we have that $Vec_G$ is equivalent (as a monoidal category) to the category of modules of $C(G)$:

$$Vec_G \simeq Mod\big(C(G)) \mathrlap{\,.}$$

(The pointwise product in $C(G)$ induces the $G$-grading and the coproduct induces the convolution in (1).)

For example, with $X \in \mathrm{Vec}$ an ungraded vector space and $g_0 \in G$ a group element, we write

(2)$$X \delta_{g_0} \in \mathrm{Vec}_G \,, \;\;\; ({X \delta_{g_0}})_g \coloneqq \left\{ \begin{array}{ll} X & \text{if}\;g = g_0 \\ 0 & \text{otherwise} \end{array} \right.$$

for the $G$-graded vector space which is concentrated on $X$ in degree $g_0$.

Now, the Drinfeld center $\mathcal{Z}({\mathrm{Vec}_G})$ is, by its general definition (Def. , but we now change the conventuion for the handedness of the half-braiding in order to end up with left group actions), the monoidal category which has:

1. as objects pairs, $(V,\beta)$, consisting of a $V \in \mathrm{Vec}_G$ and a natural isomorphism (the \emph{half-braiding})

   (3)$$\beta_{(-)} \,\colon\, (-) \widetilde\otimes V \xrightarrow{ \sim } V \widetilde\otimes (-) \mathrlap{\,,}$$

   satisfying for all $W, W' \in \mathrm{Vec}_G$ the relation

   (4)$$\beta_{W \widetilde\otimes W'} = \big({ \beta_{W} \widetilde\otimes \mathrm{id}_{W'} }\big) \circ \big({ \mathrm{id}_{W} \widetilde\otimes \beta_{W'} }\big) \mathrlap{\,,}$$

2. as morphisms $(V,\beta) \to (V',\beta')$ linear maps $f \colon V \to V'$ such that for all $W \in \mathrm{Vec}_G$ we have

   (5)$$\beta'_W \circ \big( id_W \widetilde\otimes f \big) = \big( f \widetilde\otimes id_{W} \big) \circ \beta_W \mathrlap{\,,}$$

3. as tensor product $\widetilde\otimes$ the operation which on the graded vector spaces is the convolution product $\widetilde\otimes$ from (1), extended to the half-braiding as follows:

   (6)$$(V,\beta) \widetilde\otimes (V',\beta') = \big({ V \widetilde\otimes V', (\mathrm{id}_{V} \widetilde\otimes \beta') \circ (\beta \widetilde\otimes \mathrm{id}_{V'}) }\big) \mathrlap{\,.}$$

Simple objects as inertia irreps

The above general definition of Drinfeld center applied to the case of $Vec_G$ may be further simplified:

Since every vector space is isomorphic to a direct sum of copies of $\mathbb{C}$, the linear maps $\beta_{(-)}$ (3) are determined already by their values on $\mathbb{C} \delta_{g_0}$ (2); and if we understand that $\mathbb{C} \otimes V = V$, then, by degree reasons, their $g_0 g$-components must be linear isomorphisms $g_0 \cdot (-)$ of this form:

For these, the coherence condition (4) reduces to the action property

$$g_2 \cdot \big({g_1 \cdot (-)}\big) = (g_2 g_1) \cdot (-) \mathrlap{\,.}$$

This way, the objects of $\mathcal{Z}({\mathrm{Vec}_G})$ are bijectively identified with (complex, finite-dimensional) groupoid representations of the action groupoid

$$\Lambda G \coloneqq G\sslash_{\!\mathrm{Ad}} G$$

of the adjoint action of $G$ (its inertia groupoid):

In this representation-theoretic description, the morphisms (5) of $\mathcal{Z}(\mathrm{Vec}_G)$ are equivalently the homomorphisms of groupoid representations (the intertwiners). This shows that:

Proposition

For $G$ a group, the Drinfeld center of $\mathrm{Vec}_G$ is equivalently the representation category of the inertia groupoid of $G$:

(7)$$\mathcal{Z}({\mathrm{Vec}_G}) \simeq \mathrm{Rep}({ \Lambda G }) \mathrlap{\,.}$$

The equivalence (7) implies at once that the simple objects in $\mathcal{Z}({\mathrm{Vec}_G})$ (those admitting no nontrivial direct sum decomposition) are supported on conjugacy classes of group elements

$$[g] \coloneqq \big\{ g^k \coloneqq k g k^{-1} \big\vert k \in G \big\} \mathrlap{\,,}$$

where they form an (irreducible) groupoid representation of the connected component $[g] \sslash_{\!\mathrm{Ad}} G \subset G\sslash_{\!\mathrm{Ad}} G$. Since these connected groupoids are equivalent to the delooping of their isotropy group, $[g] \sslash_{\!\mathrm{Ad}} G \simeq \mathbf{B}Z_G(g)$ (cf. this prop.), these groupoid representations are equivalently irreducible linear representations of the centralizer group $Z_G(g)$:

Proposition

The simple objects of $\mathcal{Z}({\mathrm{Vec}_G})$ are pairs $([g],\rho)$ consisting of a conjugacy class $[g]$ and an irrep of $Z_G(g)$:

$$\Big\{ \text{simple objects} \Big\} \;\simeq\; \bigsqcup_{\mathclap{ \substack{ [g] \in \\ \mathrm{Conj}(g) } }} \mathrm{Irr}\big({ [g]\sslash_{\!\mathrm{Ad}} G }\big) \;\simeq\; \bigsqcup_{\mathclap{ \substack{ [g] \in \\ \mathrm{Conj}(g) } }} \mathrm{Irr}\big({ Z_G(g) }\big) \mathrlap{\,.}$$

The fusion product

Remark

Beware that the above equivalence (7) is not a monoidal equivalence, as the standard tensor product on groupoid representations is degree-wise, not convolutive as in (1).

Instead, the fusion product in the Drinfeld center is category-theoretically given as follows:

Lemma

For a pair of objects $F_1,F_2 \in \mathrm{Rep}(\Lambda G) \simeq \mathcal{Z}({\mathrm{Vec}_G})$ (?), their fusion tensor product $\widetilde\otimes$ (?) is equivalently given by the following pull-tensor-push operation:

$$\begin{array}{ccc} ({ G\sslash_{\!\mathrm{Ad}}G }) \times ({ G\sslash_{\!\mathrm{Ad}}G }) & \xleftarrow{ (\mathrm{pr}_1,\mathrm{pr}_2) } & ({G \times G}) \sslash_{\!\mathrm{Ad}}G & \xrightarrow{ m \coloneqq (-)\cdot(-) } & G\sslash_{\!\mathrm{Ad}}G \\ ({F,F'}) &\mapsto& &\mapsto& m_!\big({ ({\mathrm{pr}_1^\ast F}) \otimes ({\mathrm{pr}_2^\ast F}) }\big) \end{array}$$

in that there is a natural isomorphism

$$F_1 \widetilde\otimes F_2 \simeq m_!\big({ ({\mathrm{pr}_1^\ast F}) \otimes ({\mathrm{pr}_2^\ast F}) }\big) \mathrlap{\,.}$$

Proof

Since the functor $m$ is a Kan fibration, the left base change $m_!$ is given by forming the direct sum over fibers. (It is immediate to check the universal property of the left adjoint explicitly.) This reproduces the definition (1). The induced $G$-action on these direct sums is the diagonal one, which under the equivalence (7) reproduces the definition (6).

To determine concretely the convolution tensor product $\widetilde{\otimes}$ of a pair of such simple objects, we first need:

Lemma

For $g,g' \in G$, with $[g] \times [g'] \subset G^2$ denoting the Cartesian product of their conjugacy classes (?), the action groupoid of the diagonal adjoint action is equivalent to the following disjoint union of delooping groupoids:

(8)$$\big({ [g] \times [g'] }\big) \sslash_{\!\mathrm{Ad}} G \;\;\simeq\;\; \bigsqcup_{\mathclap{ \substack{ [c] \in \\ \mathrm{Conj}(G) } }} \;\;\; \bigsqcup_{ \substack{ [k,k'] \in R \\ g^k g'^{k'} = c } } \mathbf{B}\big({ Z_G(g^k,g'^{l'}) }\big) \mathrlap{\,,}$$

where $R$ denotes the following double coset:

(9)$$R \coloneqq Z_G(c) \backslash G^2 / \big( Z_G(g) \times Z_G(g') \big) \mathrlap{\,.}$$

Proof

Consider the group multiplication map ${[g]} \times {[g']} \xrightarrow{ (-) \cdot (-) } G$. Since this is $G$-equivariant for the (diagonal) adjoint action, the set of $G$-orbits in $[g] \times [g']$ is partitioned into subsets labeled by conjugacy classes $[c] \in \mathrm{Conj}(G)$. An orbit in $[g] \times [g']$ is in the subset labeled by $[c]$ iff it contains an element $(g^k, g'^{k'})$ such that $g^k g'^{k'} = c$, for some $(k,k') \in G^2$. With the representative $c$ of $[c]$ held fixed, the latter pair has a unique class $[k,k'] \in G^2/\big({ Z_G(g) \times Z_G(g') }\big)$, and hence the set of orbits labeled by $[c]$ is in bijection to $R$ (9).

This shows that the disjoint union on the right of (8) is indexed by the set of orbits of the groupoid on the left. Since the disjoint union is over the delooping groupoids of the isotropy groups of these orbits, the claim follows.

It follows that:

Proposition

The fusion product of a pair of simple objects $\big({([g_1],\rho_1), ([g_2],\rho_2)}\big)$ contains any simple object $\big({[g],\rho}\big)$ with the following multiplicity:

(10)$$N_{ ([g_1],\rho_1), ([g_2],\rho_2)}^{([g],\rho)} \;=\; \sum_{\mathclap{ \substack{ [k_1,k_2] \in R \\ g_1^{k_1} g_2^{k_2} = g } }} \; \mathrm{dim}\, \mathrm{Hom}_{ Z_G(g_1)^{k_1} \cap Z_G(g_2)^{k_2} } \big({ \rho_1^{k_1} \otimes \rho_2^{k_2}, \rho }\big) \mathrlap{\,.}$$

(cf. SS26 Prop. A.8)

Proof

By Lem. we have that the fusion product is given by pull-push through

$$\big([g]\sslash_{\!\mathrm{Ad}} G\big) \times \big([g']\sslash_{\!\mathrm{Ad}} G\big) \longleftarrow \big({ [g]\times [g'] }\big)\sslash_{\!\mathrm{Ad}} G \longrightarrow G \sslash_{\!\mathrm{Ad}} G \mathrlap{\,.}$$

By Lemma , this yields on the orbit $[g]$ the isotropy representation

$$\bigoplus_{\mathclap{ \substack{ [k_1, k_2] \in R \\ g_1^{k_1} g_2^{k_2} = g } }} \, \mathrm{Ind} _{Z_G(g_1^{k_1}, g_2^{k_2})} ^{Z_G(g)} \left({ \mathrm{Res} ^{Z_G(g_1^{k_1}, g_2^{k_2})} _{Z_G(g)} \big({ \rho_1^{k_1} }\big) \otimes \mathrm{Res} ^{Z_G(g_1^{k_1}, g_2^{k_2})} _{Z_G(g)} \big({ \rho_2^{k_2} }\big) }\right) \mathrlap{\,,}$$

where we are using that on group representations the left base change $m_!$ is given by forming induced representations.

This yields the claim: By Schur's lemma, the multiplicity of the irrep $\rho$ in this expression is the dimension of the hom space from the latter to the former. Finally, by Frobenius reciprocity, $Ind \dashv Res$, the induction is left adjoint to restriction, whence we get (10) from the hom-isomorphism

Remark

Up to immediate translation, Prop. coincides with Li 2026 Thm. 3.2, proven there by different methods (Mackey theory applied to the Drinfeld double of $G$). As remarked there, this is in turn the untwisted specialization of a formula given by Goff 2012 Thm. 4.5.

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 University Press (1995, 2000) [doi:10.1017/CBO9780511613104]

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

• Bojko Bakalov, Alexandre Kirillov; section 3.2 in: Lectures on tensor categories and modular functors, University Lecture Series 21, Amer. Math. Soc. (2001) [webpage, ams:ulect/21, pdf]

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik: Tensor Categories, Mathematical Surveys and Monographs 205, AMS (2015) [ISBN:978-1-4704-3441-0, pdf]

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]

Explicit derivation of the fusion rules in the Drinfeld center of $Vec_G$:

• Hisham Sati, Urs Schreiber: §A.3.3 in: Drinfeld Center as Quantum State Monodromy over Bloch Hamiltonians around Defects [arXiv:2603.22029]