symmetric monoidal (∞,1)-category of spectra
The Drinfel’d double or quantum double construction is a construction that sends a Hopf algebra to a quasi-triangular Hopf algebra (Drinfeld 87). Or more generally, it sends a quasi-Hopf algebra to a quasi-triangulated quasi-Hopf algebra (Majid 94).
Geometrically, if the given Hopf algebra is the group algebra of a finite group $G$, then the quantum double is the groupoid convolution algebra of the corresponding inertia groupoid $\mathcal{L}\mathbf{B}G \simeq G//_{ad} G$. More generally, if the given quasi-Hopf algebra is the twisted groupoid convolution algebra of a group cohomology 3-cocycle $c \colon \mathbf{B}G \to \mathbf{B}^3 U(1)$, then the corresponding quantum double is the twisted groupoid convolution algebra of the inertia groupoid equipped with the transgressed 2-cocycle (Willerton 05)
Given a field $k$, Drinfel’d (or Drinfeld or quantum) double of a finite dimensional Hopf $k$-algebra $H$ is the tensor product algebra $D(H) = H\otimes H^*$ where $H^* = Hom_k(H, k)$ with induced canonical Hopf algebra structure. It appears that the canonical element in $D(H)$ (the image of the identity under the isomorphism $Hom_k(H,H) \cong H\otimes H^*$) is a universal $R$-element in $D(H)$ making it into a quasitriangular Hopf algebra, which is making it “almost commutative”.
This quasitriangular structure in some infinite-dimensional versions of the construction is related to the quasitriangular structure on some quantum groups.
The category of modules over a quantum double of a Hopf algebra $H$ is equivalent to the category of Yetter-Drinfeld modules of $H$-
For $H$ a Hopf algebra arising as the groupoid convolution algebra of a finite groupoid, the category of modules of its Drinfeld double is equivalently the Drinfeld center of the category of modules of the original algebra.
More generally, the analog of this statement holds for orbifolds (Hinich 05).
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 |
The original references are
A brief survey is in
The generalization of the double construction to quasi-Hopf algebras motivated by
which is reviewed in
was obtained in
The geometric interpretation of this was discussed in
The special case of the Drinfeld double of a finite group is discussed further in
A characterization of the (quasi-)Hopf algebras arising this way is in
The equivalence of category of modules over the Drinfeld double for the case of orbifolds, hence representations of the inertia orbifold, with the Drinfeld center of the category of representations of the original orbifold is discussed in
Essentially the same conclusion (with similar motivation and in terms of equivariant fibres of monoidal fibered categories with orbifolds encoded as internal groupoid objects in the base) was independently obtained by Zoran Škoda at MPI Bonn in 2004 (unpublished).