nLab
Drinfel'd double

Idea

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)

Definition

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.

Properties

Relation to Yetter-Drinfeld modules

The category of modules over a quantum double of a Hopf algebra $H$ is equivalent to the category of Yetter-Drinfeld modules of $H$-

Relation to Drinfeld center

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

Related concepts

• inertia orbifold, transgression in group cohomology

• Drinfeld center

References

The original references are

• Vladimir Drinfeld, Quantum groups In A. Gleason, editor, Proceedings of the ICM, pages 798–820, Rhode Island, 1987. AMS

• Shahn Majid, Doubles of quasitriangular Hopf algebras, Comm. Algebra 19:11. (1991), pp. 3061-3073 MR92k:16052 doi; Some remarks on the quantum double, Quantum groups and physics (Prague, 1994), Czechoslovak J. Phys. 44 (1994), no. 11-12, 1059–1071 doi

A brief survey is in

• Piotr Hajac, On and around the Drinfeld double, (pdf)

The generalization of the double construction to quasi-Hopf algebras motivated by

• Robbert Dijkgraaf, V. Pasquier, P. Roche, QuasiHopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl. 18B (1990), 60-72; Quasi-quantum groups related to orbifold models, Modern quantum field theory (Bombay, 1990), 375–383, World Sci. 1991

which is reviewed in

• A. Coste, J-M. Maillard, Representation Theory of Twisted Group Double, Annales Fond.Broglie 29 (2004) 681-694, (arXiv:hep-th/0309257)

was obtained in

• Shahn Majid, Quantum double for quasi-Hopf algebras, Lett. Math. Phys. 45 (1998), no. 1, 1–9, MR2000b:16077, doi, q-alg/9701002

• Shahn Majid, Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras, hep-th/9311184

Interpretation via transgression in group cohomology:

• Simon Willerton, The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv:math/0503266)

The special case of the Drinfeld double of a finite group is discussed further in

• Hui-Xiang Chen, Gerhard Hiss, Notes on the Drinfeld double of a finite-dimensional group algebra (pdf)

A characterization of the (quasi-)Hopf algebras arising this way is in

• Sonia Natale, On group theoretical Hopf algebras and exact factorization of finite groups (arXiv:math/0208054)

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

• Vladimir Hinich, Drinfeld double for orbifolds, Contemporary Math, 433, AMS Providence, 2007, 251-265, math.QA/0511476

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).