nLab Drinfel'd double

Idea

The Drinfel’d double- or quantum double-construction a sends a Hopf algebra to a quasi-triangular Hopf algebra [Drinfeld (1987)]; or more generally, it sends a quasi-Hopf algebra to a quasi-triangulated quasi-Hopf algebra [Majid (1994)].

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 $\mathcal{L}\mathbf{B}G \simeq G \sslash_{ad} G$ (this is made almost explicit in Dijkgraaf, Pasquier & Roche (1990), eq. 2.1.10; the categorified version of this statement is highlighted in Hinich (2007)).

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 (2005)]

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 cocommutative”.

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 finite-dimensional 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 (2007)).

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 articles:

• Vladimir Drinfeld, Quantum groups, in: A. Gleason (ed.) Proceedings of the 1986 International Congress of Mathematics 1 (1987) 798-820 [pdf]

  expanded version:

  Journal of Soviet Mathematics 41 (1988) 898–915 [doi:10.1007/BF01247086]

• Shahn Majid, Doubles of quasitriangular Hopf algebras, Comm. Algebra 19 11 (1991) 3061-3073 [MR92k:16052 doi:10.1080/00927879108824306]

• Shahn Majid, Representations, duals and quantum doubles of monoidal categories, Suppl. Rend.

  Circ. Mat. Palermo., Series II, 26 (1991) 197–206

• Shahn Majid, Some remarks on the quantum double, Czechoslovak J. Phys. 44 11-12 (1994) 1059-1071 [arXiv:hep-th/9409056, doi:10.1007/BF01690458]

Survery:

• Piotr Hajac, On and around the Drinfeld double (2009) [pdf, pdf]

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

• Robbert Dijkgraaf, Vincent Pasquier, Philippe Roche, QuasiHopf algebras, group cohomology and orbifold models, Nucl. Phys. B Proc. Suppl 18 (1990) 60–72 [doi:10.1016/0920-5632(91)90123-V]

• Robbert Dijkgraaf, Vincent Pasquier, Philippe Roche, Quasi-quantum groups related to orbifold models, International Colloquium on Modern Quantum Field Theory (Bombay, 1990), 375–383, World Sci. (1991) [cds:206306, pdf]

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, Section 1 of: The twisted Drinfeld double of a finite group via gerbes and finite groupoids, Algebr. Geom. Topol. 8 (2008) 1419–1457 [arXiv:math/0503266, doi:10.2140/agt.2008.8.1419]

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, in Quantum Groups, Contemporary Math 433 AMS (2007) 251-265 [math.QA/0511476, doi:10.1090/conm/433]

On representation theory of the quantum double of a finite group

• Shahn Majid, Leo Sean McCormack, Quantum geometric Wigner construction for $D(G)$ and braided racks, arXiv:2407.11835