nLab triangular Hopf algebra

Idea

A (quasi-)triangular Hopf algebra is a Hopf algebra whose underlying bialgebra is a (quasi-)triangular. This means that its category of modules is not just a rigid monoidal category but a braided monoidal category/symmetric monoidal category.

If this rigid symmetric monoidal category in addition satisfies a certain regularity condition (see here) then the corresponding triangular Hopf algebra is equivalent to a supercommutative Hopf algebra, hence to the formal dual of a affine algebraic supergroup. See at Deligne's theorem on tensor categories for more on this.

Definition

(…) e.g. (Gelaki, 2.1) (…)

Properties

Tannaka duality

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

• A quantum group (in the sense of Drinfeld-Jimbo) is a quasitriangular Hopf algebra.

• The Drinfeld double of a Hopf algebra is a quasi-triangulated Hopf algebra?.

Related concepts

• Yang-Baxter equation

• quasitriangulated quasi-Hopf algebra

References

• Shlomo Gelaki, On the classification of finite-dimensional triangular Hopf algebras, New directions in Hopf algebras, MSRI publications volume 43 (2002) (math/0007154)

• Wikipedia, Quasitriangular Hopf algebra

• Nicolas Andruskiewitsch, Pavel Etingof, Shlomo Gelaki, Triangular Hopf algebras with the Chevalley property (arXiv:math/0008232)

• Pavel Etingof, Shlomo Gelaki, On families of triangular Hopf algebras (arXiv:math/0110043)

Discussion in view of Deligne's theorem on tensor categories is in

• Pavel Etingof, Shlomo Gelaki, The classification of finite-dimensional triangular Hopf algebras over an algebraically closed field of characteristic 0 (arXiv:0202258)