# nLab triangular Hopf algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 algebracategory of modules
$A$$Mod_A$
$R$-algebra$Mod_R$-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
$A$$Mod_A$
$R$-2-algebra$Mod_R$-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
$A$$Mod_A$
$R$-3-algebra$Mod_R$-4-module

## References

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

Last revised on July 30, 2019 at 12:52:52. See the history of this page for a list of all contributions to it.