nLab
triangular Hopf 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
AAMod AMod_A
RR-algebraMod RMod_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
AAMod AMod_A
RR-2-algebraMod RMod_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
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

Examples

References

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

Last revised on March 31, 2015 at 16:23:47. See the history of this page for a list of all contributions to it.