symmetric monoidal (∞,1)-category of spectra
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.
(…) e.g. (Gelaki, 2.1) (…)
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 |
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?.
Shlomo Gelaki, On the classification of finite-dimensional triangular Hopf algebras, New directions in Hopf algebras, MSRI publications volume 43 (2002) (pdf)
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
Last revised on July 30, 2019 at 12:52:52. See the history of this page for a list of all contributions to it.