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 |
---|---|
-algebra | -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 |
---|---|
-2-algebra | -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 |
---|---|
-3-algebra | -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) (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
Last revised on July 19, 2024 at 23:43:15. See the history of this page for a list of all contributions to it.