symmetric monoidal (∞,1)-category of spectra
A quasi-triangular bialgebra / triangular bialgebra is a bialgebra equipped with just the right structure such as to make its category of modules into a braided monoidal category/symmetric monoidal category.
Let be an algebra in a symmetric monoidal category with symmetry ; fix and and let for be different. Then denote as the image of under the permutation which is the composition of the transpositions of tensor factors interchanging and . In the following is the monoidal category of -vector spaces.
A -bialgebra (in particular -Hopf algebra) is quasitriangular if there is an invertible element such that for any
where and
An invertible element satisfying these 3 properties is called the universal -element. As a corollary
and the quantum Yang-Baxter equation holds in the form
A quasitriangular is called triangular if .
The category of representations of a quasitrianguar bialgebra is a braided monoidal category. If is a universal -element, then is as well. If is quasitriangular, and are as well, with the universal -element being , or instead, . Any twisting of a quasitriangular bialgebra by a bialgebra 2-cocycle twists the universal -element as well; hence the twisted bialgebra is again quasitriangular. Often the -element does not exist as an element in but rather in some completion of the tensor square; we say that is essentially quasitriangular, this is true for quantized enveloping algebras in the rational form. The famous Sweedler’s Hopf algebra has a 1-parameter family of universal -matrices.
A quasitriangular structure on a bialgebra corresponds to a braided monoidal category structure on the category of modules of the underlying algebra. (For instance chapter 1, section 2 of (Carroll)).
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 |
V. G. Drinfel’d, Quantum groups, Proc. ICM 1986, Vol. 1, 2 798–820, AMS 1987.
S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. j. mod. physics A, 5, 01, pp. 1-91 (1990) doi:10.1142/S0217751X90000027
S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.
A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.
V. Chari, A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994
Robert Carroll, Calculus revisited
Last revised on September 2, 2013 at 15:35:57. See the history of this page for a list of all contributions to it.