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.
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.
|monoid/associative algebra||category of modules|
|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|
|form Drinfeld double||form Drinfeld center|
|trialgebra||Hopf monoidal category|
|monoidal category||2-category of module categories|
|Hopf monoidal category||monoidal 2-category (with some duality and strictness structure)|
|monoidal 2-category||3-category of module 2-categories|
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