nLab
quasitriangular bialgebra

Idea

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.

Definition

Let $A$ be an algebra in a symmetric monoidal category $C$ with symmetry $\tau$; fix $m,l$ and $D\in A^{\otimes k}$ and let $1\le i_r\le l$ for $1\le r\le m$ be different. Then denote $D_{i_1,\dots ,i_m}\in A^{\otimes n}$ as the image of $R\otimes 1^{\otimes (l-m)}$ under the permutation which is the composition of the $m$ transpositions $(r,i_r)$ of tensor factors interchanging $r$ and $i_r$. In the following $C$ is the monoidal category of $k$-vector spaces.

A $k$-bialgebra (in particular $k$-Hopf algebra) is quasitriangular if there is an invertible element $R\in H\otimes H$ such that for any $h\in H$

$$\tau \circ \Delta (h)=R\Delta (h)R^{-1}$$\tau\circ\Delta(h) = R\Delta(h)R^{-1}

where $\tau ={\tau }_{H,H}:H\otimes H\to H\otimes H$ and

$$(\Delta \otimes \mathrm{id})\Delta (R)=R_{13}{R}_{23}$$(\Delta\otimes id)\Delta(R)=R_{13} R_{23}

$$(\mathrm{id}\otimes \Delta )\Delta (R)=R_{13}{R}_{12}$$(id\otimes\Delta)\Delta(R)=R_{13} R_{12}

An invertible element $R$ satisfying these 3 properties is called the universal $R$-element. As a corollary

$$(ϵ\otimes \mathrm{id})R=1,(\mathrm{id}\otimes ϵ)R=\mathrm{id}$$(\epsilon\otimes id) R = 1,\,\,\,\,\,(id\otimes\epsilon)R = id

and the quantum Yang-Baxter equation holds in the form

$$R_{12}{R}_{13}{R}_{23}=R_{23}{R}_{13}{R}_{12}$$R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}

A quasitriangular $H$ is called triangular if $R_{21}:=\tau (R)=R^{-1}$.

The category of representations of a quasitrianguar bialgebra is a braided monoidal category. If $R$ is a universal $R$-element, then $R_{21}^{-1}$ is as well. If $H$ is quasitriangular, $H^{\mathrm{cop}}$ and $H_{\mathrm{op}}$ are as well, with the universal $R$-element being $R_{21}$, or instead, $R_{12}^{-1}$. Any twisting of a quasitriangular bialgebra by a bialgebra 2-cocycle twists the universal $R$-element as well; hence the twisted bialgebra is again quasitriangular. Often the $R$-element does not exist as an element in $H\otimes H$ but rather in some completion of the tensor square; we say that $H$ is essentially quasitriangular, this is true for quantized enveloping algebras $U_q(G)$ in the rational form. The famous Sweedler’s Hopf algebra has a 1-parameter family of universal $R$-matrices.

Properties

Tannaka duality

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
$A$	${\mathrm{Mod}}_A$
$R$-algebra	${\mathrm{Mod}}_R$-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
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
$A$	${\mathrm{Mod}}_A$
$R$-2-algebra	${\mathrm{Mod}}_R$-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
$A$	${\mathrm{Mod}}_A$
$R$-3-algebra	${\mathrm{Mod}}_R$-4-module

Related concepts

• quasitriangular Hopf algebra

References

• 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